1. Introduction

Entropy was originally a concept in statistical physics. In 1948, Shannon first extended this concept to the process of channel communication [1], thus creating the discipline of "information theory". The concept of graph entropy was defined by Rashevsky in 1955 for studying the relations between the topological properties of graphs and their information content [2]. In fact, topological index is topological invariant derived from molecular graphs of compounds [3], which establishes the relationship between the structure and properties of the molecule. The first topological index was introduced in 1947 by Wiener [4], and was initially used for modelling boiling points of alkane molecules.

Let \(G\) be a simple undirected graph with vertex set \(V(G)\) and edge set \(E(G)\). The distance \(d_G(u,v)\) between two vertices \(u\), \(v\) of \(G\) is the length of a shortest \((u,v)\)-path in \(G\). The Wiener index of a graph \(G\) is defined as \[W(G)=\frac{1}{2}\sum_{u\in V(G)}{D_G(u)},\] where \(D_G(u)=\sum_{v\in V(G)}{d_G(u,v)}\) is the total distance of vertex \(u\). It remains, to this day, one of the most popular and widely studied topological indices in mathematical chemistry.

In subsequent studies, scholars have proposed many topological indices, such as the Randic index [5], the Zagreb indices [6,7], the atom-boud connectivity index [8] and so on. Hundreds of different topological indices have been applied to QSAR (quantitative structure-activity relationship)/QSAR (quantitative structure-property relationship) modelings. Besides, they are also used for the discrimination of isomers [9], which is significant for the coding and the computer processing of chemical structures. In 1977, Bonchev and Trinajstic [10] introduced an molecular entropy measure based on distances, which is also called information indices, to interpret the molecular branching. They found that the information indices have greater ability for discrimination between isomers than those topological indices based on adjacency, incidence or polynomial coefficients of adjacency matrix. Dehmer et al., [11] introduced Hosoya entropy in 2014. In 2015, Mowshowitz and Dehmer [12] established the connections between the information content of a graph and Hosoya entropy. For more research in this area, readers can refer to the paper [13].

Base on the previous research, Konstantinova [14] proposed the marginal entropy of a graph \(G\) as follows:

\begin{eqnarray*} I_D(G) &=& -\sum_{u\in V(G)}{\frac{D_G(u)}{\sum_{u\in V(G)}{D_G(u)}}\log_{2}{\frac{D_G(u)}{\sum_{u\in V(G)}{D_G(u)}}}} \end{eqnarray*}\begin{eqnarray*} &=& 1+\log_{2}{W(G)}-\frac{1}{2W(G)}\sum_{u\in V(G)}{D_G(u)\log_{2}{D_G(u)}}\\ & = & -\frac{1}{\sum_{u\in V(G)}{D_G(u)}}\sum_{u\in V(G)}{D_G(u)}\log_{2}{D_G(u)}+\log_{2}{\sum_{u\in V(G)}{D_G(u)}}. \end{eqnarray*} In 2021, Sahin [15] obtained the marginal entropy of paths, stars, double stars, cycles and vertex-transitive graphs. On this basis, we give the quantitative calculation formula of marginal entropy for the complete bipartite graphs, complete multipartite graphs, firefly graphs, lollipop graphs, clique-chain graphs, Cartesian product and join of two graphs, which extends the results of Sahin.

2. Preliminaries

We first introduce several special kinds of graphs. Defined by Aouchiche et al., [16], a firefly graph \(F_{s,t,n-2s-2t-1}\) (\(s\geq 0\), \(t\geq 0\), \(n-2s-2t-1\geq 0\)) is a graph of order \(n\) that consists of \(s\) triangles, \(t\) pendent paths of length 2 and \(n-2s-2t-1\) pendent edges, sharing a common vertex. The lollipop graph \(C_{n,g}\) (shown in Figure 1), first used in [17], is obtained by attaching a vertex of cycle \(C_g\) to an end vertex of path \(P_{n-g-1}\). The class of clique-chain graphs \(G_d(a_1,a_2,\ldots,a_d,a_{d+1})\) is composed of \(d+1\) cliques \(K_{a_1}\), \(K_{a_2}\), ..., \(K_{a_{d}}\), \(K_{a_{d+1}}\), where \(n_i\geq 1\) for \(1\leq i\leq d+1\) and \(\sum_{1\leq i\leq d+1}a_i=n\), the edges between two adjacent cliques is full. The graph \(G_2(3,2,2)\) (shown in Figure 1) is an example. Obviously, the diameter of clique-chain graph \(G_d\) is \(d\).

Definition 1. For two simple graphs \(G_1\) and \(G_2\), the Cartesian product \(G_1\square G_2\) of them is defined with vertex set \(V(G_1\square G_2)=V(G_1)\times V(G_2)\) and edge set \(E(G_1\square G_2)=\big\{uv,u=(u_1,v_1),v=(u_2,v_2) \big|[u_1=u_2 \ and \ v_1v_2\in E(G_2)] \ or \ [v_1=v_2 \ and \ u_1u_2\in E(G_1)] \big\}\).

Definition 2. For two simple graphs \(G_1\) and \(G_2\), the join \(G_1\vee G_2\) of them is defined with vertex set \(V(G_1\vee G_2)=V(G_1)\cup V(G_2))\) and edge set \(E(G_1\vee G_2)=E(G_1)\cup E(G_2)\cup \big\{uv\big|u\in V(G_1),v\in V(G_2)\big\}\).

Lemma 1.([18]) Let \(G=G_1\square G_2\), for two vertices \(u=(u_1,v_1)\) and \(v=(u_2,v_2)\) of \(G\), where \(u_i\in V(G_1),v_i\in V(G_2)\), \(i=1,2\). Then we have \(d_G(u,v)=d_{G_1}(u_1,u_2)+d_{G_2}(v_1,v_2)\).

3. Main results

First, we consider the complete bipartite graph \(K_{a,b}\).

Theorem 1. The marginal entropy of the complete bipartite graph \(K_{a,b}\) is given by the following formula \[I_D(K_{a,b})=1-\frac{\log_{2}{\Big[(2a+b-2)^{(2a^2+ab-2a)}(2b+a-2)^{(2b^2+ab-2b)}\Big]}}{2a^2+2b^2+2ab-2a-2b}+\log_{2}{(a^{2}+b^{2}+ab-a-b)}.\]

Proof. Let \(A\) and \(B\) be the parts of \(K_{a,b}\) with \(a\), \(b\) vertices, respectively. We have,

• \(D_{K_{a,b}}(u)=b+2(a-1)=2a+b-2\) for \(u\in A\),
• \(D_{K_{a,b}}(u)=a+2(b-1)=2b+a-2\) for \(u\in B\).

By the definition of the marginal entropy, \(I_D(K_{a,b})\) can be computed as, \begin{eqnarray*} I_D(K_{a,b}) &=&-\frac{1}{\sum_{u\in V(K_{a,b})}{D(u)}}\sum_{u\in V(K_{a,b})}{D(u)}\log_{2}{D(u)}+\log_{2}{\sum_{u\in V(K_{a,b})}{D(u)}} \\ \nonumber &=& -\frac{1}{a(2a+b-2)+b(2b+a-2)}\Big[\sum_{u\in A}\big((2a+b-2)\log_{2}(2a+b-2)\big)+\sum_{u\in B}\big((2b+a-2)\log_{2}(2b+a-2)\big)\Big]\\ \nonumber & & +\log_{2}{\big(a(2a+b-2)+b(2b+a-2)\big)}\\ \nonumber &=& -\frac{1}{2a^2+2b^2+2ab-2a-2b}\Big[a\big((2a+b-2)\log_{2}(2a+b-2)\big)+b\big((2b+a-2)\log_{2}(2b+a-2)\big)\Big]\\ \nonumber & &+\log_{2}{(2a^2+2b^2+2ab-2a-2b)}\\ \nonumber &=& -\frac{1}{2a^2+2b^2+2ab-2a-2b}\Big[{(2a^2+ab-2a)\log_{2}{(2a+b-2)}}+{(2b^2+ab-2b)\log_{2}{(2b+a-2)}}\Big]\\ \nonumber & &+\log_{2}{\big(2(a^2+b^2+ab-a-b)\big)}\\ \nonumber &=& 1-\frac{\log_{2}{\Big[(2a+b-2)^{(2a^2+ab-2a)}(2b+a-2)^{(2b^2+ab-2b)}\Big]}}{2a^2+2b^2+2ab-2a-2b}+\log_{2}{(a^{2}+b^{2}+ab-a-b)}. \end{eqnarray*} This completes the proof.

As a generalization, we have the following result. Since bipartite graph is also special multipartite graph, when \(a_i=0\) holds for \(i\geq3\) in Theorem 2, one can get the result of Theorem 1 from Theorem 2 by simply deduction.

Theorem 2. The marginal entropy of the complete multipartite graph \(K_{a_1,a_2,\ldots,a_k}\) is given by the following formula \[I_D(K_{a_1,a_2,\ldots,a_k})=-\frac{\sum_{i=1}^{k}{\big(n+a_i-2\big)\log_{2}\big(n+a_i-2\big)}}{n^2-2n+\sum_{i=1}^{k}a_i^2}+\log_{2}{\big[n^2-2n+\sum_{i=1}^{k}a_i^2\big]}.\]

Proof. Let \(V_i\) be the part of \(K_{a_1,a_2,\ldots,a_k}\) with \(a_i\) vertices. Then for \(u\in V_i\) \[D(u)=a_1+a_2+\ldots+a_{i-1}+a_{i+1}+\ldots+a_k+2(a_i-1)=n-a_i+2(a_i-1)=n+a_i-2.\] Therefore, we have \[\sum_{u\in V(K_{a_1,\ldots,a_k})}D(u)=\sum_{i=1}^{k}a_i(n+a_i-2)=n^2-2n+\sum_{i=1}^{k}a_i^2.\] By definition of marginal entropy, \(I_D(K_{a_1,a_2,\ldots,a_k})\) can be computed as \begin{eqnarray*} I_D(K_{a_1,\ldots,a_k}) &=&-\frac{1}{\sum_{u\in V(K_{a_1,\ldots,a_k})}{D(u)}}\sum_{u\in V(K_{a_1,\ldots,a_k})}{D(u)}\log_{2}{D(u)}+\log_{2}{\sum_{u\in V(K_{a_1,\ldots,a_k})}{D(u)}} \\ \nonumber &=& -\frac{\sum_{i=1}^{k}{\big(n+a_i-2\big)\log_{2}\big(n+a_i-2\big)}}{n^2-2n+\sum_{i=1}^{k}a_i^2}+\log_{2}{\big[n^2-2n+\sum_{i=1}^{k}a_i^2\big]}. \end{eqnarray*} This completes the proof.

In the next, we present the calculation formulas of the marginal entropy for firefly graph, lollipop graph and clique-chain graph by giving a specific vertex set partition to them respectively.

Theorem 3. The marginal entropy of the \(n\)-vertex firefly graph \(F_{s,t,n-2s-2t-1}\) is given by the following formula, \begin{eqnarray*} I_D(F_{s,t,n-2s-2t-1}) &=& 1-\frac{1}{2(n^2-2n-s-3t+tn+1)}\Big[(n+t-1)\log_2(n+t-1)+2s(2n+t-4)\log_2(2n+t-4)\\ & & +(n-2s-2t-1)(2n+t-3)\log_2(2n+t-3)\Big]+t(3n+t-7)\log_2(3n+t-7)\\ & &+t(2n+t-5)\log_2(2n+t-5)+\log_2(n^2-2n-s-3t+tn+1). \end{eqnarray*}

Proof. First, denote the unique vertex with maximum degree \(n-t-1\) by \(u_0\), we partition the rest vertices of \(F_{s,t,n-2s-2t-1}\) as following:

• \(A\): the set of all the vertices in triangles,
• \(B\): the set of pendent vertex \(x\) such that \(d(x,u_0)=1\),
• \(C\): the set of pendent vertex \(x\) such that \(d(x,u_0)=2\),
• \(D=V(F_{s,t,n-2s-2t-1}) \backslash \big(A\cup B\cup C\cup\{u_0\}\big)\).

According to the definition of firefly graph, \(|A|=2s\), \(|B|=n-2s-2t-1\), \(|C|=t\), \(|D|=t\) and \(|A|+|B|+|C|+|D|+1=n\). By direct calculation, \(D(u_0)=n+t-1\); for \(u\in A\), \(D(u)=2n+t-4\); for \(u\in B\), \(D(u)=2n+t-3\); for \(u\in C\), \(D(u)=3n+t-7\); for \(u\in D\), \(D(u)=2n+t-5\). Therefore, we have \begin{eqnarray*} \sum_{u\in V(F_{s,t,n-2s-2t-1})}D(u)&=&(n+t-1)+2s(2n+t-4)+(n-2s-2t-1)(2n+t-3)+t(3n+t-7)+t(2n+t-5)\\ &=& 2n^2-4n-2s-6t+2tn+2\,, \end{eqnarray*} and then \begin{eqnarray*} I_D(F_{s,t,n-2s-2t-1}) &=& -\frac{1}{2n^2-4n-2s-6t+2tn+2}\Big[(n+t-1)\log_2(n+t-1)+2s(2n+t-4)\log_2(2n+t-4)\\ & & +(n-2s-2t-1)(2n+t-3)\log_2(2n+t-3)\Big]+t(3n+t-7)\log_2(3n+t-7)\\ & & +t(2n+t-5)\log_2(2n+t-5)+\log_2(2n^2-4n-2s-6t+2tn+2)\\ &=& 1-\frac{1}{2(n^2-2n-s-3t+tn+1)}\Big[(n+t-1)\log_2(n+t-1)+2s(2n+t-4)\log_2(2n+t-4)\\ & & +(n-2s-2t-1)(2n+t-3)\log_2(2n+t-3)\Big]+t(3n+t-7)\log_2(3n+t-7)\\ & & +t(2n+t-5)\log_2(2n+t-5)+\log_2(n^2-2n-s-3t+tn+1). \end{eqnarray*} This completes the proof.

Theorem 4. The marginal entropy of the lollipop graph \(C_{n,g}\) is given by the following formula,

• (i) If \(g\) is odd, then \begin{eqnarray*} I_D(C_{n,g})& = & -2-\log_23-\frac{12}{4n^3+5g^3-6ng^2-12g^2+12ng-10n+7g}\\ & & \times\Big[\sum_{i=1}^{n-g+1}\big[i^2-(n+1)i+\frac{1}{4}(2n^2-g^2+2n+2g-1)\big]\\ & & \times\log_2\big[i^2-(n+1)i+\frac{1}{4}(2n^2-g^2+2n+2g-1)\big]\\ & & +2\sum_{i=n-g+2}^{n-\frac{g-1}{2}}\big[(n-g)i+\frac{1}{4}(-2n^2-g^2+4ng-2n+2g-1)\big]\\ & & \times\log_2\big[(n-g)i+\frac{1}{4}(-2n^2-g^2+4ng-2n+2g-1)\big]\Big]\\ & & +\log_2\big[4n^3+5g^3-6ng^2-12g^2+12ng-10n+7g\big]. \end{eqnarray*}
• (ii) If \(g\) is even, then \begin{eqnarray*} I_D(C_{n,g})& = & -2-\log_23-\frac{12}{4n^3+5g^3-6ng^2-12g^2+12ng-4n+4g}\\ & & \times\Big[\sum_{i=1}^{n-g+1}\big[i^2-(n+1)i+\frac{1}{4}(2n^2-g^2+2n+2g)\big]\\ & & \times\log_2\big[i^2-(n+1)i+\frac{1}{4}(2n^2-g^2+2n+2g)\big]\\ & & +2\sum_{i=n-g+2}^{n-\frac{g}{2}}\big[(n-g)i+\frac{1}{4}(-2n^2-g^2+4ng-2n+2g)\big]\\ & & \times\log_2\big[(n-g)i+\frac{1}{4}(-2n^2-g^2+4ng-2n+2g)\big]\\ & & +\frac{1}{4}(2n^2+g^2-2ng+2n-2g)\log_2[\frac{1}{4}(2n^2+g^2-2ng+2n-2g)]\Big]\\ & & +\log_2\big[4n^3+5g^3-6ng^2-12g^2+12ng-4n+4g\big]. \end{eqnarray*}

Proof.

• (i) If \(g\) is odd, then we obtain \(D(v_i)\) according to the following \(A_1\) to \(A_3\).
  • (\(A_1\)) \(1\leq i\leq n-g+1\). Then \begin{eqnarray*} D(v_i) & = & \sum_{j=1}^{i-1}(i-j)+\sum_{j=i+1}^{n-g+1}(j-i)+2\sum_{j=n-g+2}^{\frac{2n-g+1}{2}}(j-i)\\ & = & \frac{i(i-1)}{2}+\frac{(n-g-i+2)(n-g-i+1)}{2}+(2n-\frac{3}{2}g-2i+\frac{5}{2})(\frac{g-1}{2})\\ & = & i^2-(n+1)i+\frac{1}{4}(2n^2-g^2+2n+2g-1). \end{eqnarray*}
  • (\(A_2\)) \(n-g+2\leq i\leq n-\frac{g-1}{2}\). Then \begin{eqnarray*} D(v_i) & = & 2\sum_{j=1}^{\frac{g-1}{2}}j+\sum_{j=1}^{n-g}(i-j)\\ \nonumber & = & (\frac{g+1}{2})(\frac{g-1}{2})+\frac{(2i-n+g-1)(n-g)}{2}\\ \nonumber & = & (n-g)i+\frac{1}{4}(-2n^2-g^2+4ng-2n+2g-1). \end{eqnarray*}
  • (\(A_3\)) \(n-\frac{g-3}{2}\leq i\leq n\), \(D(v_i)=D(v_{2n-g-i+2})\), and in this case \(n-g+2\leq 2n-g-i+2\leq n-\frac{g-1}{2}\).

    In summary, we have

    \begin{eqnarray*} \sum_{v\in V(C_{n,g})}D(v) & = & A_1+A_2+A_3\\ & = & \sum_{i=1}^{n-g+1}\big[i^2-(n+1)i+\frac{1}{4}(2n^2-g^2+2n+2g-1)\big]\\ & & +2\sum_{i=n-g+2}^{n-\frac{g-1}{2}}\big[(n-g)i+\frac{1}{4}(-2n^2-g^2+4ng-2n+2g-1)\big]\\ & = & \frac{1}{12}(4n^3+5g^3-6ng^2-12g^2+12ng-10n+7g). \end{eqnarray*} Therefore, we have \begin{eqnarray*} I_D(C_{n,g})& = & -2-\log_23-\frac{12}{4n^3+5g^3-6ng^2-12g^2+12ng-10n+7g}\\ & & \times\Big[\sum_{i=1}^{n-g+1}\big[i^2-(n+1)i+\frac{1}{4}(2n^2-g^2+2n+2g-1)\big]\\ & & \times\log_2\big[i^2-(n+1)i+\frac{1}{4}(2n^2-g^2+2n+2g-1)\big]\\ & & +2\sum_{i=n-g+2}^{n-\frac{g-1}{2}}\big[(n-g)i+\frac{1}{4}(-2n^2-g^2+4ng-2n+2g-1)\big]\\ & & \times\log_2\big[(n-g)i+\frac{1}{4}(-2n^2-g^2+4ng-2n+2g-1)\big]\Big]\\ & & +\log_2\big[4n^3+5g^3-6ng^2-12g^2+12ng-10n+7g\big]. \end{eqnarray*}
• (ii) If \(g\) is even, then we obtain \(D(v_i)\) according to the following \(B_1\) to \(B_4\).
  • (\(B_1\)) \(1\leq i\leq n-g+1\). Then \begin{eqnarray*} D(v_i) & = & \sum_{j=1}^{i-1}(i-j)+\sum_{j=i+1}^{n-g+1}(j-i)+2\sum_{j=n-g+2}^{\frac{2n-g}{2}}(j-i)+(n-\frac{g}{2}+1-i ) \end{eqnarray*} \begin{eqnarray*} & = & \frac{i(i-1)}{2}+\frac{(n-g-i+2)(n-g-i+1)}{2}\\ & & +(2n-\frac{3}{2}g-2i+2)(\frac{g}{2}-1)+(n-\frac{g}{2}+1-i )\\ & = & i^2-(n+1)i+\frac{1}{4}(2n^2-g^2+2n+2g). \end{eqnarray*}
  • (\(B_2\)) \(n-g+2\leq i\leq n-\frac{g}{2}\). Then \begin{eqnarray*} D(v_i) & = & 2\sum_{j=1}^{\frac{g}{2}-1}j+\frac{g}{2}+\sum_{j=1}^{n-g}(i-j)\\ & = & \frac{g}{2}(\frac{g}{2}-1)+\frac{g}{2}+\frac{(2i-n+g-2)(n-g)}{2}\\ & = & (n-g)i+\frac{1}{4}(-2n^2-g^2+4ng-2n+2g). \end{eqnarray*}
  • (\(B_3\)) \(i=n-\frac{g}{2}+1\). Then \begin{eqnarray*} D(v_i) & = & 2\sum_{j=1}^{\frac{g}{2}-1}j+\frac{g}{2}+\sum_{j=1}^{n-g}(i-j)\\ & = & \frac{g}{2}(\frac{g}{2}-1)+\frac{g}{2}+\frac{(2i-n+g-2)(n-g)}{2}\\ & = & \frac{1}{4}(2n^2+g^2-2ng+2n-2g). \end{eqnarray*}
  • (\(B_4\)) \(n-\frac{g}{2}+2\leq i\leq n\), \(D(v_i)=D(v_{2n-g-i+2})\), and in this case \(n-g+2\leq 2n-g-i+2\leq n-\frac{g}{2}\).

    In summary, we have

    \begin{eqnarray*} \sum_{v\in V(C_{n,g})}D(v) & = & B_1+B_2+B_3+B_4\\ & = & \sum_{i=1}^{n-g+1}\big[i^2-(n+1)i+\frac{1}{4}(2n^2-g^2+2n+2g)\big]\end{eqnarray*}\begin{eqnarray*} & & +2\sum_{i=n-g+2}^{n-\frac{g}{2}}\big[(n-g)i+\frac{1}{4}(-2n^2-g^2+4ng-2n+2g)\big]+\frac{1}{4}(2n^2+g^2-2ng+2n-2g)\\ &= & \frac{1}{12}(4n^3+5g^3-6ng^2-12g^2+12ng-4n+4g). \end{eqnarray*} Therefore, we have \begin{eqnarray*} I_D(C_{n,g})& = & -2-\log_23-\frac{12}{4n^3+5g^3-6ng^2-12g^2+12ng-4n+4g}\\ & & \times\Big[\sum_{i=1}^{n-g+1}\big[i^2-(n+1)i+\frac{1}{4}(2n^2-g^2+2n+2g)\big]\\ & & \times\log_2\big[i^2-(n+1)i+\frac{1}{4}(2n^2-g^2+2n+2g)\big]\\ & & +2\sum_{i=n-g+2}^{n-\frac{g}{2}}\big[(n-g)i+\frac{1}{4}(-2n^2-g^2+4ng-2n+2g)\big]\\ & & \times\log_2\big[(n-g)i+\frac{1}{4}(-2n^2-g^2+4ng-2n+2g)\big]\\ & & +\frac{1}{4}(2n^2+g^2-2ng+2n-2g)\log_2[\frac{1}{4}(2n^2+g^2-2ng+2n-2g)]\Big]\\ & & +\log_2\big[4n^3+5g^3-6ng^2-12g^2+12ng-4n+4g\big]. \end{eqnarray*}
  This completes the proof.

Theorem 5. Let \(G\) be the \(n\)-vertex clique-chain graph \(G_d(a_1,a_2,\ldots,a_d,a_{d+1})\). Then the marginal entropy of \(G\) is given by the following formula \[ I_D(G) = -\frac{\sum_{k=1}^{d+1}a_k\Big[\big(\sum_{i=1}^{d+1}|i-k|a_i+a_k-1\big)\log_2\big(\sum_{i=1}^{d+1}|i-k|a_i+a_k-1\big)\Big]} {\sum_{k=1}^{d+1}\sum_{i=1}^{d+1}|i-k|a_ka_i+\sum_{k=1}^{d+1}{a_k}^2+n} +\log_2\Big(\sum_{k=1}^{d+1}\sum_{i=1}^{d+1}|i-k|a_ka_i+\sum_{k=1}^{d+1}{a_k}^2+n\Big). \]

Proof. Denote the set of all vertices in clique \(K_{a_k}\) by \(V_k\), then \(V(G)=\bigcup_{k=1}^{d+1}V_k\). For \(u\in V_k\) (\(1\leq k\leq d+1\)), \[D(u)=a_k-1+\sum_{i=1}^{k-1}(k-i)a_i+\sum_{i=k+1}^{d+1}(i-k)a_i=\sum_{i=1}^{d+1}|i-k|a_i+a_k-1.\] Then, we have \[\sum_{u\in V(G)}D(u)=\sum_{k=1}^{d+1}a_k\big(\sum_{i=1}^{d+1}|i-k|a_i+a_k-1\big)=\sum_{k=1}^{d+1}\sum_{i=1}^{d+1}|i-k|a_ka_i+\sum_{k=1}^{d+1}{a_k}^2+\sum_{k=1}^{d+1}a_k\] and \[\sum_{u\in V(G)}D(u)\log_2D(u)=\sum_{k=1}^{d+1}a_k\Big[\big(\sum_{i=1}^{d+1}|i-k|+a_k-1\big)\log_2\big(\sum_{i=1}^{d+1}|i-k|+a_k-1\big)\Big].\] Therefore, recall that \(\sum_{k=1}^{d+1}a_k=n\), we have \begin{eqnarray*} I_D(G) &=& -\frac{1}{\sum_{k=1}^{d+1}\sum_{i=1}^{d+1}|i-k|a_ka_i+\sum_{k=1}^{d+1}{a_k}^2+\sum_{k=1}^{d+1}a_k} \sum_{k=1}^{d+1}a_k\Big[\big(\sum_{i=1}^{d+1}|i-k|a_i+a_k-1\big)\log_2\big(\sum_{i=1}^{d+1}|i-k|a_i+a_k-1\big)\Big]\\ & & +\log_2\Big(\sum_{k=1}^{d+1}\sum_{i=1}^{d+1}|i-k|a_ia_k+\sum_{k=1}^{d+1}{a_k}^2+\sum_{k=1}^{d+1}a_k\Big)\\ &=& -\frac{1}{\sum_{k=1}^{d+1}\sum_{i=1}^{d+1}|i-k|a_ka_i+\sum_{k=1}^{d+1}{a_k}^2+n} \sum_{k=1}^{d+1}a_k\Big[\big(\sum_{i=1}^{d+1}|i-k|a_i+a_k-1\big)\log_2\big(\sum_{i=1}^{d+1}|i-k|a_i+a_k-1\big)\Big]\end{eqnarray*}\begin{eqnarray*} & & +\log_2\Big(\sum_{k=1}^{d+1}\sum_{i=1}^{d+1}|i-k|a_ka_i+\sum_{k=1}^{d+1}{a_k}^2+n\Big). \end{eqnarray*} This completes the proof.

In the following, we provide the calculation formulas of marginal entropy under two kinds of graph operations, Cartesian product and join. For convenience, in the following discussion we let \(n\), \(n_i\) equals to \(|V(G)|\), \(|V(G_i)|\) and let \(m\), \(m_i\) equals to \(|E(G)|\), \(|E(G_i)|\) with \(i\in\{1,2\}\), respectively. Similarly, we can omit the subscript \(G\) when it does not cause ambiguity, such as \(d(u,v)=d_G(u,v)\), \(d_1(u,v)=d_{G_1}(u,v)\), \(d_2(u,v)=d_{G_2}(u,v)\), \(D_1(u)=D_{G_1}(u)\), \(V_1=V(G_1)\), \(d_1(u)=d_{G_1}(u)\) etc.

Theorem 6. Let \(G_1\) and \(G_2\) be simple connected graphs and \(G=G_1\square G_2\). Then the marginal entropy of \(G\) is given by the following formula \[ I_D(G) = 1-\frac{\sum_{u_1\in V_1}\sum_{v_1\in V_2}\Big[n_2D_1(u_1)+n_1D_2(v_1)\Big]\log_{2}{\Big[n_2D_1(u_1)+n_1D_2(v_1)\Big]}}{2n_2^2W(G_1)+2n_1^2W(G_2)} + \log_{2}\Big[{n_2^2W(G_1)+n_1^2W(G_2)}\Big]. \]

Proof. By Definition 1 and Lemma 1, for \(u=(u_1,v_1)\in V(G)=V_1\times V_2\), the total distance of \(u\) is computed in the following \begin{eqnarray*} D(u) & = & \sum_{v=(u_2,v_2)\in V(G)}d(u,v)\\ & = & \sum_{u_2\in V_1}\sum_{v_2\in V_2}\Big[d_1(u_1,u_2)+d_2(v_1,v_2)\Big]\\ & = & |V_2|\sum_{u_2\in V_1}d_1(u_1,u_2)+|V_1|\sum_{v_2\in V_2}d_2(v_1,v_2)\\ & = & n_2D_1(u_1)+n_1D_2(v_1). \end{eqnarray*} By definition of marginal entropy, \(I_D(G)\) can be computed as \begin{eqnarray*} I_D(G) &=&-\frac{1}{\sum_{u\in V(G)}{D(u)}}\sum_{u\in V(G)}{D(u)}\log_{2}{D(u)}+\log_{2}{\sum_{u\in V(G)}{D(u)}} \\ &=& -\frac{\sum_{u_1\in V_1}\sum_{v_1\in V_2}\big(n_2D_1(u_1)+n_1D_2(v_1)\big)\log_{2}\big(n_2D_1(u_1)+n_1D_2(v_1)\big)}{\sum_{u_1\in V_1}\sum_{v_1\in V_2}\big(n_2D_1(u_1)+n_1D_2(v_1)\big)}\\ & & +\log_{2}\sum_{u_1\in V_1}\sum_{v_1\in V_2}\big(n_2D_1(u_1)+n_1D_2(v_1)\big)\\ &=& -\frac{\sum_{u_1\in V_1}\sum_{v_1\in V_2}\big(n_2D_1(u_1)+n_1D_2(v_1)\big)\log_{2}\big(n_2D_1(u_1)+n_1D_2(v_1)\big)}{n_2^2\sum_{u_1\in V_1}D_1(u_1)+n_1^2\sum_{v_1\in V_2}D_2(v_1)}\\ & & +\log_{2}\Big(n_2^2\sum_{u_1\in V_1}D_1(u_1)+n_1^2\sum_{v_1\in V_2}D_2(v_1)\Big)\\ &=& 1-\frac{\sum_{u_1\in V_1}\sum_{v_1\in V_2}\big(n_2D_1(u_1)+n_1D_2(v_1)\big)\log_{2}\big(n_2D_1(u_1)+n_1D_2(v_1)\big)}{2n_2^2W(G_1)+n_1^2W(G_2)}\\ & & +\log_{2}\Big(n_2^2W(G_1)+n_1^2W(G_2)\Big). \end{eqnarray*} This completes the proof.

Theorem 7. Let \(G_1\) and \(G_2\) be simple graphs (not necessarily connected) and \(G=G_1\vee G_2\). Then the marginal entropy of \(G\) is given by the following formula \begin{eqnarray*} I_D(G)&=& 1-\frac{1}{2\big(n_1^2+n_2^2+n_1n_2-n_2-n_2-m_2-m_2\big)}\Big[\sum_{u\in V_1}{\big(2n_1+n_2-d_1(u)-2\big)\log_{2}\big(2n_1+n_2-d_1(u)-2\big)}\\ \nonumber & & +\sum_{u\in V_2}{\big(2n_2+n_1-d_2(u)-2\big)\log_{2}\big(2n_2+n_1-d_2(u)-2\big)}\Big]+\log_{2}{\Big[n_1^2+n_2^2+n_1n_2-n_2-n_2-m_2-m_2\Big]}. \end{eqnarray*}

Proof. By Definition 2, for \(u\in V(G)=V_1\cup V_2\), we have

• \(D(u)=d_1(u)+|V_2|+2\big(|V_1|-d_1(u)-1\big)=2n_1+n_2-d_1(u)-2\) for \(u\in V_1\),
• \(D(u)=d_2(u)+|V_1|+2\big(|V_2|-d_2(u)-1\big)=2n_2+n_1-d_2(u)-2\) for \(u\in V_2\).

By definition of marginal entropy, \(I_D(G)\) can be computed as \begin{eqnarray*} I_D(G) &=& -\frac{1}{\sum_{u\in V(G)}{D(u)}}\sum_{u\in V(G)}{D(u)}\log_{2}{D(u)}+\log_{2}{\sum_{u\in V(G)}{D(u)}} \\ &=& -\frac{1}{\sum_{u\in V_1}{D(u)}+\sum_{u\in V_2}{D(u)}}\Big[\sum_{u\in V_1}{D(u)}\log_{2}{D(u)}+\sum_{u\in V_2}{D(u)}\log_{2}{D(u)}\Big]\\ & & +\log_{2}{\Big[\sum_{u\in V_1}{D(u)}+\sum_{u\in V_2}{D(u)}\Big]}\\ &=& -\frac{1}{\sum_{u\in V_1}{\big(2n_1+n_2-d_1(u)-2\big)}+\sum_{u\in V_2}{\big(2n_2+n_1-d_2(u)-2}\big)}\\ & & \times\Big[\sum_{u\in V_1}{\big(2n_1+n_2-d_1(u)-2\big)\log_{2}\big(2n_1+n_2-d_1(u)-2\big)}\\ & & +\sum_{u\in V_2}{\big(2n_2+n_1-d_2(u)-2\big)\log_{2}\big(2n_2+n_1-d_2(u)-2\big)}\Big]\\ & & +\log_{2}{\Big[\sum_{u\in V_1}{\big(2n_1+n_2-d_1(u)-2\big)}+\sum_{u\in V_2}{\big(2n_2+n_1-d_2(u)-2}\big)\Big]}\\ &=& -\frac{1}{n_1(2n_1+n_2-2)-\sum_{u\in V_1}{d_1(u)}+n_2(2n_2+n_1-2)-\sum_{u\in V_2}{d_2(u)}}\\ & & \times\Big[\sum_{u\in V_1}{\big(2n_1+n_2-d_1(u)-2\big)\log_{2}\big(2n_1+n_2-d_1(u)-2\big)}\\ & & +\sum_{u\in V_2}{\big(2n_2+n_1-d_2(u)-2\big)\log_{2}\big(2n_2+n_1-d_2(u)-2\big)}\Big]\\ & & +\log_{2}{\Big[n_1(2n_1+n_2-2)-\sum_{u\in V_1}{d_1(u)}+n_2(2n_2+n_1-2)-\sum_{u\in V_2}{d_2(u)}\Big]}\\ &=& 1-\frac{1}{2\big(n_1^2+n_2^2+n_1n_2-n_2-n_2-m_2-m_2\big)}\\ & & \times\Big[\sum_{u\in V_1}{\big(2n_1+n_2-d_1(u)-2\big)\log_{2}\big(2n_1+n_2-d_1(u)-2\big)}\\ & & +\sum_{u\in V_2}{\big(2n_2+n_1-d_2(u)-2\big)\log_{2}\big(2n_2+n_1-d_2(u)-2\big)}\Big]\\ & & +\log_{2}{\Big[n_1^2+n_2^2+n_1n_2-n_2-n_2-m_2-m_2\Big]}. \end{eqnarray*} This completes the proof.

4. Summary and discussions

In this paper, the marginal entropies of the complete bipartite graphs, complete multipartite graphs, firefly graphs, lollipop graphs, clique-chain graphs, Cartesian product and join of two graphs are obtained. For some other specific types of graphs, the existing results can be useful. For example, applying Cartesian product operation to \(P_m\) and \(P_n\), \(P_m\) and \(C_n\), \(C_m\) and \(C_n\) we can get grid graph, cylinder graph and torus graph with order \(mn\), respectively. In addition, Sahin obtained the marginal entropy for paths and cycles [15], so the formulas for above three special types of graphs can be done if readers are interested. And we note that \(K_{a,b}=\overline{K_a}\vee\overline{K_b}\). Since the join of any two graphs must be connected, one can also get Theorem 1 by Theorem 7. Similarly, \(K_{a_1,a_2,\ldots,a_k}=\overline{K_{a_1}}+\ldots+\overline{K_{a_k}}\), one can also study the multiple operations of graphs.

Acknowledgments

The authors are grateful to the anonymous referee for careful reading and valuable comments which result in an improvement of the original manuscript. This work was supported by the Natural Science Foundation of Qinghai Province (No. 2021-ZJ-703).

Author Contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

It is assumed that the reader is familiar with the concept of graphs as well as that of a proper coloring of a graph. For general notation and concepts in graphs see [1,2,3]. For specific (new) notation used in this paper refer to [4,5]. By convention, if \(G\) has order \(n \geq 1\) and has no edges (\(\varepsilon(G)=0\)) then \(G\) is called a null graph denoted by, \(\mathfrak{N}_n\).

§2 deals with the introduction to Lucky \(k\)-polynomials. §2.1 presents Lucky \(k\)-polynomials for null graphs. In §3, some main results are presented in respect of complete split graphs and for generalized star graphs. Finally, in §4, a few suggestions on future research on this problem are discussed.

2. Lucky \(k\)-Polynomials

Recall from [6] that in an improper coloring an edge \(uv\) for which, \(c(u)=c(v)\) is called a bad edge. In [5] the notion of the chromatic completion number of a graph \(G\) denoted by, \(\zeta(G)\) was introduced. Also, recall from [5] that \(\zeta(G)\) is the maximum number of edges over all chromatic colorings that can be added to \(G\) without adding a bad edge.

Recall from [5] that a chromatic coloring in accordance with Lucky's theorem or an optimal near-completion \(\chi\)-partition is called a Lucky \(\chi\)-coloring or simply a Lucky coloring denoted by, \(\varphi_{\mathcal{L}}(G)\).

For \(\chi(G) \leq n\leq \lambda\) colors the number of distinct Lucky \(k\)-colorings, \(\chi(G) \leq k\leq n\) is determined by a polynomial, called the Lucky \(k\)-polynomial, \(\mathcal{L}_G(\lambda,k)\). Lastly, recall the falling factorial, \(\lambda^{(n)}= \lambda(\lambda-1)(\lambda-2)\cdots (\lambda-n+1)\).

Corollary 1. For a graph \(G\) of order \(n\geq 1\), \(\lambda \geq n\) the Lucky \(n\)-polynomial is, \begin{equation}\mathcal{L}_{G}(\lambda,n)= \lambda^{(n)}= \binom{\lambda}{n}\cdots n!. \end{equation}

Proof. For any graph of order \(n \geq 1\), it follows that any proper \(n\)-coloring necessarily has \(\theta(c_i)=1\), \(\forall~i\). Therefore the result.

A trivial upper bound is observed.

Corollary 2. For any graph \(G\) of order \(n\), \(\mathcal{L}_{K_n}(\lambda,n) \leq \mathcal{P}_G(\lambda,n)\) where \(\mathcal{P}_G(\lambda,n)\) is the chromatic polynomial of \(G\).

Theorem 1. For a graph \(G\), \(\chi(G)\leq k'\leq k\leq \lambda\), it follows that, \begin{equation} \mathcal{L}_{G}(\lambda,k') \leq \mathcal{L}_{G}(\lambda,k). \end{equation}

Proof. The result follows from the number theory result. For a \begin{equation} k'-tuple, (x_1,x_2,x_3,\dots,x_{k'}) \  such\  that\  \sum\limits_{i=1}^{k'}x_i =n \end{equation} and a \begin{equation} k-tuple, (y_1,y_2,y_3,\dots,y_k) \  such \  that\  \sum\limits_{i=1}^{k}y_i =n \end{equation} we have that, \begin{equation}\sum\limits_{i=1}^{k'-1}\prod\limits_{j=i+1}^{k'}x_ix_j \leq \sum\limits_{i=1}^{k-1}\prod\limits_{j=i+1}^{k}y_iy_j. \end{equation}

2.1. Lucky \(k\)-polynomials of null graphs

By convention let the vertices of a null graph of order \(n\geq 2\) be viewed as seated on the circumference of an imaginary circle and let the vertices be consecutively labeled \(v_i\), \(i=1,2,3,\dots,n\) in a clockwise fashion. Since \(\chi(\mathfrak{N}_n) = 1\) it is obvious that for a proper \(1\)-coloring there are exactly \(\lambda\) distinct proper \(1\)-colorings. Put differently, there are exactly \(\lambda\) distinct Lucky \(1\)-colorings. Hence, \(\mathcal{L}_{\mathfrak{N}_n}(\lambda, 1) = \lambda\). Similarly trivial, it follows that for a proper \(n\)-coloring there are \(\mathcal{L}_{\mathfrak{N}_n}(\lambda, n) = \lambda^{(n)}\) or \(\binom{\lambda}{n}n!\) such distinct Lucky \(n\)-colorings.

For the analysis of Lucky \(k\)-polynomials of null graphs of order \(n\geq 2\) and \(2\leq k \leq n-1\) we require a set theory perspective.

Case 1: As a special case we allow \(k=1\). For the set of vertices \(\{v_1,v_2\}\), we consider only Lucky \(1\)-colorings. As stated before there are \(\lambda\) such distinct Lucky \(1\)-colorings.

Case 2: For the set of vertices \(\{v_1,v_2,v_3\}\), we consider only Lucky \(2\)-colorings. For a Lucky \(2\)-coloring we consider the partitions: \begin{equation}\{\{v_1,v_2\},\{v_3\}\}, \{\{v_1,v_3\},\{v_2\}\}, \{\{v_2,v_3\},\{v_1\}\}. \end{equation} \begin{equation} Hence, \mathcal{L}_{\mathfrak{N}_3}(\lambda, 2) = 3\lambda^{(2)}= 3\lambda(\lambda-1). \end{equation}

Case 3: For the set of vertices \(\{v_1,v_2,v_3,v_4\}\), we consider both Lucky \(2\)-colorings and Lucky \(3\)-colorings. For a Lucky \(2\)-coloring we consider the partitions:

\begin{equation} \{\{v_1,v_2\},\{v_3,v_4\}\}, \{\{v_1,v_3\},\{v_2,v_4\}\}, \{\{v_1,v_4\},\{\{v_2,v_3\}\}. \end{equation} \begin{equation} Hence, \mathcal{L}_{\mathfrak{N}_4}(\lambda, 2) = 3\lambda^{(2)}= 3\lambda(\lambda-1). \end{equation} For a Lucky \(3\)-coloring we consider the partitions: \begin{equation}\{\{v_1,v_2\},\{v_3\},\{v_4\}\} , \{\{v_1,v_3\},\{v_2\},\{v_4\}\} , \{\{v_1,v_4\},\{v_2\},\{v_3\}\} ,\\ \{\{v_2,v_3\},\{v_1\},\{v_4\}\} , \{\{v_2,v_4\},\{v_1\},\{v_3\}\} , \{\{v_3,v_4\},\{v_1\},\{v_2\}\}. \end{equation} \begin{equation} Hence, \mathcal{L}_{\mathfrak{N}_4}(\lambda, 3) = 6\lambda^{(3)}= 6\lambda(\lambda-1)(\lambda-2). \end{equation}

Case 4: For the set of vertices \(\{v_1,v_2,v_3,v_4,v_5\}\), we consider Lucky \(2\)-colorings, Lucky \(3\)-colorings and Lucky \(4\)-colorings.

From Lucky's theorem [5] it follows that for a Lucky \(2\)-coloring the partitions must have the form \(\{\{3\)-elements\(\},\{2\)-elements\(\}\}\). From the partitions in Case 3 it follows that \(6\) such partitions will follow. In addition \(4\) further \(2\)-element subsets of the form \(\{v_i,v_5\}\), \(i=1,2,3,4\) together with a unique corresponding \(3\)-element subset are 4 more partitions. Hence, \(10\) such partitions exist. \begin{equation} Therefore, \mathcal{L}_{\mathfrak{N}_5}(\lambda, 2) = 10\lambda^{(2)}= 10\lambda(\lambda-1). \end{equation} From Lucky's theorem [5] it follows that for a Lucky \(3\)-coloring the partitions must have the form \(\{\{2\)-element\(\},\{2\)-element\(\},\{1\)-element\(\}\}\). From the partitions in Case 3 it follows that \(12\) such partitions will follow. In addition \(3\) further partitions of the form \(\{\{2\)-element\(\},\{2\)-element\(\},\{v_5\}\}\) are possible. The aforesaid follows from the partitions for a Lucky \(2\)-coloring in Case 3. \begin{equation} Hence, \mathcal{L}_{\mathfrak{N}_5}(\lambda, 3) = 15\lambda^{(3)}= 15\lambda(\lambda-1)(\lambda-2). \end{equation} From Lucky's theorem [5] it follows that for a Lucky \(4\)-coloring the partitions must have the form \(\{\{2\)-element\(\},\{1\)-element\(\},\{1\)-element\(\},\{1\)-element\(\}\}\). There are \(\binom{5}{2}=10\) such \(2\)-element subsets. Each will correspond with its unique triple of \(1\)-element subsets. \begin{equation} Hence, \mathcal{L}_{\mathfrak{N}_5}(\lambda, 4) = 10\lambda^{(4)}= 10\lambda(\lambda-1)(\lambda-2)(\lambda-3). \end{equation}

Observation 1. The Lucky \(k\)-polynomial for a null graph \(\mathfrak{N}_n\) has the trivial form i.e. \(\mathcal{L}_{\mathfrak{N}_n}(\lambda, k) = m_{\mathfrak{N}_n}(n,k)\cdots \lambda^{(k)}\) with \(m_{\mathfrak{N}_n}(n,k)\) some positive integer for \(k\in\{1,2,3,\dots,n\}\) and \(n=1,2,3,\dots\). Furthermore, it is conjectured that if \(G\) permits a \(k\)-coloring then the Lucky \(k\)-polynomial has the form \(\mathcal{L}_G(\lambda, k) = m_G(n,k)\cdot \lambda^{(k)}\) with \(m_G(n,k)\) some positive integer. Note that \(m_G(n,k)\leq S(n,k)\) where \(S(n,k)\) is the corresponding Stirling number of the second kind (or Stirling partition number). These subsets of specialized numbers, \(m_G(n,k)\), are called the family of Lucky numbers.

2.2. Lucky \(2\)-polynomials of null graphs

It is evident that Cases 1 to 4 present an inefficient way of determining the value of \(m_{\mathfrak{N}_n}(n,k)\). The approach in this subsection is to present a recursive result for Lucky \(2\)-colorings. We summarize the Lucky \(2\)-coloring results above as a corollary.

Corollary 3.

• (a) For \(n=1\), \(\mathcal{L}_{\mathfrak{N}_1}(\lambda,2) = 0\).
• (b) For \(n=2\), \(\mathcal{L}_{\mathfrak{N}_2}(\lambda,2) = \lambda(\lambda-1)\).
• (c) For \(n=3\), \(\mathcal{L}_{\mathfrak{N}_3}(\lambda,2) = 3\lambda(\lambda-1)\).
• (d) For \(n=4\), \(\mathcal{L}_{\mathfrak{N}_4}(\lambda,2) = 3\lambda(\lambda-1)\).
• (e) For \(n=5\), \(\mathcal{L}_{\mathfrak{N}_5}(\lambda,2) = 10\lambda(\lambda-1)\).

Theorem 2. For a null graph \(\mathfrak{N}_n\), \(n= 6,7,8,\cdots\) we have

• (a) If \(n\) is odd then, \begin{equation}\mathcal{L}_{\mathfrak{N}_n}(\lambda,2) = 2\mathcal{L}_{\mathfrak{N}_{n-1}}(\lambda,2) + \binom{n-1}{\frac{n-3}{2}}\lambda(\lambda-1). \end{equation}
• (b) If \(n\) is even then, \begin{equation}\mathcal{L}_{\mathfrak{N}_n}(\lambda,2) = \mathcal{L}_{\mathfrak{N}_{n-1}}(\lambda,2). \end{equation}

Proof.

• (a) If \(n\) is odd then \(n-1\) is even. So the number of Lucky \(2\)-colorings of \(\mathfrak{N}_{n-1}\) results from the number of partitions of the form \begin{equation}\Big\{\Big\{\frac{(n-1)}{2} -element \Big\},\Big\{\frac{(n-1)}{2} -element \Big\}\Big\}\  in \  respect \  of \Big\{v_i:i=1,2,3,\dots,v_{n-1}\Big\}. \end{equation} Hence, there are exactly \(2m_{\mathfrak{N}_{n-1}}((n-1),2)\) partitions which will be obtained from the consecutive union of \(\Big\{v_n\Big\}\) with each of the \(\frac{(n-1)}{2}\)-element subsets in each partition to obtain partitions of the form \begin{equation}\Big\{\Big\{\frac{(n+1)}{2} -element \Big\},\Big\{\frac{(n-1)}{2} -element \Big\}\Big\}\  in \  respect \  of\Big\{v_i:i=1,2,3,\dots,v_n\Big\}. \end{equation} Therefore, the term \(2\mathcal{L}_{\mathfrak{N}_{n-1}}(\lambda, 2)\) in the result. Finally the number of distinct \(\frac{(n-1)}{2}\)-element subsets which has the vertex element \(v_n\) together with each unique corresponding \(\frac{(n+1)}{2}\)-element subset must be added as \begin{equation}\Big\{\Big\{\frac{(n+1)}{2} -element \Big\},\Big\{\frac{(n-1)}{2} -element \Big\}\Big\} \end{equation} partitions. Hence, the element \(v_n\) can be added to each of the \(\binom{n-1}{\frac{n-3}{2}}\)-element subsets from the vertex set \(\Big\{v_i: i=1,2,3,\dots,v_{n-1}\Big\}\). Therefore, through immediate induction the result follows.
• (b) If \(n\) is even then \(n'=n-1\) is odd. The partitions of the vertex set \(\Big\{v_1,v_2,v_3,\dots,v_{n-1}\Big\}\) are of the form \begin{equation}\Big\{\Big\{\frac{n}{2} -element \Big\},\Big\{\lfloor \frac{(n-1)}{2}\rfloor -element \Big\}\Big\}. \end{equation} Therefore, by the union of \(\Big\{v_n\Big\}\) and each of the \(\Big\{\lfloor \frac{(n-1)}{2}\rfloor\)-element\(\Big\}\) subsets in the \(m_{\mathfrak{N}_{n-1}}((n-1),2)\) partitions, the required \(m_{\mathfrak{N}_{n-1}}((n-1),2)= m_{\mathfrak{N}_n}(n,2)\) partitions of the form \begin{equation}\Big\{\Big\{\frac{n}{2} -element \Big\},\Big\{\frac{n}{2} -element \Big\}\Big\}\  in \  respect \  of\Big\{v_i: i=1,2,3,\dots,n\Big\} \end{equation} are obtained. Therefore, through immediate induction the result follows.

2.3. Lucky \(3\)-polynomials of null graphs

In this subsection we present a recursive result for Lucky \(3\)-colorings. We summarize the Lucky \(3\)-coloring results above as a corollary.

Corollary

• (a) For \(n=1\), \(\mathcal{L}_{\mathfrak{N}_1}(\lambda,3) = 0\).
• (b) For \(n=2\), \(\mathcal{L}_{\mathfrak{N}_2}(\lambda,3) = 0\).
• (c) For \(n=3\), \(\mathcal{L}_{\mathfrak{N}_3}(\lambda,3) = \lambda(\lambda-1)(\lambda-2)\).
• (d) For \(n=4\), \(\mathcal{L}_{\mathfrak{N}_4}(\lambda,3) = 6\lambda(\lambda-1)(\lambda-2)\).
• (e) For \(n=5\), \(\mathcal{L}_{\mathfrak{N}_5}(\lambda,3) = 15\lambda(\lambda-1)(\lambda-2)\).

Partition the set of positive integers into subsets, \( X_1=\{i: i=6+3t, t=0,1,2,\dots\}\), \(X_2 = \{i: i=7+3t, t=0,1,2,\dots\}\) and \( X_3=\{i: i=8+3t, t=0,1,2,\dots\}\).

Theorem 3. For a null graph \(\mathfrak{N}_n\), \(n= 6,7,8,\cdots\), we have

• (a) If \(n \in X_1\) then, \(\mathcal{L}_{\mathfrak{N}_n}(\lambda,3) = \mathcal{L}_{\mathfrak{N}_{n-1}}(\lambda, 3)\).
• (b) If \(n \in X_2\) then, \(\mathcal{L}_{\mathfrak{N}_n}(\lambda,3) = 3\mathcal{L}_{\mathfrak{N}_{n-1}}(\lambda,3) + \binom{n-1}{\frac{n-4}{3}}\lambda(\lambda-1)(\lambda-2)\).
• (c) If \(n \in X_3\) then, \(\mathcal{L}_{\mathfrak{N}_n}(\lambda,3) = 2\mathcal{L}_{\mathfrak{N}_{n-1}}(\lambda,3) + \binom{n-1}{\frac{n-5}{3}}\lambda(\lambda-1)(\lambda-2)\).

Proof.

• (a) If \(n \in X_1\), then the partitions of the vertex set \(\Big\{v_1,v_2,v_3,\dots,v_n\Big\}\) are of the form \begin{equation}\Big\{\Big\{\frac{n}{3} -element \Big\},\Big\{\frac{n}{3} -element \Big\},\Big\{\frac{n}{3} -element \Big\}\Big\}. \end{equation} Therefore, the partitions of \(\Big\{v_1,v_2,v_3,\dots,v_{n-1}\Big\}\) are of the form \begin{equation}\Big\{\Big\{\frac{n}{3} -element \Big\},\Big\{\frac{n}{3} -element \Big\},\Big\{(\frac{n}{3}-1) -element \Big\}\Big\}. \end{equation} From the union of \(\Big\{v_n\Big\}\) and each of the \(\Big\{(\frac{n}{3}-1)\)-element\(\Big\}\Big\}\) subsets in the \(m_{\mathfrak{N}_{n-1}}((n-1),3)\) partitions, the required \(m_{\mathfrak{N}_{n-1}}((n-1),3)= m_{\mathfrak{N}_n}(n,3)\) partitions of the form \begin{equation}\Big\{\Big\{\frac{n}{3} -element \Big\},\Big\{\frac{n}{3} -element \Big\},\Big\{\frac{n}{3} -element \Big\}\Big\}\  in \  respect \  of\Big\{v_i: i=1,2,3,\dots,n\Big\} \end{equation} are obtained. Therefore, through immediate induction the result follows.
• (b) If \(n \in X_2\), then the partitions of the vertex set \(\Big\{v_1,v_2,v_3,\dots,v_n\Big\}\) are of the form \begin{equation}\Big\{\Big\{(\frac{n-1}{3}+1) -element \Big\},\Big\{\frac{(n-1)}{3} -element \Big\},\Big\{\frac{(n-1)}{3} -element \Big\}\Big\}. \end{equation} Therefore, the partitions of \(\Big\{v_1,v_2,v_3,\dots,v_{n-1}\Big\}\) are of the form \begin{equation}\Big\{\Big\{\frac{n-1}{3} -element \Big\},\Big\{\frac{(n-1)}{3} -element \Big\},\Big\{\frac{(n-1)}{3} -element \Big\}\Big\}. \end{equation} From the union of \(\Big\{v_n\Big\}\) and each of the \(\Big\{\frac{(n-1)}{3}\)-element\(\Big\}\) subsets in each partition, exactly \(3m_{\mathfrak{N}_{n-1}}((n-1),3)\) partitions of the form \begin{equation}\Big\{\Big\{(\frac{n-1}{3}+1) -element \Big\},\Big\{\frac{(n-1)}{3} -element \Big\},\Big\{\frac{(n-1)}{3} -element \Big\}\Big\} \end{equation} are obtained. Hence, the term \(3\mathcal{L}_{\mathfrak{N}_{n-1}}(\lambda, 3)\). Finally the number of distinct \(\Big\{\frac{(n-1)}{3}\)-element\(\Big\}\) subsets which has the vertex element \(v_n\) together with each unique corresponding \((\frac{(n+1)}{3}+1)\)-element subset must be added as \begin{equation}\Big\{\Big\{(\frac{(n+1)}{3}+1) -element \Big\},\Big\{\frac{(n-1)}{3} -element \Big\},\Big\{\frac{(n-1)}{3} -element \Big\}\Big\} \end{equation} partitions. Hence, the element \(v_n\) can be added to each of the \(\binom{n-1}{\lfloor \frac{n}{3}\rfloor-1}\)-element subsets from the vertex set \(\Big\{v_i: i=1,2,3,\dots,v_{n-1}\Big\}\) to obtain the term \(\binom{n-1}{\frac{n-4}{3}}\lambda(\lambda-1)(\lambda-2)\). Therefore, through immediate induction the result follows.
• (c) This result follows through similar reasoning as part (b).

We are ready to give a main result.

Theorem 4. For \(4 \leq k \leq \lambda\), let \(n\geq k\), \(X_1= \{i:i=k(t+1), t=0,1,2,\dots\}\) and \(X_2= \Bbb{N}\backslash X_1\). It follows that,

• (a) If \(\lambda \geq k > n\), then \(\mathcal{L}_{\mathfrak{N}_n}(\lambda,k) = 0\).
• (b) If \(4 \leq k \leq n \leq \lambda\) and \(n \in X_1\), then \(\mathcal{L}_{\mathfrak{N}_n}(\lambda,k) = \mathcal{L}_{\mathfrak{N}_{n-1}}(\lambda,k)\).
• (c) If \(4 \leq k \leq n \leq \lambda\) and \(n \in X_2\) let \(\frac{n}{k}= \lfloor \frac{n}{k}\rfloor + r\), \(0 < r \leq (k-1)\), then \(\mathcal{L}_{\mathfrak{N}_n}(\lambda,k) = (k-r)\mathcal{L}_{\mathfrak{N}_{n-1}}(\lambda,k) + \binom{n-1}{\frac{n-(r+k)}{k}}\lambda^{(k)}\).

Proof. Point (a) is trivial. Points (b) and (c) can be proved by similar reasoning as in the proofs of Theorems 2 and 3.

3. Lucky \(k\)-polynomials of complete split graphs

For certain graphs the Lucky \(k\)-polynomials follow trivially. Note that for a graph \(G\) the lower bound \(k \geq \chi(G)\) applies. We present a corollary without proof. Recall that a star \(S_{1,n}\) has a central vertex say \(u_1\) which is adjacent to each vertex in the independent set of vertices \(\{v_i:1\leq i\leq n\}\).

Corollary 5. For the star \(S_{1,n}\), \(n\geq 1\) and \(2\leq k \leq \lambda\), it follows that, \begin{equation}\mathcal{L}_{S_{1,n}}(\lambda,k) = \lambda \mathcal{L}_{\mathfrak{N}_n}(\lambda,k-1). \end{equation}

Recall that, a split graph is a graph in which the vertex set can be partitioned into a clique and an independent set. Note that a null graph and a star graph, \(S_{1,n}\) are relatively simple split graphs.

A complete split graph is a split graph such that each vertex in the independent set is adjacent to all the vertices of the clique (the clique is a smallest clique which permits a maximum independent set). Note that a complete graph \(K_n\) is also a complete split graph i.e. any subset of \(n-1\) vertices induces a smallest clique and the corresponding \(1\)-element subset is a maximum independent set.

Lemma 1. For a complete split graph \(G \neq K_n\), \(n\geq 3\), both the maximum independent set and the corresponding clique are unique.

Proof. Consider a clique \(Q\) and the corresponding maximum independent set \(X\) of \(G\). If it is possible to obtain another independent set say, \(X'= X\cup v_i\), \(v_i \in V(Q)\) then \(V(G)\) was not partitioned in accordance to the definition of a split graph because no \(v_j\in X\) is adjacent to \(v_i\). Similarly, \(V(Q) \cup v_k\), \(v_k\in X\) is not possible. Therefore, both the independent set and the clique are unique.

Theorem 5. Let \(X\) be the independent set in a complete split graph \(G \neq K_n\) and let the clique \(K_t\) correspond to \(\langle X \rangle\) in \(G\). Let \(t+1 \leq k \leq \lambda\). Then, \begin{equation}\mathcal{L}_G(\lambda,k) = \lambda^{(t)}\mathcal{L}_{\mathfrak{N}_{n-t}}(\lambda-t,k-t). \end{equation}

Proof. It follows that any Lucky coloring of \(K_t\) necessitates a \(t\)-coloring. From the completeness between \(K_t\) and \(\langle X\rangle\) (a \((n-t)\)-null graph) it follows that only a \((k-t)\)-coloring from amongst \(\lambda-t\) colors can be assigned to the vertices of \(X\). From Corollary 5 and Lemma 1, the result follows through immediate induction for any complete split graph.

A generalized star is defined as, a graph \(G\) which can be partitioned into an independent set \(X\) and a subgraph \(G'\) (not necessarily connected) such that each vertex \(u_i \in V(G')\) is adjacent to all vertices in \(X\). Note that a complete split graph is also a generalized star.

Lemma 2. For a generalized star \(G \neq K_n\), \(n\geq 3\) the maximum independent set \(Y\) is, either \(Y=X\) or \(Y\subseteq V(G')\) and the corresponding subgraph \(G'\) is unique.

Proof. By similar reasoning to that in the proof of Lemma 1.

Theorem 6. Let \(X\) be the independent set in a generalized star \(G \neq K_n\) and let the subgraph \(G'\) of order \(t\) correspond to \(\langle X \rangle\) in \(G\). Let \(t+1 \leq k \leq \lambda\). Then, \begin{equation}\mathcal{L}_G(\lambda,k) = max\{\mathcal{L}_{G'}(\lambda,\ell)\cdots \mathcal{L}_{\mathfrak{N}_{n-t}}(\lambda-\ell,k-\ell) \  for \  some \  \chi(G')\leq \ell \leq k-1\}. \end{equation}

Proof. Assume \(|V(G')| = t\). It follows that any Lucky coloring of \(G'\) can at most be a \(t\)-coloring. From the completeness between \(G'\) and \(\langle X\rangle\) (a \((n-t)\)-null graph) it follows that for a Lucky \(k\)-coloring any color set \(\mathcal{C}\), \(\mathcal{C}' \subseteq \mathcal{C}\) requires a \(2\)-partition into say \begin{equation}\{\{\ell -element \},\{(k-\ell) -element \}\}. \end{equation} From [5] it follows that the existence of an optimal near-completion \(\ell\)-partition of \(V(G')\) will yield a corresponding Lucky coloring of \(G'\) yielding \(\zeta(G')\). Because \(\zeta(G') + \zeta(\mathfrak{N}_{n-t})\) must be maximized and maximization is always possible, the result follows through immediate induction.

Note that, Theorem 6 can immediately be generalized to the join operation between graphs \(G\), \(H\). We state it without proof because the reasoning of proof is similar to that found in the proof of Theorem 6.

Theorem 7. For the graphs \(G\) and \(H\) it follows that, \begin{equation}\mathcal{L}_{G+H}(\lambda,k) = max\{\mathcal{L}_{G}(\lambda,\ell)\cdots \mathcal{L}_{H}(\lambda-\ell,k-\ell)\  for \  some \ \chi(G)\leq \ell \leq k-1\}. \end{equation}

4. Conclusion

From Theorem 7, it follows naturally to seek a result for the corona operation between two graphs. Other interesting problems are,

Problem 1. Find a closed formula, if such exists, for the family of Lucky numbers, \(m_G(n,k)\) for \(\chi(G) \leq k \leq \lambda\) and \(n\in \Bbb{N}\).

Problem 2. Find an efficient algorithm to find \begin{equation}\mathcal{L}_{G+H}(\lambda,k) = max\{\mathcal{L}_{G}(\lambda,\ell)\cdots \mathcal{L}_{H}(\lambda-\ell,k-\ell)\  for \  some \ \chi(G)\leq \ell \leq k-1\}. \end{equation}

Problem 3. Use Theorem 6 to formulate and proof a generalized result for complete \(q\)-partite graphs.

Problem 4. Find an efficient algorithm to find the Lucky \(k\)-polynomials of complete \(q\)-partite graphs.

Acknowledgments

The author would like to thank the anonymous referees for their constructive comments, which helped to improve on the elegance of this paper.

Conflicts of Interest:

The author declares no conflict of interest.

1. Introduction

Many uncertain theories put forward as fuzzy set [1], intuitionistic fuzzy set [2], bipolar fuzzy sets [3] and Pythagorean fuzzy set [4]. Zadeh, introduced fuzzy set, suggests that decision-makers solve uncertain problems by considering membership degree. Atanassov introduces the concept of an intuitionistic fuzzy set. It is characterized by a degree of membership and non-membership satisfying the condition that the sum of its membership degree and non-membership degree does not exceed unity. However, we may interact a problem in decision-making events where the sum of the degree of membership and non-membership of a particular attribute is exceeded one-the concept of Pythagorean fuzzy sets introduced by Yager. The theory of soft sets proposed by Molodtsov [5]. It is a tool of parameterization for coping with uncertainties. Compared with other uncertain theories, soft sets reflect the objectivity and complexity of decision-making during actual situations more accurately. It has been an outstanding achievement both in theories and applications.

Moreover, combining soft sets with other mathematical models is also a critical research area. The concept of the fuzzy soft set by Maji [6], intuitionistic fuzzy soft set [7], and Saleem Abdullah et al., initiated the concept of bipolar fuzzy soft sets [8]. Pinaki Majumdara et al., defined the concept of discussed generalized fuzzy soft sets [9]. In recent years, Peng et al., [10] has extended fuzzy soft set to Pythagorean fuzzy soft set. In 2011, Alkhazaleh et al., [11] introduced the concept of possibility fuzzy soft sets. Yager et al., [12] discuss the application for Pythagorean membership grades, complex numbers, and decision making under 2014. In 2018, Mohana et al., [13] interact bipolar Pythagorean fuzzy sets with an application under decision-making problems. Akram et al., [14] initiate the new type of models for decision making based on rough Pythagorean fuzzy bipolar soft set in 2018. Alkhazaleh et al., [15] discussed the theory of generalized interval-valued fuzzy soft set in 2012. Jana and Pal studied bipolar intuitionistic fuzzy soft sets with applications [16]. In 2019, Jana and Pal introduced Pythagorean fuzzy dombi aggregation operators [17]. Recently, Palanikumar et al., [18] discuss the application for possibility Pythagorean bipolar fuzzy soft sets.

This paper aims to extend the concept of generalized fuzzy soft sets to the parameterization of Type-II generalized Pythagorean bipolar fuzzy sets. We shall further establish a similarity measure based on the soft set model.

2. Preliminaries

Definition 1. [13] Let \(X\) be a non-empty set of the universe, Pythagorean bipolar fuzzy set(PBFS) \(A\) in \(X\) is an object having the following form: \[A= \{ x, \zeta^{+}_{A}(x), \xi^{+}_{A}(x), \zeta^{-}_{A}(x), \xi^{-}_{A}(x)| x\in X \},\] where \(\zeta^{+}_{A}(x), \xi^{+}_{A}(x) \), \(\zeta^{-}_{A}(x), \xi^{-}_{A}(x)\) represent the degree of positive membership, degree of positive non-membership, degree of negative membership and degree of negative non-membership of \(A\) respectively. Consider the mapping \(\zeta^{+}_{A}, \xi^{+}_{A}: X \rightarrow [0,1]\), \(\zeta^{-}_{A}, \xi^{-}_{A}: X \rightarrow [-1,0]\) such that \[0 \leq (\zeta^{+}_{A}(x))^{2}+(\xi^{+}_{A}(x))^{2} \leq 1\;\text{ and }\;-1 \leq -\Big[ (\zeta^{-}_{A}(x))^{2}+(\xi^{-}_{A}(x))^{2}\Big] \leq 0.\] The degree of indeterminacy is determined as \[\pi^{+}_{A}(x)=\Big[\sqrt{1-(\zeta^{+}_{A}(x))^{2}-(\xi^{+}_{A}(x))^{2}}\,\,\Big]\;\text{ and }\;\pi^{-}_{A}(x)=-\Big[\sqrt{1-(\zeta^{-}_{A}(x))^{2}-(\xi^{-}_{A}(x))^{2}}\,\,\Big].\] Then \(A= \langle \zeta^{+}_{A}, \xi^{+}_{A}, \zeta^{-}_{A}, \xi^{-}_{A}\rangle\) is called a Pythagorean bipolar fuzzy number(PBFN).

Definition 2. [13] Given that \(\alpha_{1}= (\zeta^{+}_{\alpha_{1}}, \xi^{+}_{\alpha_{1}}, \zeta^{-}_{\alpha_{1}}, \xi^{-}_{\alpha_{1}})\), \(\alpha_{2} = (\zeta^{+}_{\alpha_{2}}, \xi^{+}_{\alpha_{2}}, \zeta^{-}_{\alpha_{2}}, \xi^{-}_{\alpha_{2}})\) and \(\alpha_{3}= (\zeta^{+}_{\alpha_{3}}, \xi^{+}_{\alpha_{3}}, \zeta^{-}_{\alpha_{3}}, \xi^{-}_{\alpha_{3}})\) are any three PBFN's over \((X,E)\), then the following properties are holds:

1. \(\alpha_{1}^{c}= (\xi^{+}_{\alpha_{1}}, \zeta^{+}_{\alpha_{1}},\xi^{-}_{\alpha_{1}}, \zeta^{-}_{\alpha_{1}})\),
2. \(\alpha_{2} \cup \alpha_{3}= \big[\max(\zeta^{+}_{\alpha_{2}}, \zeta^{+}_{\alpha_{3}}), \min(\xi^{+}_{\alpha_{2}}, \xi^{+}_{\alpha_{3}}), \min(\xi^{-}_{\alpha_{2}}, , \xi^{-}_{\alpha_{3}}) \max(\zeta^{-}_{\alpha_{2}}, \zeta^{-}_{\alpha_{3}})\big]\),
3. \(\alpha_{2} \cap \alpha_{3}= \big[\min(\zeta^{+}_{\alpha_{2}}, \zeta^{+}_{\alpha_{3}}), \max(\xi^{+}_{\alpha_{2}}, \xi^{+}_{\alpha_{3}}), \max(\zeta^{-}_{\alpha_{2}}, \zeta^{-}_{\alpha_{3}}), \min(\xi^{-}_{\alpha_{2}}, \xi^{-}_{\alpha_{3}})\big]\),
4. \(\alpha_{2} \geq \alpha_{3}\) if and only if \(\zeta^{+}_{\alpha_{2}} \geq \zeta^{+}_{\alpha_{3}}\) and \(\xi^{+}_{\alpha_{2}} \leq \xi^{+}_{\alpha_{3}}\) and \(\zeta^{-}_{\alpha_{2}} \leq \zeta^{-}_{\alpha_{3}}\) and \(\xi^{-}_{\alpha_{2}} \geq \xi^{-}_{\alpha_{3}}\),
5. \(\alpha_{2} = \alpha_{3}\) if and only if \(\zeta^{+}_{\alpha_{2}} = \zeta^{+}_{\alpha_{3}}\) and \(\xi^{+}_{\alpha_{2}} = \xi^{+}_{\alpha_{3}}\) and \(\zeta^{-}_{\alpha_{2}} = \zeta^{-}_{\alpha_{3}}\) and \(\xi^{-}_{\alpha_{2}} = \xi^{-}_{\alpha_{3}}\).

Definition 3. [8] Let \(X\) be a non-empty set of the universe and \(E\) be a set of parameter. The pair \((\mathscr{M}, A)\) is called a bipolar fuzzy soft set(BFSS) on \(X\) if \(A \subseteq E\) and \(\mathscr{M}: A \rightarrow \mathscr{GBM}^{X},\) where \(\mathscr{GBM}^{X}\) is the set of all bipolar fuzzy subsets of \(X\).

Definition 4. [14] Let \(X\) be a non-empty set of the universe and \(E\) be a set of parameter. The pair \((\mathscr{M}, A)\) is called a Pythagorean bipolar fuzzy soft set(PBFSS) on \(X\) if \(A \subseteq E\) and \(\mathscr{M}: A \rightarrow P\mathscr{GBM}^{X},\) where \(P\mathscr{GBM}^{X}\) is the set of all Pythagorean bipolar fuzzy subsets of \(X\).

Definition 5. [9] Let \(X=\{x_{1}, x_{2},...,x_{n}\}\) be a non-empty set of the universe and \(E=\{e_{1}, e_{2},...,e_{m}\}\) be a set of parameter. The pair \((X,E)\) is a soft universe. Consider the mapping \(\mathscr{M}: E \rightarrow I^{X}\) and \(\xi\) be a fuzzy subset of \(E\), i.e. \(\xi : E \rightarrow I=[0,1]\), where \(I^{X}\) is the collection of all fuzzy subsets of \(X\). Let \(\mathscr{M}_{\xi}: E \rightarrow I^{X} \times I\) be a function defined as \(\mathscr{M}_{\xi}(e)= (\mathscr{M}(e)(x), \xi(e)), \forall x \in X\). Then \(\mathscr{M}_{\xi}\) is called a generalized fuzzy soft set(GFSS) on \((X,E)\). Here for each parameter \(e_{i}\), \(\mathscr{M}_{\xi}(e_{i})= (\mathscr{M}(e_{i})(x), \xi(e_{i}))\) indicates not only the degree of belongingness of the elements of \(X\) in \(\mathscr{M}(e_{i})\) but also the degree of possibility of such belongingness which is represented by \(\xi(e_{i})\). So we can write \(\mathscr{M}_{\xi}(e_{i})\) as follows: \[\mathscr{M}_{\xi}(e_{i}) = \left(\Big\{\frac {x_{1}} {\mathscr{M}(e_{i})(x_{1})}, \frac {x_{2}} {\mathscr{M}(e_{i})(x_{2})},...,\frac {x_{n}} {\mathscr{M}(e_{i})(x_{n})} \Big\}, \xi(e_{i})\right),\] where \(\mathscr{M}(e_{i})(x_{1})\), \(\mathscr{M}(e_{i})(x_{2})\),...,\(\mathscr{M}(e_{i})(x_{n})\) are the degrees of belongingness and \(\xi(e_{i})\) is the degree of possibility of such belongingness.

Definition 6. [11] Let \(X=\{x_{1}, x_{2},...,x_{n}\}\) be a non-empty set of the universe and \(E=\{e_{1}, e_{2},...,e_{m}\}\) be a set of parameter. The pair \((X,E)\) is a soft universe. Consider the mapping \(\mathscr{M}: E \rightarrow \mathscr{M}(X)\) and \(\xi\) be a fuzzy subset of \(E\), i.e. \(\xi : E \rightarrow \mathscr{M}(X)\). Let \(\mathscr{M}_{\xi}: E \rightarrow \mathscr{M}(X) \times \mathscr{M}(X)\) be a function defined as \(\mathscr{M}_{\xi}(e)= (\mathscr{M}(e)(x), \xi(e)(x)), \forall x \in X\). Then \(\mathscr{M}_{\xi}\) is called a possibility fuzzy soft set(PFSS) on \((X,E)\). Here for each parameter \(e_{i}\), \(\mathscr{M}_{\xi}(e_{i})= (\mathscr{M}(e_{i})(x), \xi(e_{i})(x))\) indicates not only the degree of belongingness of the elements of \(X\) in \(\mathscr{M}(e_{i})\) but also the degree of possibility of such belongingness which is represented by \(\xi(e_{i})\). So we can write \(\mathscr{M}_{\xi}(e_{i})\) as follows: \[\mathscr{M}_{\xi}(e_{i}) = \Big\{\left(\frac {x_{1}} {\mathscr{M}(e_{i})(x_{1})}, \xi(e_{i})(x_{1})\right),\left( \frac {x_{2}} {\mathscr{M}(e_{i})(x_{2})} , \xi(e_{i})(x_{2})\right),...,\left(\frac {x_{n}} {\mathscr{M}(e_{i})(x_{n})}, \xi(e_{i})(x_{n})\right) \Big\}.\]

3. Type-II generalized Pythagorean bipolar fuzzy soft sets

Definition 7. Let \(X=\{x_{1}, x_{2}, ...,x_{n}\}\) be a non-empty set of the universe and \( E=\{e_{1}, e_{2}, ...,e_{m}\}\) be a set of parameter. The pair \((X,E)\) is called a soft universe. Suppose that \(\mathscr{M} : E \rightarrow P\mathscr{GBM}^{X}\) and \(p\) is a Pythagorean bipolar fuzzy subset of \(E\). That is \(p : E \rightarrow I\) where \(I\) denotes the collection of all Pythagorean bipolar fuzzy subsets of \(X\). If \(\mathscr{M}^{\mathscr{GB}}_{p} : E \rightarrow P\mathscr{GBM}^{X} \times I\) is a function defined as \[\mathscr{M}^{\mathscr{GB}}_{p}(e) = \big\langle \mathscr{GBM}(e)(x), p(e)\big\rangle, x\in X ,\] then \(\mathscr{M}^{\mathscr{GB}}_{p}\) is a Type-II generalized Pythagorean bipolar fuzzy soft set(Type-II GPBFSS) on \((X,E)\). For each parameter \(e\),

\(\mathscr{M}^{\mathscr{GB}}_{p}(e)=\)

\( \left( \Bigg\{\frac {x_{1}} { \zeta^{+}_{\mathscr{M}(e)}(x_{1}), \xi^{+}_{\mathscr{M}(e)}(x_{1}), \zeta^{-}_{\mathscr{M}(e)}(x_{1}), \xi^{-}_{\mathscr{M}(e)}(x_{1})},...,\frac {x_{n}} {\zeta^{+}_{\mathscr{M}(e)}(x_{n}), \xi^{+}_{\mathscr{M}(e)}(x_{n}), \zeta^{-}_{\mathscr{M}(e)}(x_{n}), \xi^{-}_{\mathscr{M}(e)}(x_{n})} \Bigg\}, \left(\zeta^{+}_{p(e)}, \xi^{+}_{p(e)},\zeta^{-}_{p(e)}, \xi^{-}_{p(e)}\right)\right) \).

Example 1. Let \(X = \{x_{1}, x_{2}, x_{3}\}\) be a set of three leptospirosis patients and \(E = \{e_{1}\) = high fever, \(e_{2}\) = headache, \(e_{3}\) = chills\(\}\) is a set of parameters. Suppose that \(\mathscr{M}^{\mathscr{GB}}_{p} : E \rightarrow P\mathscr{GBM}^{X} \times I\) is given by \begin{equation*} \mathscr{M}^{\mathscr{GB}}_{p}(e_{1})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.6 ,0.7 , -0.3 , -0.8 )}\\ \frac{x_{2}}{(0.9 , 0.4,-0.7 , -0.5 )}\\ \frac{x_{3}}{(0.8 ,0.5 ,-0.2 ,-0.9)} \end{array}\right\} (0.6 ,0.5 ,-0.8 , -0.3 ) \right); \end{equation*} \begin{equation*} \mathscr{M}^{\mathscr{GB}}_{p}(e_{2})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.7 ,0.4 ,-0.2 , -0.8 )}\\ \frac{x_{2}}{(0.3 , 0.9,-0.7 ,-0.4 )}\\ \frac{x_{3}}{(0.5 , 0.6 ,-0.2 , -0.9)} \end{array}\right\}(0.9 ,0.2 ,-0.7 ,-0.4 ) \right); \end{equation*} \begin{equation*} \mathscr{M}^{\mathscr{GB}}_{p}(e_{3})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.3 ,0.7 ,-0.8 , -0.4 )}\\ \frac{x_{2}}{(0.8 , 0.4,-0.7 , -0.3 )}\\ \frac{x_{3}}{(0.9, 0.2 ,-0.5 , -0.6 )} \end{array}\right\}(0.6 ,0.5 ,-0.7 , -0.3 ) \right). \end{equation*}

Definition 8. Let \(X\) be a non-empty set of the universe and \(E\) be a set of parameter. Suppose that \(\mathscr{M}^{\mathscr{GB}}_{p}\) and \(\mathscr{N}^{\mathscr{GB}}_{q}\) are two Type-II GPBFSS's on \((X, E)\). Now \(\mathscr{M}^{\mathscr{GB}}_{p} \subseteq \mathscr{N}^{\mathscr{GB}}_{q}\) if and only if

• (i) \( \mathscr{M}(e)(x) \subseteq \mathscr{N}(e)(x)\) implies \begin{equation*} \left\{\begin{array}{1r} \zeta^{+}_{\mathscr{M}(e)}(x) \leq \zeta^{+}_{\mathscr{N}(e)}(x),\,\,\,\, \xi^{+}_{\mathscr{M}(e)}(x) \geq \xi^{+}_{\mathscr{N}(e)}(x)\\ \zeta^{-}_{\mathscr{M}(e)}(x) \geq \zeta^{-}_{\mathscr{N}(e)}(x),\,\,\,\, \xi^{-}_{\mathscr{M}(e)}(x) \leq \xi^{-}_{\mathscr{N}(e)}(x) \end{array}\right\} , \end{equation*}
• (ii) \(p(e) \subseteq q(e)\) implies \begin{equation*} \left\{\begin{array}{1r} \zeta^{+}_{p(e)} \leq \zeta^{+}_{q(e)}, \,\,\,\, \xi^{+}_{p(e)} \geq \xi^{+}_{q(e)}\\ \zeta^{-}_{p(e)} \geq \zeta^{-}_{q(e)}, \,\,\,\, \xi^{-}_{p(e)} \leq \xi^{-}_{q(e)} \end{array}\right\} \end{equation*} \(\forall e \in E\).

Example 2. Consider the Type-II GPBFSS \(\mathscr{M}^{\mathscr{GB}}_{p}\) over \((X, E)\) in Example 1. Let \(\mathscr{N}^{\mathscr{GB}}_{q}\) be another Type-II GPBFSS over \((X, E)\) defined as: \begin{equation*} \mathscr{N}^{\mathscr{GB}}_{q}(e_{1})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.7, 0.5 ,-0.6-0.7 )}\\ \frac{x_{2}}{(0.9,0.2,-0.8-0.4)} \\ \frac{x_{3}}{(0.9, 0.1 ,-0.5-0.8)} \end{array}\right\} (0.6, 0.4 ,-0.9-0.2 ) \right); \end{equation*} \begin{equation*} \mathscr{N}^{\mathscr{GB}}_{q}(e_{2})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.8, 0.3 ,-0.4-0.6 )}\\ \frac{x_{2}}{(0.6,0.7,-0.8 -0.3 )} \\ \frac{x_{3}}{(0.7,0.4 ,-0.3 -0.7)} \end{array}\right\}(0.9, 0.1 ,-0.8-0.3 ) \right); \end{equation*} \begin{equation*} \mathscr{N}^{\mathscr{GB}}_{q}(e_{3})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.5, 0.6 ,-0.9-0.3 )} \\ \frac{x_{2}}{(0.8,0.3,-0.8-0.2 )}\\ \frac{x_{3}}{(0.9, 0.1 ,-0.7-0.5 )} \end{array}\right\}(0.7, 0.4 ,-0.9-0.1 ) \right). \end{equation*}

Definition 9. Let \(X\) be a non-empty set of the universe and \(E\) be a set of parameter. Let \(\mathscr{M}^{\mathscr{GB}}_{p}\) be a Type-II GPBFSS on \((X, E)\). The complement of \(\mathscr{M}^{\mathscr{GB}}_{p}\) is denoted by \(\mathscr{M}^{\mathscr{GB}}_{{p}^{c}}\) and is defined by \[\mathscr{M}^{\mathscr{GB}}_{{p}^{c}} = \Big\langle \mathscr{GBM}^{c}(e)(x), {p}^{c}(e)\Big\rangle,\] where \(\mathscr{GBM}^{c}(e)(x) = \left(\xi^{+}_{\mathscr{M}(e)}(x),\zeta^{+}_{\mathscr{M}(e)}(x), \xi^{-}_{\mathscr{M}(e)}(x),\zeta^{-}_{\mathscr{M}(e)}(x)\right)\) and \(p^{c}(e)= \left(\xi^{+}_{p(e)},\zeta^{+}_{p(e)}, \xi^{-}_{p(e)},\zeta^{-}_{p(e)}\right)\). It is true that \(\mathscr{M}^{\mathscr{GB}}_{({p}^{c})^{c}}= \mathscr{M}^{\mathscr{GB}}_{p}\).

Definition 10. Let \(X\) be a non-empty set of the universe and \(E\) be a set of parameter. Let \(\mathscr{M}^{\mathscr{GB}}_{p}\) and \(\mathscr{N}^{\mathscr{GB}}_{q}\) be two Type-II GPBFSSs on \((X, E)\). The union and intersection of \(\mathscr{M}^{\mathscr{GB}}_{p}\) and \(\mathscr{N}^{\mathscr{GB}}_{q}\) over \((X, E)\) are denoted by \(\mathscr{M}^{\mathscr{GB}}_{p}\cup \mathscr{N}^{\mathscr{GB}}_{q}\) and \(\mathscr{M}^{\mathscr{GB}}_{p}\cap \mathscr{N}^{\mathscr{GB}}_{q}\) respectively and are defined by \[V_{v} : E \rightarrow P\mathscr{GBM}^{X} \times I\;\text{ and }\; W_{w} : E \rightarrow P\mathscr{GBM}^{X} \times I\] such that \[V_{v}(e)(x)= (V(e)(x), v(e))\;\text{ and }\;W_{w}(e)(x)= (W(e)(x), w(e)),\] where \(V(e)(x)= \mathscr{M}(e)(x)\cup \mathscr{N}(e)(x)\), \(v(e)= p(e)\cup q(e)\), \(W(e)(x)= \mathscr{M}(e)(x)\cap \mathscr{N}(e)(x)\) and \(w(e)= p(e)\cap q(e)\), for all \(x\in X\).

Example 3. Let \(\mathscr{M}^{\mathscr{GB}}_{p}\) and \(\mathscr{N}^{\mathscr{GB}}_{q}\) be the two Type-II GPBFSS's on \((X, E)\). By the Example 1 in \(\mathscr{M}^{\mathscr{GB}}_{p}\) and \(\mathscr{N}^{\mathscr{GB}}_{q}\) is defined as, \begin{equation*} \mathscr{N}^{\mathscr{GB}}_{q}(e_{1})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.3, 0.4 ,-0.2,-0.3 )}\\ \frac{x_{2}}{(0.4,0.5,-0.6, -0.2 )}\\ \frac{x_{3}}{(0.6, 0.2 ,-0.1,-0.4)} \end{array}\right\} (0.5, 0.4 ,-0.3,-0.1 ) \right); \end{equation*} \begin{equation*} \mathscr{N}^{\mathscr{GB}}_{q}(e_{2})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.8,0.7,-0.4,-0.3 )}\\ \frac{x_{2}}{(0.6,0.4,-0.3,-0.8 )}\\ \frac{x_{3}}{(0.5,0.3 ,-0.5, -0.4)} \end{array}\right\}(0.2, 0.1 ,-0.3,-0.5 ) \right); \end{equation*} \begin{equation*} \mathscr{N}^{\mathscr{GB}}_{q}(e_{3})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.6, 0.4,-0.4,-0.1 )}\\ \frac{x_{2}}{(0.7,0.9,-0.6, -0.4 )}\\ \frac{x_{3}}{(0.2, 0.6,-0.3,-0.2 )} \end{array}\right\} (0.5, 0.6,-0.3,-0.4 ) \right). \end{equation*} Now, \(\mathscr{M}^{\mathscr{GB}}_{p} \cup \mathscr{N}^{\mathscr{GB}}_{q}\) can be written as: \begin{equation*} \mathscr{M}^{\mathscr{GB}}_{p} \cup \mathscr{N}^{\mathscr{GB}}_{q}(e_{1})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.6 ,0.4 ,-0.3 , -0.3 )}\\ \frac{x_{2}}{(0.9 , 0.4,-0.7 , -0.2 )}\\ \frac{x_{3}}{(0.8 ,0.2 ,-0.2 ,-0.4)} \end{array}\right\} (0.6 ,0.4 ,-0.8 , -0.1 ) \right); \end{equation*} \begin{equation*} \mathscr{M}^{\mathscr{GB}}_{p} \cup \mathscr{N}^{\mathscr{GB}}_{q}(e_{2})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.8 ,0.4 ,-0.4 , -0.3 )}\\ \frac{x_{2}}{(0.6 , 0.4,-0.4 , -0.4 )}\\ \frac{x_{3}}{(0.5 , 0.3 ,-0.5 ,-0.4)} \end{array}\right\} (0.9 ,0.1 ,-0.7 , -0.4 ) \right); \end{equation*} \begin{equation*} \mathscr{M}^{\mathscr{GB}}_{p} \cup \mathscr{N}^{\mathscr{GB}}_{q}(e_{3})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.6 ,0.4 ,-0.8 , -0.1 )}\\ \frac{x_{2}}{(0.8 , 0.4,-0.7 , -0.3 )}\\ \frac{x_{3}}{(0.9 , 0.2 ,-0.5 , -0.2 )} \end{array}\right\} (0.6 ,0.5 ,-0.7 , -0.3 ) \right). \end{equation*} Now, \(\mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{N}^{\mathscr{GB}}_{q}\) can be written as: \begin{equation*} \mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{N}^{\mathscr{GB}}_{q}(e_{1})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.3 ,0.7 ,-0.2 , -0.8 )}\\ \frac{x_{2}}{(0.4 , 0.5,-0.6 , -0.5 )}\\ \frac{x_{3}}{(0.6 ,0.5 ,-0.1 ,-0.9)} \end{array}\right\} (0.5 ,0.5 ,-0.3 , -0.3 ) \right); \end{equation*} \begin{equation*} \mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{N}^{\mathscr{GB}}_{q}(e_{2})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.7 ,0.7 ,-0.2 , -0.8 )}\\ \frac{x_{2}}{(0.3 , 0.9,-0.3 , -0.8 )}\\ \frac{x_{3}}{(0.5 , 0.6 ,-0.2 ,-0.9)} \end{array}\right\}(0.2 ,0.2 ,-0.3 , -0.5 ) \right); \end{equation*} \begin{equation*} \mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{N}^{\mathscr{GB}}_{q}(e_{3})= \left(\left\{\begin{array}{1r} \frac{x_{1}}{(0.3 ,0.7 ,-0.4 , -0.4 )}\\ \frac{x_{2}}{(0.7 , 0.9,-0.6 , -0.4 )}\\ \frac{x_{3}}{(0.2 , 0.6 ,-0.3 , -0.6 )} \end{array}\right\} (0.5 ,0.6 ,-0.3 , -0.4 ) \right). \end{equation*}

Definition 11. A Type-II GPBFSS \(\rho^{\mathscr{GB}}_{\epsilon}(e)(x) = \Big\langle\rho(e)(x), \epsilon(e)\Big\rangle\) is said to a null Type-II GPBFSS \(\rho^{\mathscr{GB}}_{\epsilon}: E \rightarrow P\mathscr{GBM}^{X} \times I\), where \( \rho^{+}(e)(x) = (0,1)\), \(\epsilon^{+}(e) = (0, 1)\) and \( \rho^{-}(e)(x) = (-1,0)\) and \(\epsilon^{-}(e) = (-1, 0), \,\, \forall \, x \in X\).

Definition 12. A Type-II GPBFSS \(\pi^{\mathscr{GB}}_{\sigma}(e)(x) = \Big\langle\pi(e)(x),\sigma(e)\Big\rangle\) is said to a absolute Type-II GPBFSS \(\pi^{\mathscr{GB}}_{\sigma}: E \rightarrow P\mathscr{GBM}^{X} \times I\), where \( \pi^{+}(e)(x) = (1, 0)\), \(\sigma^{+}(e) = (1, 0)\) and \( \pi^{-}(e)(x) = (0, -1)\) and \(\sigma^{-}(e)= (0, -1)\), \(\forall \, x \in X\).

Theorem 13. Let \(\mathscr{M}^{\mathscr{GB}}_{p}\) be a Type-II GPBFSS on \((X, E)\). Then the following properties are holds:

1. \(\mathscr{M}^{\mathscr{GB}}_{p} = \mathscr{M}^{\mathscr{GB}}_{p} \cup \mathscr{M}^{\mathscr{GB}}_{p}\) and \(\mathscr{M}^{\mathscr{GB}}_{p} = \mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{M}^{\mathscr{GB}}_{p}\),
2. \(\mathscr{M}^{\mathscr{GB}}_{p} \subseteq \mathscr{M}^{\mathscr{GB}}_{p} \cup \mathscr{M}^{\mathscr{GB}}_{p}\) and \( \mathscr{M}^{\mathscr{GB}}_{p} \subseteq \mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{M}^{\mathscr{GB}}_{p}\),
3. \(\mathscr{M}^{\mathscr{GB}}_{p} \cup \rho^{\mathscr{GB}}_{\epsilon} = \mathscr{M}^{\mathscr{GB}}_{p}\) and \(\mathscr{M}^{\mathscr{GB}}_{p} \cap \rho^{\mathscr{GB}}_{\epsilon} = \rho^{\mathscr{GB}}_{\epsilon} \),
4. \(\mathscr{M}^{\mathscr{GB}}_{p} \cup \pi^{\mathscr{GB}}_{\sigma} = \pi^{\mathscr{GB}}_{\sigma}\) and \(\mathscr{M}^{\mathscr{GB}}_{p} \cap \pi^{\mathscr{GB}}_{\sigma} = \mathscr{M}^{\mathscr{GB}}_{p}\).

Remark 1. Let \(\mathscr{M}^{\mathscr{GB}}_{p}\) be a Type-II GPBFSS on \((X, E)\). If \(\mathscr{M}^{\mathscr{GB}}_{p} \neq \pi^{\mathscr{GB}}_{\sigma} \) or \(\mathscr{M}^{\mathscr{GB}}_{p} \neq \rho^{\mathscr{GB}}_{\epsilon}\), then \(\mathscr{M}^{\mathscr{GB}}_{p} \cup \mathscr{M}^{\mathscr{GB}}_{{p}^{c}} \neq \pi^{\mathscr{GB}}_{\sigma}\) and \(\mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{M}^{\mathscr{GB}}_{{p}^{c}} \neq \rho^{\mathscr{GB}}_{\epsilon}\).

Theorem 14. Let \(\mathscr{M}^{\mathscr{GB}}_{p}\), \(\mathscr{N}^{\mathscr{GB}}_{q}\) and \(\mathscr{O}^{\mathscr{GB}}_{r}\) are three Type-II GPBFSS's over \((X, E)\), then the following properties hold:

1. \(\mathscr{M}^{\mathscr{GB}}_{p} \cup \mathscr{N}^{\mathscr{GB}}_{q} = \mathscr{N}^{\mathscr{GB}}_{q} \cup \mathscr{M}^{\mathscr{GB}}_{p}\),
2. \(\mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{N}^{\mathscr{GB}}_{q} = \mathscr{N}^{\mathscr{GB}}_{q} \cap \mathscr{M}^{\mathscr{GB}}_{p}\),
3. \(\mathscr{M}^{\mathscr{GB}}_{p} \cup (\mathscr{N}^{\mathscr{GB}}_{q} \cup \mathscr{O}^{\mathscr{GB}}_{r}) = (\mathscr{M}^{\mathscr{GB}}_{p} \cup \mathscr{N}^{\mathscr{GB}}_{q}) \cup \mathscr{O}^{\mathscr{GB}}_{r}\),
4. \(\mathscr{M}^{\mathscr{GB}}_{p} \cap (\mathscr{N}^{\mathscr{GB}}_{q} \cap \mathscr{O}^{\mathscr{GB}}_{r}) = (\mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{N}^{\mathscr{GB}}_{q}) \cap \mathscr{O}^{\mathscr{GB}}_{r}\),
5. \((\mathscr{M}^{\mathscr{GB}}_{p} \cup \mathscr{N}^{\mathscr{GB}}_{q})^{c} = \mathscr{M}^{\mathscr{GB}}_{{p}^{c}} \cap \mathscr{N}^{\mathscr{GB}}_{{q}^{c}}\),
6. \((\mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{N}^{\mathscr{GB}}_{q})^{c} = \mathscr{M}^{\mathscr{GB}}_{{p}^{c}} \cup \mathscr{N}^{\mathscr{GB}}_{{q}^{c}}\),
7. \((\mathscr{M}^{\mathscr{GB}}_{p} \cup \mathscr{N}^{\mathscr{GB}}_{q}) \cap \mathscr{M}^{\mathscr{GB}}_{p} = \mathscr{M}^{\mathscr{GB}}_{p}\),
8. \((\mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{N}^{\mathscr{GB}}_{q}) \cup \mathscr{M}^{\mathscr{GB}}_{p} = \mathscr{M}^{\mathscr{GB}}_{p}\),
9. \(\mathscr{M}^{\mathscr{GB}}_{p} \cup (\mathscr{N}^{\mathscr{GB}}_{q} \cap \mathscr{O}^{\mathscr{GB}}_{r}) = (\mathscr{M}^{\mathscr{GB}}_{p} \cup \mathscr{N}^{\mathscr{GB}}_{q}) \cap (\mathscr{M}^{\mathscr{GB}}_{p} \cup \mathscr{O}^{\mathscr{GB}}_{r})\),
10. \(\mathscr{M}^{\mathscr{GB}}_{p} \cap (\mathscr{N}^{\mathscr{GB}}_{q} \cup \mathscr{O}^{\mathscr{GB}}_{r}) = (\mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{N}^{\mathscr{GB}}_{q}) \cup (\mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{O}^{\mathscr{GB}}_{r})\).

Proof. The proof follows from Definition 9 and 10.

Definition 15. Let \((\mathscr{M}^{\mathscr{GB}}_{p},A)\) and \((\mathscr{N}^{\mathscr{GB}}_{q},B)\) be two Type-II GPBFSS's on \((X, E)\). Then the operations AND is denoted by \((\mathscr{M}^{\mathscr{GB}}_{p},A) \wedge (\mathscr{N}^{\mathscr{GB}}_{q},B)\) and is defined by \[(\mathscr{M}^{\mathscr{GB}}_{p},A) \wedge (\mathscr{N}^{\mathscr{GB}}_{q},B) = (\mathscr{O}^{\mathscr{GB}}_{r}, A \times B),\] where \(\mathscr{O}^{\mathscr{GB}}_{r} (\gamma, \delta) = \left(\mathscr{O}(\gamma, \delta)(x), r(\gamma, \delta)\right)\) such that \[\mathscr{O}(\gamma, \delta) = \mathscr{M}(\gamma) \cap \mathscr{N}(\delta)\;\text{ and }\;r(\gamma, \delta) = p(\gamma) \cap q(\delta),\] for all \((\gamma, \delta)\in A \times B \).

Definition 16. Let \((\mathscr{M}^{\mathscr{GB}}_{p},A)\) and \((\mathscr{N}^{\mathscr{GB}}_{q},B)\) be two Type-II GPBFSS's on \((X, E)\), then the operations OR is denoted by \((\mathscr{M}^{\mathscr{GB}}_{p},A) \vee (\mathscr{N}^{\mathscr{GB}}_{q},B)\) and is defined by \[(\mathscr{M}^{\mathscr{GB}}_{p},A) \vee (\mathscr{N}^{\mathscr{GB}}_{q},B) = (\mathscr{O}^{\mathscr{GB}}_{r}, A \times B),\] where \(\mathscr{O}^{\mathscr{GB}}_{r} (\gamma, \delta) = \left(\mathscr{O}(\gamma, \delta)(x), r(\gamma, \delta)\right)\) such that \[\mathscr{O}(\gamma, \delta) = \mathscr{M}(\gamma) \cup \mathscr{N}(\delta)\;\text{ and }\;r(\gamma, \delta) = p(\gamma) \cup q(\delta),\] for all \((\gamma, \delta)\in A \times B \).

Theorem 17. Let \((\mathscr{M}^{\mathscr{GB}}_{p},A)\) and \((\mathscr{N}^{\mathscr{GB}}_{q},B)\) be two Type-II GPBFSS's on \((X, E)\), then

1. \(((\mathscr{M}^{\mathscr{GB}}_{p},A) \wedge (\mathscr{N}^{\mathscr{GB}}_{q},B))^{c} = (\mathscr{M}^{\mathscr{GB}}_{p^c},A) \vee (\mathscr{N}^{\mathscr{GB}}_{q^c},B)\),
2. \(((\mathscr{M}^{\mathscr{GB}}_{p},A) \vee (\mathscr{N}^{\mathscr{GB}}_{q},B))^{c} = (\mathscr{M}^{\mathscr{GB}}_{p^c},A) \wedge (\mathscr{N}^{\mathscr{GB}}_{q^c},B)\).

Proof.

1. Suppose that \[(\mathscr{M}^{\mathscr{GB}}_{p},A) \wedge (\mathscr{N}^{\mathscr{GB}}_{q},B) = (\mathscr{O}^{\mathscr{GB}}_{r}, A \times B).\] Now, \[\mathscr{O}^{\mathscr{GB}}_{{r}^{c}}(\gamma, \delta)= ( \mathscr{O}^{c}(\gamma, \delta)(x), r^{c}(\gamma, \delta)),\] for all \((\gamma, \delta) \in A \times B\). By Theorem 14 and Definition 15, \[\mathscr{O}^{c}(\gamma, \delta)= (\mathscr{M}(\gamma) \cap \mathscr{N}(\delta))^{c}= \mathscr{M}^{c}(\gamma) \cup \mathscr{N}^{c}(\delta)\] and \[r^{c}(\gamma, \delta)= (p(\gamma) \cap q(\delta))^{c}= p^{c}(\gamma) \cup q^{c}(\delta).\] On the other hand, given that \[(\mathscr{M}^{\mathscr{GB}}_{p^c},A) \vee (\mathscr{N}^{\mathscr{GB}}_{q^c},B)= (\sigma_{o}, A \times B),\] where \(\sigma_{o}(\gamma, \delta) = (\sigma(\gamma, \delta)(x), o(\gamma, \delta))\) such that \[\sigma(\gamma, \delta)= \mathscr{M}^{c}(\gamma) \cup \mathscr{N}^{c}(\delta)\] and \[o(\gamma, \delta)= p^{c}(\gamma) \cup q^{c}(\delta)\] for all \((\gamma, \delta) \in A \times B\). Thus, \[\mathscr{O}^{c}_{r}=\sigma_{o}.\] Hence \[((\mathscr{M}^{\mathscr{GB}}_{p},A) \wedge (\mathscr{N}^{\mathscr{GB}}_{q},B))^{c} = (\mathscr{M}^{\mathscr{GB}}_{p^c},A) \vee (\mathscr{N}^{\mathscr{GB}}_{q^c},B).\]
2. Suppose that \[(\mathscr{M}^{\mathscr{GB}}_{p},A) \vee (\mathscr{N}^{\mathscr{GB}}_{q},B) = (\mathscr{O}^{\mathscr{GB}}_{r}, A \times B).\] Now, \[\mathscr{O}^{\mathscr{GB}}_{{r}^{c}}(\gamma, \delta)= ( \mathscr{O}^{c}(\gamma, \delta)(x), r^{c}(\gamma, \delta)),\] for all \((\gamma, \delta) \in A \times B\). By Theorem 14 and Definition 16, \[\mathscr{O}^{c}(\gamma, \delta)= (\mathscr{M}(\gamma) \cup \mathscr{N}(\delta))^{c}= \mathscr{M}^{c}(\gamma) \cap \mathscr{N}^{c}(\delta)\] and \[r^{c}(\gamma, \delta)= (p(\gamma) \cup q(\delta))^{c}= p^{c}(\gamma) \cap q^{c}(\delta).\] On the other hand, given that \[(\mathscr{M}^{\mathscr{GB}}_{p^c},A) \wedge (\mathscr{N}^{\mathscr{GB}}_{q^c},B)= (\sigma_{o}, A \times B),\] where \(\sigma_{o}(\gamma, \delta) = (\sigma(\gamma, \delta)(x), o(\gamma, \delta))\) such that \[\sigma(\gamma, \delta)= \mathscr{M}^{c}(\gamma) \cap \mathscr{N}^{c}(\delta)\] and \[o(\gamma, \delta)= p^{c}(\gamma) \cap q^{c}(\delta)\] for all \((\gamma, \delta) \in A \times B\). Thus, \[\mathscr{O}^{c}_{r}=\sigma_{o}.\] Hence \[((\mathscr{M}^{\mathscr{GB}}_{p},A) \vee (\mathscr{N}^{\mathscr{GB}}_{q},B))^{c} = (\mathscr{M}^{\mathscr{GB}}_{p^c},A) \wedge (\mathscr{N}^{\mathscr{GB}}_{q^c},B).\]

4. Similarity measure between two PPBFSS's

Definition 18. Let \(X\) be a non-empty set of the universe and \(E\) be a set of parameter. Suppose that \(\mathscr{M}^{\mathscr{GB}}_{p}\) and \(\mathscr{N}^{\mathscr{GB}}_{q}\) are two Type-II GPBFSS's on \((X, E)\). The similarity measure between two Type-II GPBFSS's \(\mathscr{M}^{\mathscr{GB}}_{p}\) and \(\mathscr{N}^{\mathscr{GB}}_{q}\) is denoted by \(Sim(\mathscr{M}^{\mathscr{GB}}_{p}, \mathscr{N}^{\mathscr{GB}}_{q})\) and is defined as \[Sim(\mathscr{M}^{\mathscr{GB}}_{p}, \mathscr{N}^{\mathscr{GB}}_{q})= \Big[\Delta^{\mathscr{GB}}(\mathscr{M}, \mathscr{N}) \cdot \Upsilon^{\mathscr{GB}}(p,q)\Big]\] such that \[\Delta^{\mathscr{GB}}(\mathscr{M}, \mathscr{N})= \frac{\mathbb{T}^{\mathscr{GB}}(\mathscr{M}(e)(x), \mathscr{N}(e)(x)) \,\, + \,\, \mathbb{S}^{\mathscr{GB}}(\mathscr{M}(e)(x), \mathscr{N}(e)(x))}{2}\] and \[\Upsilon^{\mathscr{GB}}(p, q)= 1- \frac{\sum|\alpha_{1}-\alpha_{2}|}{{\sum|\alpha_{1}+\alpha_{2}|}},\] where \(\mathbb{T}^{\mathscr{GB}}(\mathscr{M}(e)(x), \mathscr{N}(e)(x))\) can be written as \[\frac{1}{m} \sum^{m}_{j=1} \frac{\sum^{n}_{i=1} \Big[\big[\zeta^{+}_{\mathscr{M}(e_{i})}(x_{j}) \,\, \cdot\,\, \zeta^{+}_{\mathscr{N}(e_{i})}(x_{j})\big] + \big[\zeta^{-}_{\mathscr{M}(e_{i})}(x_{j}) \,\, \cdot\,\, \zeta^{-}_{\mathscr{N}(e_{i})}(x_{j})\big]\Big]} {\sum^{n}_{i=1} \Bigg[\Big[\,1- \sqrt{\big[(1-\zeta^{2+}_{\mathscr{M}(e_{i})}(x_{j})) \,\, \cdot\,\, (1-\zeta^{2+}_{\mathscr{N}(e_{i})}(x_{j}))\big]}\,\Big] + \Big[\,1- \sqrt{\big[(1-\zeta^{2-}_{\mathscr{M}(e_{i})}(x_{j})) \,\, \cdot\,\, (1-\zeta^{2-}_{\mathscr{N}(e_{i})}(x_{j}))\big]}\,\Big]\Bigg]}\] and \(\mathbb{S}^{\mathscr{GB}}(\mathscr{M}(e)(x), \mathscr{N}(e)(x))\) can be written as \[\frac{1}{m} \sum^{m}_{j=1} \sqrt{1- \frac{\sum^{n}_{i=1} \Big[\big|\xi^{2+}_{\mathscr{M}(e_{i})}(x_{j}) \,\, - \,\, \xi^{2+}_{\mathscr{N}(e_{i})}(x_{j})\big | + \big|\xi^{2-}_{\mathscr{M}(e_{i})}(x_{j}) \,\, - \,\, \xi^{2-}_{\mathscr{N}(e_{i})}(x_{j})\big|\Big]} {\sum^{n}_{i=1} \Bigg[\Big[1+\big[(\xi^{2+}_{\mathscr{M}(e_{i})}(x_{j})) \,\, \cdot\,\, (\xi^{2+}_{\mathscr{N}(e_{i})}(x_{j}))\,\big]\Big] + \Big[1+\big[(\xi^{2-}_{\mathscr{M}(e_{i})}(x_{j})) \,\, \cdot\,\, (\xi^{2-}_{\mathscr{N}(e_{i})}(x_{j}))\,\big]\Big]\Bigg]}},\] \[\alpha_{1}= \frac{\zeta^{2+}_{p(e_{i})} + \zeta^{2-}_{p(e_{i})}}{\big[\zeta^{2+}_{p(e_{i})} \,\, + \,\, \xi^{2+}_{p(e_{i})}\big] + \big[\zeta^{2-}_{p(e_{i})} \,\, + \,\, \xi^{2-}_{p(e_{i})}\big]}\] and \[\alpha_{2}= \frac{\zeta^{2+}_{q(e_{i})} + \zeta^{2-}_{q(e_{i})}}{\big[\zeta^{2+}_{q(e_{i})} \,\, + \,\, \xi^{2+}_{q(e_{i})}\big] + \big[\zeta^{2-}_{q(e_{i})} \,\, + \,\, \xi^{2-}_{q(e_{i})}\big]}.\]

Theorem 19. Let \(\mathscr{M}^{\mathscr{GB}}_{p},\,\, \mathscr{N}^{\mathscr{GB}}_{q}\) and \(\mathscr{O}^{\mathscr{GB}}_{r}\) be the any three Type-II GPBFSS's over \((X, E)\). Then the following statements hold:

1. \(Sim(\mathscr{M}^{\mathscr{GB}}_{p}, \mathscr{N}^{\mathscr{GB}}_{q})\) = \(Sim(\mathscr{N}^{\mathscr{GB}}_{q}, \mathscr{M}^{\mathscr{GB}}_{p})\),
2. \(0 \leq Sim(\mathscr{M}^{\mathscr{GB}}_{p}, \mathscr{N}^{\mathscr{GB}}_{q}) \leq 1 \),
3. \(\mathscr{M}^{\mathscr{GB}}_{p} = \mathscr{N}^{\mathscr{GB}}_{q} \implies Sim(\mathscr{M}^{\mathscr{GB}}_{p}, \mathscr{N}^{\mathscr{GB}}_{q})=1\),
4. \(\mathscr{M}^{\mathscr{GB}}_{p} \subseteq \mathscr{N}^{\mathscr{GB}}_{q} \subseteq \mathscr{O}^{\mathscr{GB}}_{r} \implies Sim(\mathscr{M}^{\mathscr{GB}}_{p}, \mathscr{O}^{\mathscr{GB}}_{r}) \leq Sim(\mathscr{N}^{\mathscr{GB}}_{q}, \mathscr{O}^{\mathscr{GB}}_{r})\),
5. \(\mathscr{M}^{\mathscr{GB}}_{p} \cap \mathscr{N}^{\mathscr{GB}}_{q} = \{\phi\} \Leftrightarrow Sim(\mathscr{M}^{\mathscr{GB}}_{p}, \mathscr{N}^{\mathscr{GB}}_{q}) = 0\).

Proof.

The statements (1), (2) and (5) are trivial.

(3) Given that \(\mathscr{M}^{\mathscr{GB}}_{p}= \mathscr{N}^{\mathscr{GB}}_{q}\). Now, \begin{align*} \mathbb{T}^{\mathscr{GB}}(\mathscr{M}(e)(x_{j}),\mathscr{N}(e)(x_{j})) %&=& \frac{\sum^{n}_{i=1} \big[\zeta^{2+}_{\mathscr{M}(e_{i})}(x_{j}) + \zeta^{2-}_{\mathscr{M}(e_{i})}(x_{j})\big] } %{\sum^{n}_{i=1} \Big[\big[1-1+ \zeta^{2+}_{\mathscr{M}(e_{i})}(x_{j})\big] + \big[1-1+ \zeta^{2-}_{\mathscr{M}(e_{i})}(x_{j})\big] \Big]}\\ &= \frac{\sum^{n}_{i=1} \big[\zeta^{2+}_{\mathscr{M}(e_{i})}(x_{j}) + \zeta^{2-}_{\mathscr{M}(e_{i})}(x_{j})\big] } {\sum^{n}_{i=1} \big[\zeta^{2+}_{\mathscr{M}(e_{i})}(x_{j}) + \zeta^{2-}_{\mathscr{M}(e_{i})}(x_{j}) \big]}= 1. \end{align*} Hence, \(\mathbb{T}^{\mathscr{GB}}(\mathscr{M}(e)(x),\mathscr{N}(e)(x))=\frac{1}{m}[1+1+...+1 \text{(m times)}] =1\). Now, \(\mathbb{S}^{\mathscr{GB}}(\mathscr{M}(e)(x_{j}),\mathscr{N}(e)(x_{j}))=\sqrt{(1-0)}= 1 \). Hence, \(\mathbb{S}^{\mathscr{GB}}(\mathscr{M}(e)(x),\mathscr{N}(e)(x))=\frac{1}{m}[1+1+...+1 \text{(m times)}] =1\). Thus, \(\Delta^{\mathscr{GB}}(\mathscr{M}, \mathscr{N})= \frac{1+1}{2}= 1\) and \(\Upsilon^{\mathscr{GB}}(p, q) = 1\). Hence \(Sim(\mathscr{M}^{\mathscr{GB}}_{p}, \mathscr{N}^{\mathscr{GB}}_{q})=1\). This proves (3).

(4) Given that

\[\mathscr{M}^{\mathscr{GB}}_{p} \subseteq \mathscr{N}^{\mathscr{GB}}_{q} \implies \left\{\begin{array}{1r} \zeta^{+}_{\mathscr{M}(e)}(x_{j}) \leq \zeta^{+}_{\mathscr{N}(e)}(x_{j}),\,\, \xi^{+}_{\mathscr{M}(e)}(x_{j}) \geq \xi^{+}_{\mathscr{N}(e)}(x_{j}),\,\,\\ \zeta^{-}_{\mathscr{M}(e)}(x_{j}) \geq \zeta^{-}_{\mathscr{N}(e)}(x_{j}),\,\, \xi^{-}_{\mathscr{M}(e)}(x_{j}) \leq \xi^{-}_{\mathscr{N}(e)}(x_{j}),\\ \zeta^{+}_{p(e)}(x_{j}) \leq \zeta^{+}_{q(e)}(x_{j}), \,\, \xi^{+}_{p(e)}(x_{j}) \geq \xi^{+}_{q(e)}(x_{j}), \,\,\\ \zeta^{-}_{p(e)}(x_{j}) \geq \zeta^{-}_{q(e)}(x_{j}), \,\, \xi^{-}_{p(e)}(x_{j}) \leq \xi^{-}_{q(e)}(x_{j}),\\ \end{array}\right\} \] \[ \mathscr{M}^{\mathscr{GB}}_{p} \subseteq \mathscr{O}^{\mathscr{GB}}_{r} \implies \left\{\begin{array}{1r} \zeta^{+}_{\mathscr{M}(e)}(x_{j}) \leq \zeta^{+}_{\mathscr{O}(e)}(x_{j}),\,\, \xi^{+}_{\mathscr{M}(e)}(x_{j}) \geq \xi^{+}_{\mathscr{O}(e)}(x_{j}),\,\,\\ \zeta^{-}_{\mathscr{M}(e)}(x_{j}) \geq \zeta^{-}_{\mathscr{O}(e)}(x_{j}),\,\, \xi^{-}_{\mathscr{M}(e)}(x_{j}) \leq \xi^{-}_{\mathscr{O}(e)}(x_{j}), \\ \zeta^{+}_{p(e)}(x_{j}) \leq \zeta^{+}_{r(e)}(x_{j}), \,\, \xi^{+}_{p(e)}(x_{j}) \geq \xi^{+}_{r(e)}(x_{j}), \,\,\\ \zeta^{-}_{p(e)}(x_{j}) \geq \zeta^{-}_{r(e)}(x_{j}), \,\, \xi^{-}_{p(e)}(x_{j}) \leq \xi^{-}_{r(e)}(x_{j}),\\ \end{array}\right\} \] \[ \mathscr{N}^{\mathscr{GB}}_{q} \subseteq \mathscr{O}^{\mathscr{GB}}_{r} \implies \left\{\begin{array}{1r} \zeta^{+}_{\mathscr{N}(e)}(x_{j}) \leq \zeta^{+}_{\mathscr{O}(e)}(x_{j}),\,\, \xi^{+}_{\mathscr{N}(e)}(x_{j}) \geq \xi^{+}_{\mathscr{O}(e)}(x_{j}),\,\,\\ \zeta^{-}_{\mathscr{N}(e)}(x_{j}) \geq \zeta^{-}_{\mathscr{O}(e)}(x_{j}),\,\, \xi^{-}_{\mathscr{N}(e)}(x_{j}) \leq \xi^{-}_{\mathscr{O}(e)}(x_{j}), \\ \zeta^{+}_{q(e)}(x_{j}) \leq \zeta^{+}_{r(e)}(x_{j}), \,\, \xi^{+}_{q(e)}(x_{j}) \geq \xi^{+}_{r(e)}(x_{j}), \,\,\\ \zeta^{-}_{q(e)}(x_{j}) \geq \zeta^{-}_{r(e)}(x_{j}), \,\, \xi^{-}_{q(e)}(x_{j}) \leq \xi^{-}_{r(e)}(x_{j}),\\ \end{array}\right\} \] Clearly, \(\zeta^{+}_{\mathscr{M}(e)}(x_{j}) \cdot \zeta^{+}_{\mathscr{O}(e)}(x_{j}) \leq \zeta^{+}_{\mathscr{N}(e)}(x_{j}) \cdot\zeta^{+}_{\mathscr{O}(e)}(x_{j})\;\text{ and }\;\zeta^{-}_{\mathscr{M}(e)}(x_{j}) \cdot \zeta^{-}_{\mathscr{O}(e)}(x_{j}) \leq \zeta^{-}_{\mathscr{N}(e)}(x_{j}) \cdot\zeta^{-}_{\mathscr{O}(e)}(x_{j}) \) implies that Clearly, \((\zeta^{+}_{\mathscr{M}(e)}(x_{j}))^{2} \leq (\zeta^{+}_{\mathscr{N}(e)}(x_{j}))^{2}\) and \((\zeta^{-}_{\mathscr{M}(e)}(x_{j}))^{2} \leq (\zeta^{-}_{\mathscr{N}(e)}(x_{j}))^{2}\) implies that \[\Big[(1-(\zeta^{+}_{\mathscr{M}(e)}(x_{j}))^{2}) \cdot (1-(\zeta^{+}_{\mathscr{O}(e)}(x_{j}))^{2})\Big] \geq \Big[(1-(\zeta^{+}_{\mathscr{N}(e)}(x_{j}))^{2}) \cdot (1-(\zeta^{+}_{\mathscr{O}(e)}(x_{j}))^{2})\Big]\] and Similarly, Adding Eqs (2) and (3) we get, \begin{align*} &\Big[1-\sqrt{\big[(1-(\zeta^{+}_{\mathscr{M}(e)}(x_{j}))^{2}) \cdot (1-(\zeta^{+}_{\mathscr{O}(e)}(x_{j}))^{2})\big]}\,\,\Big] + \Big[1-\sqrt{\big[(1-(\zeta^{-}_{\mathscr{M}(e)}(x_{j}))^{2}) \cdot (1-(\zeta^{-}_{\mathscr{O}(e)}(x_{j}))^{2})\big]}\,\,\Big]\\ &\leq \Big[1-\sqrt{\big[(1-(\zeta^{+}_{\mathscr{N}(e)}(x_{j}))^{2}) \cdot (1-(\zeta^{+}_{\mathscr{O}(e)}(x_{j}))^{2})\big]}\,\,\Big] + \Big[1-\sqrt{\big[(1-(\zeta^{-}_{\mathscr{N}(e)}(x_{j}))^{2}) \cdot (1-(\zeta^{-}_{\mathscr{O}(e)}(x_{j}))^{2})\big]}\,\,\Big]. \end{align*} Hence, \begin{align*} &\sum^{n}_{i=1}\Bigg[ \Big[1-\sqrt{\big[(1-(\zeta^{+}_{\mathscr{M}(e_{i})}(x_{j}))^{2}) \cdot (1-(\zeta^{+}_{\mathscr{O}(e_{i})}(x_{j}))^{2})\big]}\,\,\Big] + \Big[1-\sqrt{\big[(1-(\zeta^{-}_{\mathscr{M}(e_{i})}(x_{j}))^{2}) \cdot (1-(\zeta^{-}_{\mathscr{O}(e_{i})}(x_{j}))^{2})\big]}\,\,\Big] \Bigg]\\ & \leq \sum^{n}_{i=1}\Bigg[\Big[1-\sqrt{\big[(1-(\zeta^{+}_{\mathscr{N}(e_{i})}(x_{j}))^{2}) \cdot (1-(\zeta^{+}_{\mathscr{O}(e_{i})}(x_{j}))^{2})\big]}\,\,\Big]+ \Big[1-\sqrt{\big[(1-(\zeta^{-}_{\mathscr{N}(e_{i})}(x_{j}))^{2}) \cdot (1-(\zeta^{-}_{\mathscr{O}(e_{i})}(x_{j}))^{2})\big]}\,\,\Big]\Bigg] \end{align*} and Dividing Eq. (1) by Eq. (4), \begin{align*} &\frac{1}{m} \sum^{m}_{j=1}\frac{\sum^{n}_{i=1} \Big[\big[\zeta^{+}_{\mathscr{M}(e_{i})}(x_{j}) \,\, \cdot\,\, \zeta^{+}_{\mathscr{O}(e_{i})}(x_{j})\big] + \big[\zeta^{-}_{\mathscr{M}(e_{i})}(x_{j}) \,\, \cdot\,\, \zeta^{-}_{\mathscr{O}(e_{i})}(x_{j})\big]\Big]} {\sum^{n}_{i=1} \Bigg[\Big[\,1- \sqrt{\big[(1-\zeta^{2+}_{\mathscr{M}(e_{i})}(x_{j})) \,\, \cdot\,\, (1-\zeta^{2+}_{\mathscr{O}(e_{i})}(x_{j}))\big]}\,\Big] + \Big[\,1- \sqrt{\big[(1-\zeta^{2-}_{\mathscr{M}(e_{i})}(x_{j})) \,\, \cdot\,\, (1-\zeta^{2-}_{\mathscr{O}(e_{i})}(x_{j}))\big]}\,\Big]\Bigg]}\end{align*} Clearly, \(\xi^{2+}_{\mathscr{M}(e)}(x_{j}) \geq \xi^{2+}_{\mathscr{N}(e)}(x_{j})\geq \xi^{2+}_{\mathscr{O}(e)}(x_{j})\,\, \text{and} \,\, \xi^{2-}_{\mathscr{M}(e)}(x_{j}) \geq \xi^{2-}_{\mathscr{N}(e)}(x_{j})\geq \xi^{2-}_{\mathscr{O}(e)}(x_{j}).\) Thus, \[\Big[\xi^{2+}_{\mathscr{M}(e)}(x_{j}) - \xi^{2+}_{\mathscr{O}(e)}(x_{j})\Big] \geq \Big[\xi^{2+}_{\mathscr{N}(e)}(x_{j}) - \xi^{2+}_{\mathscr{O}(e)}(x_{j}) \Big]\] and \[\Big[\xi^{2-}_{\mathscr{M}(e)}(x_{j}) - \xi^{2-}_{\mathscr{O}(e)}(x_{j})\Big] \geq \Big[\xi^{2-}_{\mathscr{N}(e)}(x_{j}) - \xi^{2-}_{\mathscr{O}(e)}(x_{j}) \Big].\] Hence Also, \[\Big[\xi^{2+}_{\mathscr{M}(e)}(x_{j}) \cdot \xi^{2+}_{\mathscr{O}(e)}(x_{j})\Big] \geq \Big[\xi^{2+}_{\mathscr{N}(e)}(x_{j}) \cdot \xi^{2+}_{\mathscr{O}(e)}(x_{j})\Big]\] and \[\Big[\xi^{2-}_{\mathscr{M}(e)}(x_{j}) \cdot \xi^{2-}_{\mathscr{O}(e)}(x_{j})\Big] \geq \Big[\xi^{2-}_{\mathscr{N}(e)}(x_{j}) \cdot \xi^{2-}_{\mathscr{O}(e)}(x_{j})\Big].\] Hence Dividing Eq. (6) by Eq. (7), we get \begin{eqnarray*} && \frac{\sum^{n}_{i=1} \Bigg[\Big|\xi^{2+}_{\mathscr{M}(e_{i})}(x_{j}) - \xi^{2+}_{\mathscr{O}(e_{i})}(x_{j})\Big| + \Big|\xi^{2-}_{\mathscr{M}(e_{i})}(x_{j}) - \xi^{2-}_{\mathscr{O}(e_{i})}(x_{j})\Big|\Bigg]}{\sum^{n}_{i=1} \Bigg[\Big[1+ \big[\xi^{2+}_{\mathscr{M}(e_{i})}(x_{j}) \cdot \xi^{2+}_{\mathscr{O}(e_{i})}(x_{j})\big]\Big] + \Big[1+ \big[\xi^{2-}_{\mathscr{M}(e_{i})}(x_{j}) \cdot \xi^{2-}_{\mathscr{O}(e_{i})}(x_{j})\big]\Big] \Bigg]} \\ && \geq \frac{\sum^{n}_{i=1} \Bigg[ \Big|\xi^{2+}_{\mathscr{N}(e_{i})}(x_{j}) - \xi^{2+}_{\mathscr{O}(e_{i})}(x_{j})\Big| + \Big|\xi^{2-}_{\mathscr{N}(e_{i})}(x_{j}) - \xi^{2-}_{\mathscr{O}(e_{i})}(x_{j})\Big|\Bigg]} {\sum^{n}_{i=1} \Bigg[\Big[1+ \big[\xi^{2+}_{\mathscr{N}(e_{i})}(x_{j}) \cdot \xi^{2+}_{\mathscr{O}(e_{i})}(x_{j})\big]\Big] + \Big[1+ \big[\xi^{2-}_{\mathscr{N}(e_{i})}(x_{j}) \cdot \xi^{2-}_{\mathscr{O}(e_{i})}(x_{j})\big]\Big]\Bigg]} \end{eqnarray*} and \begin{eqnarray*} &&\frac{1}{m} \sum^{m}_{j=1} \sqrt{1-\frac{\sum^{n}_{i=1} \Bigg[\Big|\xi^{2+}_{\mathscr{M}(e_{i})}(x_{j}) - \xi^{2+}_{\mathscr{O}(e_{i})}(x_{j})\Big| + \Big|\xi^{2-}_{\mathscr{M}(e_{i})}(x_{j}) - \xi^{2-}_{\mathscr{O}(e_{i})}(x_{j})\Big|\Bigg]}{\sum^{n}_{i=1} \Bigg[\Big[1+ \big[\xi^{2+}_{\mathscr{M}(e_{i})}(x_{j}) \cdot \xi^{2+}_{\mathscr{O}(e_{i})}(x_{j})\big]\Big] + \Big[1+ \big[\xi^{2-}_{\mathscr{M}(e_{i})}(x_{j}) \cdot \xi^{2-}_{\mathscr{O}(e_{i})}(x_{j})\big]\Big] \Bigg]}}\end{eqnarray*} Adding Eqs (5) and (8) and divided by 2, Clearly \(\alpha_{1} \leq \alpha_{2} \leq \alpha_{3}\), where \[\alpha_{1}= \frac{\zeta^{2+}_{p(e_{i})} + \zeta^{2-}_{p(e_{i})}}{\big[\zeta^{2+}_{p(e_{i})} \,\, + \,\, \xi^{2+}_{p(e_{i})}\big] + \big[\zeta^{2-}_{p(e_{i})} \,\, + \,\, \xi^{2-}_{p(e_{i})}\big] },\] \[\alpha_{2}= \frac{\zeta^{2+}_{q(e_{i})} + \zeta^{2-}_{q(e_{i})}}{\big[\zeta^{2+}_{q(e_{i})} \,\, + \,\, \xi^{2+}_{q(e_{i})}\big] + \big[\zeta^{2-}_{q(e_{i})} \,\, + \,\, \xi^{2-}_{q(e_{i})}\big] },\] \[\alpha_{3}= \frac{\zeta^{2+}_{r(e_{i})} + \zeta^{2-}_{r(e_{i})}}{\big[\zeta^{2+}_{r(e_{i})} \,\, + \,\, \xi^{2+}_{r(e_{i})}\big] + \big[\zeta^{2-}_{r(e_{i})} \,\, + \,\, \xi^{2-}_{r(e_{i})}\big] }.\] Clearly, \[\alpha_{1}- \alpha_{3}\leq \alpha_{2}- \alpha_{3}.\] Thus, Since Dividing Eq. (10) by Eq. (11), we get \[\frac{-\sum^{n}_{i=1}\big|\alpha_{1} - \alpha_{3}\big|} {\sum^{n}_{i=1}\big|\alpha_{1} + \alpha_{3}\big|} \leq \frac{-\sum^{n}_{i=1}\big|\alpha_{2}- \alpha_{3}\big|}{\sum^{n}_{i=1}\big|\alpha_{2}+ \alpha_{3}\big|} \implies 1-\frac{\sum^{n}_{i=1}\big|\alpha_{1} - \alpha_{3}\big|} {\sum^{n}_{i=1}\big|\alpha_{1} + \alpha_{3}\big|} \leq 1-\frac{\sum^{n}_{i=1}\big|\alpha_{2}- \alpha_{3}\big|}{\sum^{n}_{i=1}\big|\alpha_{2}+ \alpha_{3}\big|}.\] Hence Multiplying Eqs (9) and (12), \[\Delta^{\mathscr{GB}}(\mathscr{M}, \mathscr{O}) \cdot \Upsilon^{\mathscr{GB}}(p, r) \leq \Delta^{\mathscr{GB}}(\mathscr{N},\mathscr{O}) \cdot \Upsilon^{\mathscr{GB}}(q, r).\] Hence \[Sim(\mathscr{M}^{\mathscr{GB}}_{p}, \mathscr{O}^{\mathscr{GB}}_{r}) \leq Sim(\mathscr{N}^{\mathscr{GB}}_{q}, \mathscr{O}^{\mathscr{GB}}_{r}).\] This proves (4).

Example 4. Calculate the similarity measure between the two Type-II GPBFSS's namely \(\mathscr{M}^{\mathscr{GB}}_{p}\) and \(\mathscr{N}^{\mathscr{GB}}_{q}\). Let \(X=\{x_{1}, x_{2}, x_{3}\}\) and \(E = \{e_{1}, e_{2}, e_{3}\}\) can be defined as below:

\(\mathscr{M}^{\mathscr{GB}}_{p}(e)\)	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)
\(\mathscr{M}(e)(x_{1})\)	\((0.6,0.65,-0.3,-0.8)\)	\((0.9,0.35,-0.7,-0.5)\)	\((0.8,0.45,-0.2,-0.9)\)
\(\mathscr{M}(e)(x_{2})\)	\((0.55,0.55,-0.35,-0.75)\)	\((0.85,0.25,-0.75,-0.45)\)	\((0.75,0.35,-0.25,-0.85)\)
\(\mathscr{M}(e)(x_{3})\)	\((0.45,0.55,-0.55,-0.7)\)	\((0.75,0.25,-0.45,-0.65)\)	\((0.65,0.35,-0.15,-0.75)\)
\(p(e)\)	\((0.6,0.5,-0.8,-0.3)\)	\((0.8,0.3,-0.6,-0.5)\)	\((0.7,0.4,-0.8,-0.6)\)

\(\mathscr{N}^{\mathscr{GB}}_{q}(e)\)	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)
\(\mathscr{N}(e)(x_{1})\)	\((0.3,0.35,-0.2,-0.3)\)	\((0.4,0.45,-0.6,-0.2)\)	\((0.6,0.25,-0.1,-0.4)\)
\(\mathscr{N}(e)(x_{2})\)	\((0.25,0.3,-0.25,-0.2)\)	\((0.35,0.4,-0.65,-0.15)\)	\((0.55,0.2,-0.15,-0.35)\)
\(\mathscr{N}(e)(x_{3})\)	\((0.35,0.4,-0.2,-0.45)\)	\((0.55,0.45,-0.35,-0.25)\)	\((0.6,0.4,-0.25,-0.55)\)
\(q(e)\)	\((0.5,0.35,-0.3,-0.1)\)	\((0.6,0.15,-0.4,-0.2)\)	\((0.4,0.25,-0.5,-0.6)\)

Now, \[\mathbb{T}^{\mathscr{GB}}(\mathscr{M}(e)(x_{1}),\mathscr{N}(e)(x_{1}))=\frac{1.52}{1.876484}=0.810025,\] \[\mathbb{T}^{\mathscr{GB}}(\mathscr{M}(e)(x_{2}),\mathscr{N}(e)(x_{2}))=\frac{1.46}{1.778538}=0.820899\] and \[\mathbb{T}^{\mathscr{GB}}(\mathscr{M}(e)(x_{3}),\mathscr{N}(e)(x_{3}))=\frac{1.265}{1.390973}=0.909436.\] Hence, \[\mathbb{T}^{\mathscr{GB}}(\mathscr{M}(e)(x),\mathscr{N}(e)(x))=\frac{0.810025+0.820899+0.909436}{3}=0.846787.\] Now,\[\mathbb{S}^{\mathscr{GB}}(\mathscr{M}(e)(x_{1}),\mathscr{N}(e)(x_{1}))=\sqrt{1-\frac{1.93}{6.286419}}=0.83246,\] \[\mathbb{S}^{\mathscr{GB}}(\mathscr{M}(e)(x_{2}),\mathscr{N}(e)(x_{2}))=\sqrt{1-\frac{1.695}{6.157688}}=0.851313\] and \[\mathbb{S}^{\mathscr{GB}}(\mathscr{M}(e)(x_{3}),\mathscr{N}(e)(x_{3}))=\sqrt{1-\frac{1.2275}{6.376444}}=0.898607.\] Hence\[\mathbb{S}^{\mathscr{GB}}(\mathscr{M}(e)(x),\mathscr{N}(e)(x))=\frac{0.83246+0.851313+0.898607}{3}=0.860793.\] Thus, \[\Delta^{\mathscr{GB}}(\mathscr{M},\mathscr{N})=\frac{0.846787+0.860793}{2}=0.85379\]and \[\Upsilon^{\mathscr{GB}}(p,q)=1-\frac{0.365483}{4.282159}=0.91465.\] Hence, \[ Sim(\mathscr{M}^{\mathscr{GB}}_{p},\mathscr{N}^{\mathscr{GB}}_{q})=0.85379 \times 0.91465=0.780919.\]

5. Application for medical diagnosis

Decision making problems are a big part of human society and applied widely to practical fields like education, economics, management, engineering and Hospital. However, with the development of science and technology, the uncertainty also plays a dominant role at some point of the decision making analysis. In this application, we present a method for a medical diagnosis problem based on the proposed similarity measure of Type-II GPBFSS's. This technique of similarity measure between two Type-II GPBFSS's can be applied to detect whether an ill person is suffering from a certain disease or not. We first give the following remark;

Remark 2. Let \(\mathscr{M}_{p}\) and \(\mathscr{N}_{q}\) be two Type-II GPBFSS's over the same soft universe \((X,E)\). We call the two Type-II GPBFSS 's to be significantly similar if \(Sim(\mathscr{M}_{p}, \mathscr{N}_{q}) \geq 70\).

We first construct a Type-II GPBFSS for the illness with the help of a medical person and a Type-II GPBFSS for the ill person. Then, we calculate the similarity measure between two Type-II GPBFSS's. If they are significantly similar, then we infer that the person may have disease, and otherwise not.

5.1. Algorithm

The algorithm for the selection of the best choice is given as:

1. Input the Type-II GPBFSS \(\mathscr{M}^{\mathscr{GB}}_{p}\) in tabular form.
2. Input the set of choice parameters \(A \subseteq E\).
3. Compute the values of \(\mathbb{T}^{\mathscr{GB}}\) and \(\mathbb{S}^{\mathscr{GB}}\).
4. Calculate the \(\Delta^{\mathscr{GB}}\) value by taking \(\frac{\mathbb{T}^{\mathscr{GB}}+ \mathbb{S}^{\mathscr{GB}}}{2}\).
5. Determine the value \(\Upsilon^{\mathscr{GB}}= 1- \frac{\sum|\alpha_{1}-\alpha_{2}|}{{\sum|\alpha_{1}+\alpha_{2}|}}\).
6. Compute similarity measure \(\Delta^{\mathscr{GB}} \cdot \Upsilon^{\mathscr{GB}}\).
7. Select similarity measure using suitable criteria for significantly similar.
8. Finally, decision is to choose as the solution to the problem.
9. End.

5.2. Case study

Suppose that there are five patients \(\mathscr{P}_{1}, \mathscr{P}_{2}, \mathscr{P}_{3}, \mathscr{P}_{4}\) and \(\mathscr{P}_{5}\) in a hospital with certain symptoms of Scrub Typhus. Let the universal set contain only three elements. That is \(X = \{x_1 :\) severe, \(x_2 \): mild, \(x_3\) : no} and the set of parameters \(E\) is the set of certain symptoms of Scrub Typhus is represented by \(E = \{e_1 :\) Fever and chills, \(e_2 \): headache, \(e_3\) : muscle pain, \(e_4\) : mental changes, \(e_5\): enlarged lymph nodes}. Table 1 represents the Scrub Typhus prepared with the help of a medical person.

Table 1. Type-II GPBFSS for pneumonia(Scrub Typhus).

\(\mathscr{L}^{\mathscr{GB}}_{p}(e)\)	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)
\(\mathscr{L}(e)(x_{1})\)	\((0.92,0.18,-0.91,-0.25)\)	\((0.83,0.25,-0.72,-0.35)\)	\((0.91,0.35,-0.85,-0.25)\)
\(\mathscr{L}(e)(x_{2})\)	\((0.91,0.15,-0.9,-0.2)\)	\((0.82,0.23,-0.7,-0.3)\)	\((0.9,0.33,-0.81,-0.22)\)
\(\mathscr{L}(e)(x_{3})\)	\((0.85,0.25,-0.8,-0.25)\)	\((0.8,0.3,-0.75,-0.35)\)	\((0.7,0.4,-0.8,-0.2)\)
\(l(e)\)	\((1,0,-1,0)\)	\((1,0,-1,0)\)	\((1,0,-1,0)\)
	\(e_{4}\)	\(e_{5}\)
	\((0.84,0.15,-0.92,-0.35)\)	\((0.93,0.25,-0.73,-0.36)\)
	\((0.8,0.13,-0.9,-0.31)\)	\((0.9,0.21,-0.71,-0.34)\)
	\((0.75,0.35,-0.85,-0.3)\)	\((0.85,0.25,-0.75,-0.4)\)
	\((1,0,-1,0)\)	\((1,0,-1,0)\)

We construct the Type-II GPBFSS's for five patients under consideration as in Tables 2-6.

Table 2. Type-II GPBFSS for the ill person \(\mathscr{P}_{1}\).

\(\mathscr{A}^{\mathscr{GB}}_{p}(e)\)	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)
\(\mathscr{A}(e)(x_{1})\)	\((0.75,0.45,-0.5,-0.85)\)	\((0.45,0.75,-0.5,-0.75)\)	\((0.75,0.65,-0.6,-0.45)\)
\(\mathscr{A}(e)(x_{2})\)	\((0.6,0.4,-0.7-0.35)\)	\((0.5,0.3,0.4-0.65)\)	\((0.7,0.5,-0.45-0.55)\)
\(\mathscr{A}(e)(x_{3})\)	\((0.75,0.3,-0.55,-0.3)\)	\((0.55,0.7,-0.45,-0.5)\)	\((0.45,0.5,-0.7,-0.3)\)
\(p_{1}(e)\)	\((0.8,0.15,-0.55,-0.45)\)	\((0.7,0.25,-0.65,-0.55)\)	\((0.6,0.5,-0.4,-0.6)\)
	\(e_{4}\)	\(e_{5}\)
	\((0.35,0.65,-0.7,-0.55)\)	\((0.65,0.55,-0.3,-0.65)\)
	\((0.6,0.55,-0.6-0.45)\)	\((0.8,0.45,-0.5-0.65)\)
	\((0.55,0.6,-0.6,-0.4)\)	\((0.65,0.5,-0.45,-0.5)\)
	\((0.4,0.6,-0.6,-0.3)\)	\((0.5,0.7,-0.5,-0.4)\)

Table 3. Type-II GPBFSS for the ill person\(\mathscr{P}_{2}\).

\(\mathscr{B}^{\mathscr{GB}}_{p}(e)\)	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)
\(\mathscr{B}(e)(x_{1})\)	\((0.6,0.45,-0.7,-0.65)\)	\((0.55,0.35,-0.65,-0.55)\)	\((0.7,0.4,-0.75,-0.45)\)
\(\mathscr{B}(e)(x_{2})\)	\((0.62,0.4,-0.7,-0.35)\)	\((0.7,0.65,-0.45,-0.5)\)	\((0.65,0.5,-0.5,-0.55)\)
\(\mathscr{B}(e)(x_{3})\)	\((0.75,0.35,-0.65,-0.45)\)	\((0.5,0.6,-0.55,-0.63)\)	\((0.55,0.65,-0.84,-0.4)\)
\(p_{2}(e)\)	\((0.8,0.15,-0.55,-0.5)\)	\((0.7,0.35,-0.7,-0.45)\)	\((0.65,0.65,-0.45,-0.75)\)
	\(e_{4}\)	\(e_{5}\)
	\((0.5,0.65,-0.6,-0.5)\)	\((0.75,0.45,-0.7,-0.65)\)
	\((0.7,0.45,-0.6,-0.45)\)	\((0.8,0.35,-0.55,-0.6)\)
	\((0.65,0.45,-0.73,-0.55)\)	\((0.8,0.55,-0.52,-0.6)\)
hline	\((0.45,0.75,-0.7,-0.35)\)	\((0.3,0.85,-0.45,-0.65)\)

Table 4. Type-II GPBFSS for the ill person \(\mathscr{P}_{3}\).

\(\mathscr{C}^{\mathscr{GB}}_{p}(e)\)	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)
\(\mathscr{C}(e)(x_{1})\)	\((0.8,0.45,-0.75,-0.6)\)	\((0.7,0.35,-0.7,-0.65)\)	\((0.75,0.25,-0.8,-0.6)\)
\(\mathscr{C}(e)(x_{2})\)	\((0.8,0.55,-0.75,-0.4)\)	\((0.75,0.6,-0.5,-0.65)\)	\((0.7,0.6,-0.65,-0.5)\)
\(\mathscr{C}(e)(x_{3})\)	\((0.82,0.25,-0.65,-0.45)\)	\((0.7,0.55,-0.55,-0.65)\)	\((0.65,0.4,-0.8,-0.4)\)
\(p_{3}(e)\)	\((0.8,0.25,-0.65,-0.45)\)	\((0.7,0.25,-0.7,-0.4)\)	\((0.6,0.65,-0.6,-0.55)\)
	\(e_{4}\)	\(e_{5}\)
	\((0.8,0.3,-0.75,-0.55)\)	\((0.6,0.4,-0.6,-0.75)\)
	\((0.65,0.7,-0.7,-0.4)\)	\((0.85,0.55,-0.6,-0.55)\)
	\((0.6,0.3,-0.75,-0.6)\)	\((0.8,0.35,-0.5,-0.65)\)
	\((0.5,0.45,-0.85,-0.35)\)	\((0.4,0.5,-0.65,-0.5)\)

Table 5. Type-II GPBFSS for the ill person \(\mathscr{P}_{4}\).

\(\mathscr{D}^{\mathscr{GB}}_{p}(e)\)	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)
\(\mathscr{D}(e)(x_{1})\)	\((0.7,0.45,-0.6,-0.75)\)	\((0.5,0.75,-0.65,-0.75)\)	\((0.6,0.6,-0.7,-0.4)\)
\(\mathscr{D}(e)(x_{2})\)	\((0.6,0.7,-0.8,-0.4)\)	\((0.55,0.6,-0.5,-0.7)\)	\((0.7,0.65,-0.5,-0.6)\)
\(\mathscr{D}(e)(x_{3})\)	\((0.8,0.45,-0.6,-0.4)\)	\((0.65,0.8,-0.5,-0.6)\)	\((0.55,0.6,-0.85,-0.4)\)
\(p_{4}(e)\)	\((0.85,0.2,-0.65,-0.5)\)	\((0.8,0.3,-0.75,-0.6)\)	\((0.75,0.6,-0.5,-0.85)\)
	\(e_{4}\)	\(e_{5}\)
	\((0.75,0.55,-0.65,-0.5)\)	\((0.7,0.35,-0.4,-0.6)\)
	\((0.6,0.7,-0.8,-0.5)\)	\((0.8,0.55,-0.6,-0.7)\)
	\((0.6,0.7,-0.7,-0.55)\)	\((0.7,0.6,-0.5,-0.6)\)
	\((0.5,0.75,-0.8,-0.45)\)	\((0.55,0.8,-0.65,-0.4)\)

Table 6. Type-II GPBFSS for the ill person \(\mathscr{P}_{5}\).

\(\mathscr{E}^{\mathscr{GB}}_{p}(e)\)	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)
\(\mathscr{E}(e)(x_{1})\)	\((0.85,0.4,-0.6,-0.8)\)	\((0.5,0.7,-0.55,-0.7)\)	\((0.8,0.65,-0.75,-0.4)\)
\(\mathscr{E}(e)(x_{2})\)	\((0.75,0.6,-0.8,-0.45)\)	\((0.5,0.65,-0.5,-0.7)\)	\((0.75,0.4,-0.65,-0.6)\)
\(\mathscr{E}(e)(x_{3})\)	\((0.65,0.45,-0.6,-0.4)\)	\((0.55,0.7,-0.55,-0.6)\)	\((0.45,0.6,-0.8,-0.45)\)
\(p_{5}(e)\)	\((0.9,0.2,-0.6,-0.5)\)	\((0.75,0.45,-0.7,-0.45)\)	\((0.7,0.65,-0.55,-0.8)\)
	\(e_{4}\)	\(e_{5}\)
	\((0.45,0.6,-0.8,-0.5)\)	\((0.7,0.55,-0.4,-0.65)\)
	\((0.55,0.75,-0.8,-0.5)\)	\((0.6,0.5,-0.45,-0.75)\)
	\((0.6,0.7,-0.7,-0.55)\)	\((0.75,0.65,-0.5,-0.65)\)
	\((0.45,0.55,-0.8,-0.4)\)	\((0.65,0.35,-0.6,-0.75)\)

The generalized bipolar fuzzy values in Tables 2-6 are provided by the experts,depending on their assessment of the alternatives against the criteria under consideration. We calculate the similarity measure of Type-II GPBFSS's in Tables 2-6 with the one in Table 1. Calculating the similarity measure for \(\mathscr{P}_{1}\) to \(\mathscr{P}_{5}\) ill persons are given below the Table 7.

Table 7. Similarity measure for \(\mathscr{P}_{1}\) to \(\mathscr{P}_{5}\) ill persons.

	\(\mathbb{T}^{\mathscr{GB}}(x_{1})\)	\(\mathbb{S}^{\mathscr{GB}}(x_{1})\)	\(\mathbb{T}^{\mathscr{GB}}(x_{2})\)	\(\mathbb{S}^{\mathscr{GB}}(x_{2})\)	\(\mathbb{T}^{\mathscr{GB}}(x_{3})\)	\(\mathbb{S}^{\mathscr{GB}}(x_{3})\)
\(\mathscr{(L,A)}\)	\(0.823604\)	\(0.823796\)	\(0.774984\)	\(0.904968\)	\(0.903511\)	\(0.934002\)
\(\mathscr{(L,B)}\)	\(0.89267\)	\(0.90181\)	\(0.903051\)	\(0.909339\)	\(0.953667\)	\(0.905286\)
\(\mathscr{(L,C)}\)	\(0.945782\)	\(0.899062\)	\(0.95394\)	\(0.870195\)	\(0.970414\)	\(0.929305\)
\(\mathscr{(L,D)}\)	\(0.879251\)	\(0.862084\)	\(0.919538\)	\(0.830471\)	\(0.94947\)	\(0.875255\)
\(\mathscr{(L,E)}\)	\(0.897963\)	\(0.84803\)	\(0.9162\)	\(0.841527\)	\(0.934313\)	\(0.874316\)
	\(\mathbb{T}^{\mathscr{GB}}\)	\(\mathbb{S}^{\mathscr{GB}}\)	\(\Delta^{\mathscr{GB}}\)	\(\Upsilon^{\mathscr{GB}}\)	Similarity
	\(0.834033\)	\(0.887589\)	\(0.860811\)	\(0.742551\)	\(0.639196\)
	\(0.916463\)	\(0.905479\)	\(0.910971\)	\(0.68785\)	\(0.626611\)
	\(0.956712\)	\(0.899521\)	\(0.928116\)	\(0.8097\)	\(0.751495\)
	\(0.916086\)	\(0.855937\)	\(0.870668\)	\(0.744906\)	\(0.648565\)
	\(0.916159\)	\(0.854624\)	\(0.885392\)	\(0.769917\)	\(0.681678\)

We find that the similarity measure of the first two patients and last two patients are \(< 0.70\), but the similarity measure of third patient \(\mathscr{P}_{3}\) is \(\mathscr{(L,{P}}_{3})={ 0.751495} \geq 0.70\). Hence these two Type-II GPBFSS's are significantly similar. Therefore, we conclude that the patient \(\mathscr{P}_{3}\) is suffering from Scrub Typhus.

6. Conclusion

The main goal of this work is to present a Type-II GPBFSS and studied some of its properties. Similarity measure of two Type-II GPBFSS's is discussed and an application of this to medical diagnosis has been shown. In the future direction, we will apply the generalized cubic fuzzy soft sets and generalized spherical fuzzy soft sets theory.

Acknowledgments

The author is obliged the thankful to the reviewer for the numerous and significant suggestions that raised the consistency of the ideas presented in this paper.

Conflicts of Interest

The author declares no conflict of interest.

1. Introduction

Plane trees (or ordered trees) have computer science and mathematics applications alike. These are rooted trees with the property that all the children of the vertices are ordered. These trees have been generalized by labeling the vertices to obtain \(k\)-plane trees, which we now define. When the vertices of the plane trees are labeled with integers in the set \(\{1,2,\ldots,k\}\) such that there are no edges \((i,j)\) with the property that \(i+j>k+1\) then such trees are called \(k\)-plane trees. These trees were introduced and studied by Gu, Prodinger, and Wagner in [1]. The aforementioned authors showed that the number of \(k\)-plane trees with root labelled by \(h\) on \(n>1\) vertices is given by

and the total number of \(k\)-plane trees with \(n>1\) vertices is \[\frac{k}{n} \binom{(k+1)(n-1)}{n-1}.\] In [2], Okoth and Wagner showed that the total number of \(k\)-plane trees on \(n>1\) vertices whose root is labelled by \(k\) such that there are \(v_j\) vertices labelled by \(j\in\{1,2,\ldots,k\}\) is Here, empty sums and empty products are considered to be \(0\) and 1 respectively.

Setting \(k=2\) and \(h=2\) in Eq. (1), we find that the number of \(2\)-plane trees on \(n\) vertices whose root is labelled by 2 is given by

\[\frac{1}{2n-1} \binom{3n-3}{n-1}.\] This formula also counts the number of noncrossing trees on \(n\) vertices and ternary trees on \(n-1\) vertices among other structures, see sequence https://oeis.org/A001764 in [3]. A bijection between the set of 2-plane trees and the set of ternary trees was constructed by Gu and Prodinger in an earlier paper [4]. Of course, setting \(k=h=1\) in Eq. (1), we recover Catalan's number that counts the number of plane trees with \(n\) vertices.

A \(k\)-noncrossing tree is a noncrossing tree whose vertices are labelled with integers in the set \(\{1,2,\ldots,k\}\) such that if an edge \((i,j)\) is an ascent in the path from the root then \(i+j\leq k+1\). These trees were first studied by Pang and Lv [5]. When \(k=2\) we obtain \(2\)-noncrossing trees, introduced and enumerated by Yan and Liu [6]. The author recently refined the formula of Yan and Liu in [7]. Similarly, when \(k=1\) we recover noncrossing trees.

In 2015, Okoth and Wagner [8] obtained counting formulas for the number of noncrossing trees whose edges are locally oriented. By local orientation, we mean that the edges are oriented from vertices of the lower label towards vertices of the higher label. The said authors coined the words locally-oriented noncrossing trees for these trees. In the same paper, they called a vertex of in-degree 0 (resp. outdegree 0) as source (resp. sink). A \(k\)-noncrossing tree whose edges have local orientation will be referred to as locally-oriented \(k\)-noncrossing tree.

A noncrossing increasing tree [9] is a noncrossing tree whose labels increase along the paths from the root of the tree. The number of these trees on \(n\) vertices is also given by the Catalan number \[\dfrac{1}{n}{2n-2\choose n-1}.\] In [10], Okoth introduced and enumerated 2-noncrossing increasing trees by root degree and the number of vertices labeled by 2. These are 2-noncrossing trees whose labels increase as one moves away from the root. In the same spirit, a \(k\)-noncrossing increasing tree is a \(k\)-noncrossing tree whose labels increase as one moves away from the root.

Lattice paths have been studied for centuries. For example, a Dyck path of length \(n\) is a path from point \((0,0)\) to point \((n,0)\) which consists of up steps and down steps such that the path does not go below the \(x\)-axis. In this paper, we also consider lattice paths starting at \((0,0)\) and consisting of unit vertical and unit horizontal steps such that the path either touches or lies above the line \(x=ky.\) We shall call these lattice paths as restricted lattice paths.

This paper is organized as follows: We construct bijections between the set of \(k\)-plane trees and the set of \(k\)-noncrossing increasing trees in §2, locally oriented \((k-1)\)-noncrossing trees in §3, Dyck paths in §4 and restricted lattice paths in §5 respectively. We conclude the paper in §6 and expose some problems.

2. \(k\)-Noncrossing increasing trees

In the sequel, we show that the number of \(k\)-noncrossing increasing trees on \(n\) vertices such that the root is labeled by \(h\) is given by Eq. (1), the same formula that enumerates \(k\)-plane trees by root label and several vertices.

Theorem 1. There is a bijection between the set of \(k\)-plane trees on \(n\) vertices whose root is labelled by \(h\) and the set of \(k\)-noncrossing increasing trees on \(n\) vertices whose root is labelled by \(h\).

Proof. Let \(T\) be a \(k\)-plane tree on \(n\) vertices whose root is labeled by \(h\). We obtain a corresponding \(k\)-noncrossing increasing tree \(T'\) on \(n\) vertices whose root is labelled by \(h\) as follows: Each vertex in \(T\) becomes a vertex of \(T'\) and the children of a given vertex in \(T\) become right wing of a butterfly rooted at a corresponding vertex in \(T'\) such that if \(v_1, v_2,\ldots,v_i\) are the left-to-right children of the root \(v\) in \(T\) then \(v_1\) receives the highest label, followed by \(v_2\), and so on in \(T'\). Vertex \(v_i\) receives the least label. If a vertex is labeled by \(j\) in plane tree \(T\), the corresponding vertex in the noncrossing tree \(T'\) is labeled by \(j\). Therefore, the root of the \(k\)-noncrossing increasing tree is labeled by \(h\).

We now obtain the reverse procedure: Let \(T'\) be a \(k\)-noncrossing increasing tree with root \(r\) labeled by \(h\) and have \(n\) vertices. Let \(r_{1}< r_{2}< \cdots< r_{d}\) be the vertices of \(T'\) incident to \(r\). Vertex \(r\) corresponds to the root in the \(k\)-plane tree \(T\) and vertices \(r_{1},r_{2},\ldots, r_{d}\) correspond to the right-to-left children of the root in the \(k\)-plane tree. (So if the root in the \(k\)-noncrossing increasing tree is labeled by \(h\), the corresponding \(k\)-plane tree has its root labeled by \(h\).) Now, consider vertex \(r_1\) and all the vertices incident to it apart from \(r\), which is already in the \(k\)-plane tree. Draw these vertices as the children of \(r_1\). Again the ordering of the children is done with the vertex, with the largest label being the leftmost vertex. We repeat the process, and the algorithm stops when all the vertices have been drawn in the \(k\)-plane tree.

An example is given in Figure 1.

3. Locally oriented \((k-1)\)-noncrossing trees

In this section, we use number of vertices of each kind and the number of sources labelled by \(1\) as the statistics of enumeration. In the following theorem, we show that Eq. (2) which counts the number of \(k\)-plane trees with root labelled by \(k\), on \(n\) vertices, \(v_i\) of which are labelled by \(i\) for \(1\leq i\leq k\), also counts the number of locally oriented \((k-1)\)-noncrossing trees on \(n\) vertices such that the root is labelled by \(1\), the total number of sources labelled by \(1\) is \(v_k\), the total number of non-source vertices labelled by 1 is \(v_1\) and the number of vertices labelled by \(i\) is \(v_i\) for \(2\leq i\leq k-1\).

Theorem 2. There is a bijection between the set of \(k\)-plane trees with root labelled by \(k\) on \(n\) vertices, \(v_i\) of which are labelled by \(i\) for \(1\leq i\leq k\), and the set of locally oriented \((k-1)\)-noncrossing trees with root labelled by 1 on \(n\) vertices such that the total number of sources labelled by 1 is \(v_k\), non-source vertices labelled by 1 is \(v_1\) and for \(i\in\{1,2,\ldots,k-1\}\), there are \(v_i\) vertices labelled by \(i\).

Proof. By Theorem 1, it suffices to construct a bijection between the set of \(k\)-noncrossing increasing trees with root labelled by \(k\) on \(n\) vertices, \(v_i\) of which are labelled by \(i\) for \(1\leq i\leq k\), and the set of locally oriented \((k-1)\)-noncrossing trees with root labelled by 1 on \(n\) vertices such that the total number of sources labelled by 1 is \(v_k\), non-source vertices labelled by 1 is \(v_1\) and for \(i\in\{1,2,\ldots,k-1\}\), there are \(v_i\) vertices labelled by \(i\).

Consider a \(k\)-noncrossing increasing tree on \(n\) vertices with root labelled by \(k\). For \(i\in\{1,2,\ldots,k\}\), let the number of vertices labelled by \(i\) be \(v_i\). We obtain the corresponding locally oriented \((k-1)\)-noncrossing tree by the following steps:

• (i) Relabel the root of the \(k\)-noncrossing increasing tree by 1.
• (ii) Starting at the root, we visit the vertices of the \(k\)-noncrossing increasing tree in clockwise direction. Let \(v\) be the first vertex labelled by \(k\) (other than the root). Let \(u\) be the vertex incident to \(v\) such that \(u< v\). Vertex \(u\) is labelled by 1 so as to satisfy the ascent rule.
• (iii) Insert a new vertex \(v'\) labelled by 1 just before \(u\). Delete the edge \(uv\) and create a new edge \(uv'\). Adjoin \(v\) and the butterflies rooted at \(v\) as the right wing of the butterfly rooted at \(v'.\)
• (iv) Repeat the procedure until there are no vertices labelled by \(k\).
• (v) Rename the vertices in clockwise direction as \(1,2,\ldots,n\) and use local orientation to give directions to the edges.

The tree obtained is a locally oriented \((k-1)\)-noncrossing tree on \(n\) vertices with \(v_k\) sources labelled by 1 (all the vertices labelled by \(k\) in the \(k\)-noncrossing increasing tree become sources labelled by 1), \(v_1\) non-source vertices labelled by 1 (these were the vertices labelled by 1 in the \(k\)-noncrossing increasing tree) and \(v_i\) vertices labelled by \(i\) for \(i\in\{1,2,\ldots,k-1\}\) (the vertices labelled by \(i\) for \(2\leq i\leq k-1\) retain their labels in the locally oriented \((k-1)\)-noncrossing tree).

We now obtain the reverse procedure:

• (i) Relabel the root of the locally oriented \(k\)-noncrossing tree as \(k\).
• (ii) Starting at the root, we visit the vertices of the locally oriented \(k\)-noncrossing tree in clockwise direction. Let \(v\) be a source labelled by 1 with the largest vertex label. Let \(u\) be the vertex incident to \(v\) such that \(u>v\) and \(u\) is the vertex with the largest label.
• (iii) Insert a new vertex \(u'\) after \(v\). Label the new vertex by \(k.\) Delete the edge \(uv\) and create a new edge \(vu'\). Adjoin \(u\) and the butterflies rooted at \(u\) as the right wing of the butterfly rooted at \(u'.\)
• (iv) Rename the vertices in clockwise direction as \(1,2,\ldots,n\).
• (v) Repeat the procedure until there is no source labelled by 1.

We obtain a \(k\)-noncrossing increasing tree on \(n\) vertices of which \(v_i\) vertices are labeled by \(i\), and the root is labeled by \(k\). That is, a source labeled by 1, a non-source vertex labeled by 1, and a vertex labeled by \(i\) for \(2\leq i\leq k-1\) in a locally oriented \((k-1)\)-noncrossing tree corresponds to a vertex labeled by \(k\), a vertex labeled by \(1\) and a vertex labeled by \(i\) for \(2\leq i\leq k-1\) respectively in the \(k\)-noncrossing increasing tree whose root is labeled by \(k\).

For an example of the bijection, see Figure 2.

Setting \(k=2\) in Eq. (2), we recover the formula for the number of locally oriented noncrossing trees on \(n\) vertices with \(i\) sources obtained by Okoth and Wagner in [8]. In [7], the present author obtained the number of locally oriented \(2\)-noncrossing trees on \(n\) vertices whose root is labeled by 2 with a given number of vertices labeled by 2 and number of sources labeled by 1.

4. Dyck paths

In this section, we show that there is a bijection between the set of \(k\)-plane trees, with root labeled by 1, on \(n\) vertices and the set of Dyck paths from \((0,0)\) to \((2n-2,0)\) where the up steps are labeled with integers in the set \(\{1,2,3,\ldots,k\}\) such that if an up step labeled \(i\) is followed by another up step labeled \(j\) then the coherence condition \(i+j\leq k+1\) must be satisfied.

Let \(T\) be a \(k\)-plane tree with root labeled by 1 on \(n\) vertices. We obtain the Dyck path \(D\) of length \(2n-2\) as follows: Starting to the left of a given plane tree \(T\), we move around the tree, always moving away from the root on the left-hand side of an edge and moving towards the root on the right-hand side of an edge. Each edge corresponds to one-up step and one down step is as follows:

• (i) Draw an up step (labelled by \(i\)) if the vertex visited, as one traverses an edge on the left hand side of the edge, is labelled by \(i\).
• (ii) Draw a down step if one traverses an edge on the right hand side of the edge (towards the root).

We obtain a Dyck path \(D\) of length \(2n-2\) with up steps labeled by \(i\in\{1,2,\ldots,k\}\) and satisfying the condition \(i+j\leq k+1\) for any adjacent labels \(i\) and \(j\). Since the condition is satisfied in the \(k\)-plane tree, it must be true for Dyck paths obtained here.

For a bijection, we obtain the reverse procedure: Let \(D\) be a Dyck path of length \(2n-2\) such that its up steps are labelled with integers in the set \(\{1,2,\ldots,k\}\) and that label \(i\) is never followed by label \(j\) if \(i+j>k+1.\) We obtain a corresponding \(k\)-plane tree with a root labeled by 1 on \(n\) vertices by the following procedure: Draw a vertex \(v\) and label it 1. Starting at the beginning of the Dyck path, if the up step is labeled \(i\) then draw an edge connecting \(v\) to a new vertex labeled by \(i\). Continue the process with the Dyck path until you get a down step. For each down step, move up an edge (on the right-hand side of the edge, in the tree already drawn, towards the root). Continue until you find an up step. Let \(u\) be the vertex reached before the up step. If the up step is labeled \(j\) then draw an edge connecting \(u\) to a new vertex labeled by \(j\). Repeat the procedure until all paths in the Dyck path are visited. The tree obtained is a \(k\)-plane tree with roots labeled by 1 on \(n\) vertices.

Figure 3 below gives an example of the bijection.

Based on the bijection above, we give the following theorem:

Theorem 3. The number of Dyck paths from \((0,0)\) to \((2n-2,0)\), consisting of \((n-1)\) down steps of unit length and \((n-1)\) up steps of unit length labelled by integers in the set \(\{1,2,\ldots,k\}\) such that if label \(i\) is followed by label \(j\) then \(i+j\leq k+1\), is given by \begin{align*} \dfrac{1}{n}{(k+1)n-2\choose n-1}. \end{align*}

We note that the case of \(k=2\) in Theorem 3 was obtained in [10] and of \(k=1\) gives the equivalent result for unlabelled Dyck paths, given by Catalan numbers.

5. Restricted lattice paths

Consider lattice paths made up of \(kn-k\) vertical steps of unit length and \(n-1\) horizontal steps, also of unit length, such that the paths lie above or touch the line \(x=ky\). In this section, we construct a bijection between the set of these paths and the set of \(k\)-plane trees with roots labelled by \(k\) on \(n\) vertices.

Let \(T\) be a \(k\)-plane tree with root labelled by \(1\) on \(n\) vertices. We obtain the corresponding lattice path with \(n-1\) vertical steps and \(kn-k\) horizontal steps such that the paths never go below the line \(x=ky\). We use the procedure of Gu, Prodinger and Wagner in [1] to move from a \(k\)-plane tree to a Dyck path. We shall then construct a bijection beween the set of the Dyck paths and the set of the lattice paths defined here. Before we present our bijection and for the benefit of clarity, we present the bijection obtained by the said authors in the next subsection.

5.1. Bijection between \(k\)-plane trees and Dyck paths

We begin by giving the procedure of obtaining a Dyck path from a \(k\)-plane tree with root labelled by \(k\). Starting to the left of a given tree \(T\), we move around the tree, always moving away from the root on the left hand side of an edge and towards the root on the right hand side of an edge, as was in §3. Each edge contributes a total of \(k\) down steps and one up step as follows:

• (i) Draw \(i-1\) down steps (\((1,-1)\) steps), followed by an up step (\((k,k)\) step) if the vertex visited, as one traverses an edge on the left hand side of the edge, is labelled by \(i\).
• (ii) Draw \(k-i+1\) down steps (\((1,-1)\) steps)) if one traverses an edge on the right hand side of the edge and the initial vertex is labelled by \(i\).

The resultant path is a Dyck path \(D\) from \((0,0)\) to \((2kn-2k, 0).\)

Let us now obtain the reverse procedure: Let \(D\) be a Dyck path from \((0,0)\) to \((2kn-2k,0)\). We obtain a corresponding \(k\)-plane tree on \(n\) vertices whose root is labelled by \(k\) by the following procedure: We visit all up steps and down steps of the Dyck path starting at point \((0,0)\).

• (i) Draw a root (call it \(u\)) of the \(k\)-plane tree and label it by \(k\).
• (ii)If an up step is preceded by a down step of length \(i-1\) then draw an edge from \(u\) to new vertex \(v\), labelled by \(i\). Next, if the following up step is preceded by another down step of length \(j-1\) for \(1\leq j\leq k\) then draw an edge from \(v\) to a new vertex \(w\) labelled by \(j\).
• (iii) Otherwise, if the terminal vertex in the tree is labelled by \(i\) then for consecutive \(k-i+1\) down steps, move up the edge of the tree on the right hand side of the edge towards the root.
• (iv) Repeat steps (ii) and (iii) until all the steps in the Dyck path are visited.

We obtain a \(k\)-plane tree on \(n\) vertices whose root is labelled by \(k\).

An example to explain the bijection is given in Figure 4.

5.2. Bijection between Dyck paths and restricted lattice paths

Let \(D\) be the Dyck path defined in §5.1. We obtain a lattice path from \((0,0)\) to \((kn-k, n-1)\) with \(n-1\) vertical steps by the following steps.

• (i) For each up step, draw a vertical step.
• (ii) For each down step, draw a horizontal step.

The lattice path obtained is never goes below the line \(x=ky\).

We obtain the reverse procedure: Let \(L\) be a lattice path comprising of \(n-1\) unit vertical steps and \(kn-k\) unit horizontal steps such that the lattice path lies above or touches the line \(x=ky\). We obtain its corresponding Dyck path as follows:

• (i) For each horizontal step, draw a down step.
• (ii) For each vertical step, draw an up step.

An example is given in Figure 5 which corresponds to the 4-plane tree and Dyck path in the last subsection.

6. Conclusion

In this paper, we have constructed bijections between the set of \(k\)-plane trees and the sets of \(k\)-noncrossing increasing trees, locally oriented \((k-1)\)-noncrossing trees, Dyck paths, and some restricted lattice paths. It would be interesting to enumerate these structures from a generating function approach. Setting \(k=2\) and \(h=1\) in Eq. (1), we obtain the number of \(2\)-plane trees on \(n\) vertices whose root is labelled by 1. It is given by the sequence \(1,2,7,30,\ldots\) which is recorded as https://oeis.org/A006013 in the Neil Sloane's celebrated On-Line Encyclopedia of Integer Sequences (OEIS) [3]. Among the structures listed to be counted by the same formula include \(S\)-Motzkin paths introduced and enumerated by Prodinger and his coauthors in [11], \((3/2)\)-ary trees introduced and studied by Knuth in his Annual Christmas Tree Lecture [12], pattern avoidance patterns, Dyck paths, restricted lattice paths, and plane trees with \(n\) internal vertices and \(n\) leaves. It would be interesting to construct bijections between the set of \(k\)-plane trees and the sets of generalizations of \(S\)-Motzkin paths, \((3/2)\)-ary trees, pattern avoidance patterns among other structures listed under sequence https://oeis.org/A006013 in [3].

Conflicts of Interest:

The author declares no conflict of interest.

1. Introduction

In this paper, we consider a pair of graphs that traditionally are denoted by \(S\) and \(T\). These are constructed by starting with any two vertex-disjoint graphs \(G_1\) and \(G_2\). Let \(a\) and \(b\) be two distinct vertices of \(G_1\), and let \(c\) and \(d\) be two distinct vertices of \(G_2\). Then \(S\) is the graph obtained from \(G_1\) and \(G_2\) by connecting \(a\) with \(c\) and \(b\) with \(d\). The graph \(T\) is obtained analogously, by connecting \(a\) with \(d\) and \(b\) with \(c\), see Figure 1.

In 1982, Polansky and Zander discovered a remarkable property of the graphs \(S\) and \(T\) [1]. They established that the characteristic polynomials of \(S\) and \(T\) are related as \[ \phi(T,\lambda) - \phi(S,\lambda) = \big[ \phi(G_1-a,\lambda) - \phi(G_1-b,\lambda) \big] \big[ \phi(G_2-c,\lambda) - \phi(G_2-d,\lambda) \big]\,. \] In the special case when \(G_1 \cong G_2\), \[ \phi(T,\lambda) - \phi(S,\lambda) = \big[ \phi(G_1-a,\lambda) - \phi(G_1-b,\lambda) \big]^2\,, \] which means that the inequality holds for all real values of the variable \(\lambda\).

The inequality (1) implies certain regularities for the distribution of the eigenvalues of \(S\) and \(T\) [2,3,4] and have appropriate (experimentally verifiable) consequences on the distribution of the molecular orbital energy levels [5]. The authors of [1] called this a '' topological effect on molecular orbitals'' and used the acronym TEMO. Eventually, TEMO was extensively investigated; a detailed bibliography of this research can be found in the books [6,7].

After the discovery of the regularities between the eigenvalues of \(S\) and \(T\), a number of other TEMO-like relations for these pairs of graphs was discovered [8,9,10,11,12,13,14,15,16].

2. TEMO for Sombor index

The Sombor index (\(SO\)) is a recently conceived vertex-degree-based graph invariant [17], that already attracted much attention (see, e.g. [18,19,20,21,22]). It is defined as where \(\delta_u\) is the degree (= number of first neighbors) of the vertex \(u\), \(uv\) denotes the edge connecting the vertices \(u\) and \(v\), and the summation goes over all edges of the underlying graph \(G\).

In what follows, we establish a TEMO-like property of the Sombor index, i.e., investigate the relation between \(SO(S)\) and \(SO(T)\).

Denote by \(\delta_a,\delta_b,\delta_c,\delta_d\) the degrees of the vertices \(a,b,c,d\) of the graphs \(S\) and \(T\) (see Fig. 1). It is obvious that if either \(\delta_a=\delta_b\) or \(\delta_c=\delta_d\) or both, then \(SO(S)=SO(T)\). Therefore, we consider the case \(\delta_a \neq \delta_b\) and \(\delta_c \neq \delta_d\). Without loss of generality, we may assume that \(\delta_a > \delta_b\) and \(\delta_c > \delta_d\).

Theorem 1. Let \(G_1\) and \(G_2\) be arbitrary vertex-disjoint graphs and \(a,b,c,d\) their vertices as indicated in Figure 1. If \(\delta_a > \delta_b\) and \(\delta_c > \delta_d\), then \(SO(S) < SO(T)\).

Note that the degree of the vertex \(a\) in the graph \(G_1\) is \(\delta_a-1\), etc.

Proof. Observe first that \begin{eqnarray*} SO(S) & = & \sqrt{\delta_a^2+\delta_c^2} + \sqrt{\delta_b^2+\delta_d^2} + SO^\ast\,, \\ SO(T) & = & \sqrt{\delta_a^2+\delta_d^2} + \sqrt{\delta_b^2+\delta_c^2} + SO^\ast\,, \end{eqnarray*} where \(SO^\ast\) is the sum of the terms \(\sqrt{\delta_u^2 + \delta_v^2}\) over other edges of \(S\) or \(T\). Thus, \[ SO(S)-SO(T) = \sqrt{\delta_a^2+\delta_c^2} + \sqrt{\delta_b^2+\delta_d^2} - \sqrt{\delta_a^2+\delta_d^2} - \sqrt{\delta_b^2+\delta_c^2}\,. \] It needs to be demonstrated that

In order to achieve this goal, consider \[ Q = \big( \delta_a^2-\delta_b^2 \big)\big( \delta_c^2-\delta_d^2 \big)\,, \] which by the assumptions made in the statement of Theorem 1 is evidently positive-valued. \begin{eqnarray*} Q > 0 & \Longleftrightarrow & \delta_a^2\,\delta_c^2 + \delta_b^2\,\delta_d^2 > \delta_a^2\,\delta_d^2 + \delta_b^2\,\delta_c^2 \\ & \Longleftrightarrow & \delta_a^2\,\delta_b^2 + \delta_c^2\,\delta_d^2 + \delta_a^2\,\delta_c^2 + \delta_b^2\,\delta_d^2 > \delta_a^2\,\delta_b^2 + \delta_c^2\,\delta_d^2 + \delta_a^2\,\delta_d^2 + \delta_b^2\,\delta_c^2 \\ & \Longleftrightarrow & \big(\delta_a^2+\delta_d^2 \big)\big(\delta_b^2+\delta_c^2 \big) > \big(\delta_a^2+\delta_c^2 \big)\big(\delta_b^2+\delta_d^2 \big) \\ & \Longleftrightarrow & 2 \sqrt{\big(\delta_a^2+\delta_d^2 \big)\big(\delta_b^2+\delta_c^2 \big)} > 2 \sqrt{\big(\delta_a^2+\delta_c^2 \big)\big(\delta_b^2+\delta_d^2 \big)} \\ & \Longleftrightarrow & \big( \delta_a^2+\delta_d^2 \big) + \big( \delta_b^2+\delta_c^2 \big) + 2 \sqrt{\big(\delta_a^2+\delta_d^2 \big)\big(\delta_b^2+\delta_c^2 \big)} > \big( \delta_a^2+\delta_c^2 \big) + \big( \delta_b^2+\delta_d^2 \big) + 2 \sqrt{\big(\delta_a^2+\delta_c^2 \big)\big(\delta_b^2+\delta_d^2 \big)} \\ & \Longleftrightarrow & \Big( \sqrt{\delta_a^2+\delta_d^2} + \sqrt{\delta_b^2+\delta_c^2} \Big)^2 > \Big( \sqrt{\delta_a^2+\delta_c^2} + \sqrt{\delta_b^2+\delta_d^2} \Big)^2 \end{eqnarray*} which directly implies the inequality (3).

3. More TEMO-type relations

In an analogous, yet slightly easier, manner, we can verify the following TEMO-type results.

Using the notation of Eq. (2), the second Zagreb index \(M_2\), the Randic index \(R\), the reciprocal Randic index \(RR\), and the nirmala index \(N\) are, respectively, defined as [23,24,25,26]

\begin{eqnarray*} M_2 = M_2(G) & = & \sum_{uv \in \mathbf E(G)} \delta_u\,\delta_v \,,\\ R = R(G) & = & \sum_{uv \in \mathbf E(G)} \frac{1}{\sqrt{\delta_u\,\delta_v}}\,, \\ RR = RR(G) & = & \sum_{uv \in \mathbf E(G)} \sqrt{\delta_u\,\delta_v} \,,\\ N = N(G) & = & \sum_{uv \in \mathbf E(G)} \sqrt{\delta_u+\delta_v}\,. \end{eqnarray*}

Theorem 2. Let \(G_1\) and \(G_2\) be arbitrary vertex-disjoint graphs and \(a,b,c,d\) their vertices as indicated in Figure 1. If \(\delta_a > \delta_b\) and \(\delta_c > \delta_d\), then

• (a) \(M_2(S) > M_2(T)\),
• (b) \(R(S) > R(T)\),
• (c) \(RR(S) > RR(T)\),
• (d) \(N(S) < N(T)\).

Analogous relations hold also for the reduced versions of these indices, in which \(\delta\) is replaced by \(\delta-1\).

Conflicts of Interest:

The author declares no conflict of interest.

1. Introduction and preliminaries

In this paper, we introduce the binomial transform of the generalized fifth order Jacobsthal sequence and we investigate, in detail, two special cases which we call them the binomial transform of the fifth order Jacobsthal and fifth order Jacobsthal-Lucas sequences. We investigate their properties in the next sections. In this section, we present some properties of the generalized \((r,s,t,u,v)\) sequence (generalized Pentanacci) sequence.

The generalized \((r,s,t,u,v)\) sequence (the generalized Pentanacci sequence or 5-step Fibonacci sequence)

\begin{equation*} \{W_{n}\}_{n\geq 0}=\{W_{n}(W_{0},W_{1},W_{2},W_{3},W_{4};r,s,t,u,v)\}_{n\geq 0} \end{equation*} is defined by the fifth-order recurrence relations where the initial values \(W_{0},W_{1},W_{2},W_{3},W_{4}\ \)are arbitrary complex (or real) numbers and \(r,s,t,u,v\) are real numbers. Pentanacci sequence has been studied by many authors and more detail can be found in the extensive literature dedicated to these sequences, see for example [1,2,3,4,5]. The sequence \( \{W_{n}\}_{n\geq 0}\) can be extended to negative subscripts by defining \begin{equation*} W_{-n}=-\frac{u}{v}W_{-(n-1)}-\frac{t}{v}W_{-(n-2)}-\frac{s}{v}W_{-(n-3)}- \frac{r}{v}W_{-(n-4)}+\frac{1}{v}W_{-(n-5)} \end{equation*} for \(n=1,2,3,....\) Therefore, recurrence (1) holds for all integer \(n.\)

As \(\{W_{n}\}\) is a fifth order recurrence sequence (difference equation), it's characteristic equation is

whose roots are \(\alpha ,\beta ,\gamma ,\delta ,\lambda .\) Note that we have the following identities: \begin{align*} \alpha +\beta +\gamma +\delta +\lambda &=r, \\ \alpha \beta + \alpha \lambda +\alpha \gamma +\beta \lambda +\alpha \delta +\beta \gamma +\lambda \gamma +\beta \delta +\lambda \delta +\gamma \delta &=-s, \\ \alpha \beta \lambda +\alpha \beta \gamma +\alpha \lambda \gamma + \alpha \beta \delta +\alpha \lambda \delta +\beta \lambda \gamma +\alpha \gamma \delta +\beta \lambda \delta +\beta \gamma \delta +\lambda \gamma \delta &=t, \\ \alpha \beta \lambda \gamma +\alpha \beta \lambda \delta +\alpha \beta \gamma \delta +\alpha \lambda \gamma \delta +\beta \lambda \gamma \delta &=-u \\ \alpha \beta \gamma \delta \lambda &=v. \end{align*} Generalized Pentanacci numbers can be expressed, for all integers \(n,\) using Binet's formula.

Theorem 1. (Binet's formula of generalized \((r,s,t,u,v)\) numbers (generalized Pentanacci numbers))

where \begin{align*} { p}_{1} &{ =}{ W}_{4}{ -(\beta +\gamma +\delta +\lambda )W}_{3}{ +(\beta \lambda +\beta \gamma +\lambda \gamma +\beta \delta +\lambda \delta +\gamma \delta )W}_{2}{ -(\beta \lambda \gamma +\beta \lambda \delta +\beta \gamma \delta +\lambda \gamma \delta )W}_{1} { +(\beta \lambda \gamma \delta )W}_{0}{ ,} \\ { p}_{2} &{ =}{ W}_{4}{ -(\alpha +\gamma +\delta +\lambda )W}_{3}{ +( \alpha \lambda +\alpha \gamma +\alpha \delta +\lambda \gamma +\lambda \delta +\gamma \delta )W}_{2}{ -(\alpha \lambda \gamma +\alpha \lambda \delta +\alpha \gamma \delta +\lambda \gamma \delta )W}_{1}{ +(\alpha \lambda \gamma \delta )W}_{0}{ ,} \\ { p}_{3} &{ =}{ W}_{4}{ -(\alpha +\beta +\delta +\lambda )W}_{3}{ +(\alpha \beta + \alpha \lambda +\beta \lambda +\alpha \delta +\beta \delta +\lambda \delta )W}_{2}{ -(\alpha \beta \lambda + \alpha \beta \delta +\alpha \lambda \delta +\beta \lambda \delta )W}_{1}{ +(\alpha \beta \lambda \delta )W}_{0} { ,} \\ { p}_{4} &{ =}{ W}_{4}{ -(\alpha +\beta +\gamma +\lambda )W}_{3}{ +(\alpha \beta + \alpha \lambda +\alpha \gamma +\beta \lambda +\beta \gamma +\lambda \gamma )W}_{2}{ -(\alpha \beta \lambda +\alpha \beta \gamma +\alpha \lambda \gamma +\beta \lambda \gamma )W}_{1}{ +(\alpha \beta \lambda \gamma )W}_{0}{ ,} \\ { p}_{5} &{ =}{ W}_{4}{ -(\alpha +\beta +\gamma +\delta )W}_{3}{ +(\alpha \beta +\alpha \gamma +\alpha \delta +\beta \gamma +\beta \delta +\gamma \delta )W}_{2}{ -(\alpha \beta \gamma + \alpha \beta \delta +\alpha \gamma \delta +\beta \gamma \delta )W}_{1}{ +(\alpha \beta \gamma \delta )W}_{0}{ .} \end{align*}

Usually, it is customary to choose \(r,s,t,u,v\) so that the Eq. (2) has at least one real (say \(\alpha \)) solutions. Eq. (3) can be written in the following form: \begin{equation*} W_{n}=A_{1}\alpha ^{n}+A_{2}\beta ^{n}+A_{3}\gamma ^{n}+A_{4}\delta ^{n}+A_{5}\lambda ^{n}\,, \end{equation*} where \begin{align*} A_{1} &=\frac{p_{1}}{(\alpha -\beta )(\alpha -\gamma )(\alpha -\delta )(\alpha -\lambda )}, \\ A_{2} &=\frac{p_{2}}{(\beta -\alpha )(\beta -\gamma )(\beta -\delta )(\beta -\lambda )}, \\ A_{3} &=\frac{p_{3}}{(\gamma -\alpha )(\gamma -\beta )(\gamma -\delta )(\gamma -\lambda )}, \\ A_{4} &=\frac{p_{4}}{(\delta -\alpha )(\delta -\beta )(\delta -\gamma )(\delta -\lambda )}, \\ A_{5} &=\frac{p_{5}}{(\lambda -\alpha )(\lambda -\beta )(\lambda -\gamma )(\lambda -\delta )}. \end{align*} Next, we give the ordinary generating function \(\sum\limits_{n=0}^{\infty }W_{n}x^{n}\) of the sequence \(W_{n}.\)

Lemma 1. Suppose that \(f_{W_{n}}(x)=\sum\limits_{n=0}^{\infty }W_{n}x^{n}\) is the ordinary generating function of the generalized \( (r,s,t,u,v)\) sequence \(\{W_{n}\}_{n\geq 0}.\) Then, \(\sum\limits_{n=0}^{\infty }W_{n}x^{n}\) is given by

We next find Binet formula of generalized \((r,s,t,u,v)\) numbers \(\{W_{n}\}\) by the use of generating function for \(W_{n}.\)

Theorem 2. (Binet's formula of generalized \((r,s,t,u,v)\) numbers)

where \begin{align*} { q}_{1} &{ =}{ W}_{0}{ \alpha }^{4}{ +(W}_{1}{ -rW}_{0}{ )\alpha }^{3}{ +(W}_{2}{ -rW}_{1}{ -sW }_{0}{ )\alpha }^{2}{ +(W}_{3}{ -rW} _{2}{ -sW}_{1}{ -tW}_{0}{ )\alpha +(W} _{4}{ -rW}_{3}{ -sW}_{2}{ -tW}_{1} { -vW}_{0}{ ),} \\ { q}_{2} &{ =}{ W}_{0}{ \beta }^{4}{ +(W}_{1}{ -rW}_{0}{ )\beta }^{3}{ +(W}_{2}{ -rW}_{1}{ -sW} _{0}{ )\beta }^{2}{ +(W}_{3}{ -rW}_{2} { -sW}_{1}{ -tW}_{0}{ )\beta +(W}_{4} { -rW}_{3}{ -sW}_{2}{ -tW}_{1} { -vW}_{0}{ ),} \\ { q}_{3} &{ =}{ W}_{0}{ \gamma }^{4}{ +(W}_{1}{ -rW}_{0}{ )\gamma }^{3}{ +(W}_{2}{ -rW}_{1}{ -sW }_{0}{ )\gamma }^{2}{ +(W}_{3}{ -rW} _{2}{ -sW}_{1}{ -tW}_{0}{ )\gamma +(W} _{4}{ -rW}_{3}{ -sW}_{2}{ -tW}_{1} { -vW}_{0}{ ),} \\ { q}_{4} &{ =}{ W}_{0}{ \delta }^{4}{ +(W}_{1}{ -rW}_{0}{ )\delta }^{3}{ +(W}_{2}{ -rW}_{1}{ -sW }_{0}{ )\delta }^{2}{ +(W}_{3}{ -rW} _{2}{ -sW}_{1}{ -tW}_{0}{ )\delta +(W} _{4}{ -rW}_{3}{ -s3W}_{2}{ -tW}_{1} { -vW}_{0}{ ),} \\ { q}_{5} &{ =}{ W}_{0}{ \lambda }^{4}{ +(W}_{1}{ -rW}_{0}{ )\lambda }^{3}{ +(W}_{2}{ -rW}_{1}{ -sW}_{0}{ )\lambda }^{2}{ +(W}_{3}{ -rW}_{2}{ -sW}_{1}{ -tW}_{0}{ )\lambda +(W}_{4}{ -rW}_{3}{ -sW}_{2} { -tW}_{1}{ -vW}_{0}{ ).} \end{align*}

Matrix formulation of \(W_{n}\) can be given as [6]: In fact, Kalman give the formula in the following form \begin{equation*} \left( \begin{array}{c} W_{n} \\ W_{n+1} \\ W_{n+2} \\ W_{n+3} \\ W_{n+4} \end{array} \right) =\left( \begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ r & s & t & u & v \end{array} \right) ^{n}\left( \begin{array}{c} W_{0} \\ W_{1} \\ W_{2} \\ W_{3} \\ W_{4} \end{array} \right) . \end{equation*} Next, we consider two special cases of the generalized \((r,s,t,u,v)\) sequence \(\{W_{n}\}\) which we call them \((r,s,t,u,v)\) and Lucas \((r,s,t,u,v)\) sequences. \((r,s,t,u,v)\) sequence\(\{G_{n}\}_{n\geq 0}\) and Lucas \( (r,s,t,u,v) \) sequence \(\{H_{n}\}_{n\geq 0}\) are defined, respectively, by the fifth-order recurrence relations The sequences \(\{G_{n}\}_{n\geq 0}\ \) and \(\{H_{n}\}_{n\geq 0}\) can be extended to negative subscripts by defining \begin{eqnarray*} G_{-n} &=&-\frac{u}{v}G_{-(n-1)}-\frac{t}{v}G_{-(n-2)}-\frac{s}{v}G_{-(n-3)}- \frac{r}{v}G_{-(n-4)}+\frac{1}{v}G_{-(n-5)}, \\ H_{-n} &=&-\frac{u}{v}H_{-(n-1)}-\frac{t}{v}H_{-(n-2)}-\frac{s}{v}H_{-(n-3)}- \frac{r}{v}H_{-(n-4)}+\frac{1}{v}H_{-(n-5)}, \end{eqnarray*} for \(n=1,2,3,...\) respectively. Therefore, recurrences (7) and (8) hold for all integers \(n.\)

For more details on the generalized \((r,s,t,u,v)\) numbers, see [4].

Some special cases of \((r,s,t,u,v)\) sequence \(\{G_{n}(0,1,r,r^{2}+s, r^{3}+2sr+t ;r,s,t,u,v)\}\) and Lucas \((r,s,t,u,v)\) sequence \(\{H_{n}(5,r, 2s+r^{2},r^{3}+3sr+3t,r^{4}+4r^{2}s+4tr+2s^{2}+4u ;r,s,t,u,v)\}\) are as follows:

1. \(G_{n}(0,1,1,2,4;1,1,1,1,1)=P_{n},\) Pentanacci sequence,
2. \(H_{n}(5,1,3,7,15;1,1,1,1,1)=Q_{n},\) Pentanacci-Lucas sequence,
3. \(G_{n}(0,1,2,5,13;2,1,1,1,1)=P_{n},\) fifth-order Pell sequence,
4. \(H_{n}(5,2,6,17,46;2,1,1,1,1)=Q_{n},\) fifth-order Pell-Lucas sequence.

For all integers \(n,\) \((r,s,t,u,v)\) and Lucas \((r,s,t,u,v)\) numbers (using initial conditions in (3) or (5)) can be expressed using Binet's formulas as \begin{eqnarray*} G_{n} &=&\frac{\alpha ^{n+3}}{(\alpha -\beta )(\alpha -\gamma )(\alpha -\delta )(\alpha -\lambda )}+\frac{\beta ^{n+3}}{(\beta -\alpha )(\beta -\gamma )(\beta -\delta )(\beta -\lambda )}+\frac{\gamma ^{n+3}}{(\gamma -\alpha )(\gamma -\beta )(\gamma -\delta )(\gamma -\lambda )} \\ &&+\frac{\delta ^{n+3}}{(\delta -\alpha )(\delta -\beta )(\delta -\gamma )(\delta -\lambda )}+\frac{\lambda ^{n+3}}{(\lambda -\alpha )(\lambda -\beta )(\lambda -\gamma )(\lambda -\delta )}, \\ H_{n} &=&\alpha ^{n}+\beta ^{n}+\gamma ^{n}+\delta ^{n}+\lambda ^{n}, \end{eqnarray*} respectively.

Lemma 1 gives the following results as particular examples (generating functions of \((r,s,t,u,v)\), Lucas \((r,s,t,u,v)\) and modified \((r,s,t,u,v)\) numbers).

Corollary 1. Generating functions of \((r,s,t,u,v)\), Lucas \((r,s,t,u,v)\) and modified \( (r,s,t,u,v)\) numbers are \begin{eqnarray*} \sum\limits_{n=0}^{\infty }G_{n}x^{n} &=&\frac{x}{ 1-rx-sx^{2}-tx^{3}-ux^{4}-vx^{5}}, \\ \sum\limits_{n=0}^{\infty }H_{n}x^{n} &=&\frac{5-4rx-3sx^{2}-2tx^{3}-ux^{4}}{ 1-rx-sx^{2}-tx^{3}-ux^{4}-vx^{5}}, \end{eqnarray*} respectively.

The following theorem shows that the generalized Pentanacci sequence \(W_{n}\) at negative indices can be expressed by the sequence itself at positive indices.

Theorem 3. For \(n\in \mathbb{Z},\) for the generalized Pentanacci sequence (or generalized \((r,s,t,u,v)\)-sequence or 5-step Fibonacci sequence) we have the following: \begin{eqnarray*} W_{-n}&=&\frac{1}{24} v^{-n}(W_{0}H_{n}^{4}-4W_{n}H_{n}^{3}+3W_{0}H_{2n}^{2}+12H_{n}^{2}W_{2n}-6W_{0}H_{n}^{2}H_{2n}-6W_{0}H_{4n}-8W_{n}H_{3n}-12H_{2n}W_{2n}\\ &&-24H_{n}W_{3n}+24W_{4n}+8W_{0}H_{n}H_{3n}+12W_{n}H_{n}H_{2n}) \\ &=& v^{-n}(W_{4n}-H_{n}W_{3n}+\frac{1}{2}(H_{n}^{2}-H_{2n})W_{2n}- \frac{1}{6}(H_{n}^{3}+2H_{3n}-3H_{2n}H_{n})W_{n}\\ &&+\frac{1}{24} (H_{n}^{4}+3H_{2n}^{2}-6H_{n}^{2}H_{2n}-6H_{4n}+8H_{3n}H_{n})W_{0}). \end{eqnarray*}

Proof. For the proof, see [5], Theorem 1.

Using Theorem 3, we have the following corollary, see [5], Corollary 4.

Corollary 2. For \(n\in \mathbb{Z},\) we have \begin{equation*} H_{-n}=\frac{1}{24} v^{-n}(H_{n}^{4}+3H_{2n}^{2}-6H_{n}^{2}H_{2n}-6H_{4n}+8H_{3n}H_{n}). \end{equation*}

Note that \(G_{-n}\) and \(H_{-n}\) can be given as follows by using \(G_{0}=0\) and \(H_{0}=5\) in Theorem 3: \begin{eqnarray*} G_{-n} &=& v^{-n}(G_{4n}-H_{n}G_{3n}+\frac{1}{2} (H_{n}^{2}-H_{2n})G_{2n}-\frac{1}{6}(H_{n}^{3}+2H_{3n}-3H_{2n}H_{n})G_{n}), \\ H_{-n} &=&\frac{1}{24} v^{-n}(H_{n}^{4}+3H_{2n}^{2}-6H_{n}^{2}H_{2n}-6H_{4n}+8H_{3n}H_{n}), \end{eqnarray*} respectively.

Next, we consider the case \(r=1,\) \(s=1,t=1,u=1,v=2\) and in this case we write \(V_{n}=W_{n}.\) A generalized fifth order Jacobsthal sequence \( \{V_{n}\}_{n\geq 0}=\{V_{n}(V_{0},V_{1},V_{2},V_{3},V_{4})\}_{n\geq 0}\) is defined by the fifth order recurrence relations

with the initial values \(V_{0}=c_{0},V_{1}=c_{1},V_{2}=c_{2},\) \(V_{3}=c_{3},\) \(V_{4}=c_{4}\) not all being zero.

The sequence \(\{V_{n}\}_{n\geq 0}\) can be extended to negative subscripts by defining

\begin{equation*} V_{-n}=-\frac{1}{2}V_{-(n-1)}-\frac{1}{2}V_{-(n-2)}-\frac{1}{2}V_{-(n-3)}- \frac{1}{2}V_{-(n-4)}+\frac{1}{2}V_{-(n-5)} \end{equation*} for \(n=1,2,3,....\) Therefore, recurrence (9) holds for all integer \(n.\) For more information on the generalized fifth order Jacobsthal numbers, see [7].

The first few generalized fifth order Jacobsthal numbers with positive subscript and negative subscript are given in the Table 1

Table 1. A few generalized fifth order Jacobsthal numbers.

\(n\)	\(V_{n}\)	\(V_{-n}\)
\(0\)	\(V_{0}\)	\(...\)
\(1\)	\(V_{1}\)	\(\frac{1}{2}V_{4}-\frac{1}{2}V_{1}-\frac{1}{2}V_{2}-\frac{1}{ 2}V_{3}-\frac{1}{2}V_{0}\)
\(2\)	\(V_{2}\)	\(\frac{3}{4}V_{3}-\frac{1}{4}V_{1}-\frac{1}{4}V_{2}-\frac{1}{ 4}V_{0}-\frac{1}{4}V_{4}\)
\(3\)	\(V_{3}\)	\(\frac{7}{8}V_{2}-\frac{1}{8}V_{1}-\frac{1}{8}V_{0}-\frac{1}{ 8}V_{3}-\frac{1}{8}V_{4}\)
\(4\)	\(V_{4}\)	\(\frac{15}{16}V_{1}-\frac{1}{16}V_{0}-\frac{1}{16}V_{2}- \frac{1}{16}V_{3}-\frac{1}{16}V_{4}\)
\(5\)	\(2V_{0}+V_{1}+V_{2}+V_{3}+V_{4}\)	\(\frac{31}{32}V_{0}-\frac{1}{32} V_{1}-\frac{1}{32}V_{2}-\frac{1}{32}V_{3}-\frac{1}{32}V_{4}\)
\(6\)	\(2V_{0}+3V_{1}+2V_{2}+2V_{3}+2V_{4}\)	\(\frac{31}{64}V_{4}-\frac{33}{64 }V_{1}-\frac{33}{64}V_{2}-\frac{33}{64}V_{3}-\frac{33}{64}V_{0}\)
\(7\)	\(4V_{0}+4V_{1}+5V_{2}+4V_{3}+4V_{4}\)	\(\frac{95}{128}V_{3}-\frac{33}{ 128}V_{1}-\frac{33}{128}V_{2}-\frac{33}{128}V_{0}-\frac{33}{128}V_{4}\)

Eq. (3) can be used to obtain Binet's formula of generalized fifth order Jacobsthal numbers. Generalized fifth order Jacobsthal numbers can be expressed, for all integers \(n,\) using Binet's formula

\begin{eqnarray*} { V}_{n} &{ =}&\frac{p_{1}\alpha ^{n}}{(\alpha -\beta )(\alpha -\gamma )(\alpha -\delta )(\alpha -\lambda )}{ + }\frac{p_{2}\beta ^{n}}{(\beta -\alpha )(\beta -\gamma )(\beta -\delta )(\beta -\lambda )}{ +}\frac{p_{3}\gamma ^{n}}{(\gamma -\alpha )(\gamma -\beta )(\gamma -\delta )(\gamma -\lambda )} \\ &&{ +}\frac{p_{4}\delta ^{n}}{(\delta -\alpha )(\delta -\beta )(\delta -\gamma )(\delta -\lambda )}{ +}\frac{p_{5}\lambda ^{n} }{(\lambda -\alpha )(\lambda -\beta )(\lambda -\gamma )(\lambda -\delta )} { ,} \end{eqnarray*} where \begin{align*} { p}_{1} &{ =}{ V}_{4}{ -(\beta +\gamma +\delta +\lambda )V}_{3}{ +(\beta \lambda +\beta \gamma +\lambda \gamma +\beta \delta +\lambda \delta +\gamma \delta )V}_{2}{ -(\beta \lambda \gamma +\beta \lambda \delta +\beta \gamma \delta +\lambda \gamma \delta )V}_{1} { +(\beta \lambda \gamma \delta )V}_{0}{ ,} \\ { p}_{2} &{ =}{ V}_{4}{ -(\alpha +\gamma +\delta +\lambda )V}_{3}{ +( \alpha \lambda +\alpha \gamma +\alpha \delta +\lambda \gamma +\lambda \delta +\gamma \delta )V}_{2}{ -(\alpha \lambda \gamma +\alpha \lambda \delta +\alpha \gamma \delta +\lambda \gamma \delta )V}_{1}{ +(\alpha \lambda \gamma \delta )V}_{0}{ ,} \\ { p}_{3} &{ =}{ V}_{4}{ -(\alpha +\beta +\delta +\lambda )V}_{3}{ +(\alpha \beta + \alpha \lambda +\beta \lambda +\alpha \delta +\beta \delta +\lambda \delta )V}_{2}{ -(\alpha \beta \lambda + \alpha \beta \delta +\alpha \lambda \delta +\beta \lambda \delta )V}_{1}{ +(\alpha \beta \lambda \delta )V}_{0} { ,} \\ { p}_{4} &{ =}{ V}_{4}{ -(\alpha +\beta +\gamma +\lambda )V}_{3}{ +(\alpha \beta + \alpha \lambda +\alpha \gamma +\beta \lambda +\beta \gamma +\lambda \gamma )V}_{2}{ -(\alpha \beta \lambda +\alpha \beta \gamma +\alpha \lambda \gamma +\beta \lambda \gamma )V}_{1}{ +(\alpha \beta \lambda \gamma )V}_{0}{ ,} \\ { p}_{5} &{ =}{ V}_{4}{ -(\alpha +\beta +\gamma +\delta )V}_{3}{ +(\alpha \beta +\alpha \gamma +\alpha \delta +\beta \gamma +\beta \delta +\gamma \delta )V}_{2}{ -(\alpha \beta \gamma + \alpha \beta \delta +\alpha \gamma \delta +\beta \gamma \delta )V}_{1}{ +(\alpha \beta \gamma \delta )V}_{0}{ .} \end{align*} As \(\{V_{n}\}\) is a fifth order recurrence sequence (difference equation), it's characteristic equation is The roots \(\alpha ,\beta ,\gamma ,\delta \) and \(\lambda \) of Eq. (10) are given by: \begin{eqnarray*} \alpha &=&2, \end{eqnarray*}\begin{eqnarray*} \beta &=&\frac{1}{4}(\sqrt{5}-1)+\frac{\sqrt{2\sqrt{5}+10}}{4}i, \\ \gamma &=&\frac{1}{4}(\sqrt{5}-1)-\frac{\sqrt{2\sqrt{5}+10}}{4}i, \\ \delta &=&-\frac{1}{4}(\sqrt{5}+1)+\frac{\sqrt{-2\sqrt{5}+10}}{4}i, \\ \lambda &=&-\frac{1}{4}(\sqrt{5}+1)-\frac{\sqrt{-2\sqrt{5}+10}}{4}i. \end{eqnarray*} Note that we have the following identities: \begin{eqnarray*} \alpha +\beta +\gamma +\delta +\lambda &=&1, \\ \alpha \beta + \alpha \lambda +\alpha \gamma +\beta \lambda +\alpha \delta +\beta \gamma +\lambda \gamma +\beta \delta +\lambda \delta +\gamma \delta &=&-1, \\ \alpha \beta \lambda +\alpha \beta \gamma +\alpha \lambda \gamma + \alpha \beta \delta +\alpha \lambda \delta +\beta \lambda \gamma +\alpha \gamma \delta +\beta \lambda \delta +\beta \gamma \delta +\lambda \gamma \delta &=&1, \\ \alpha \beta \lambda \gamma +\alpha \beta \lambda \delta +\alpha \beta \gamma \delta +\alpha \lambda \gamma \delta +\beta \lambda \gamma \delta &=&-1 ,\\ \alpha \beta \gamma \delta \lambda &=&2. \end{eqnarray*} Now we consider four special cases of the sequence \(\{V_{n}\}\). Fifth-order Jacobsthal sequence \(\{J_{n}\}_{n\geq 0}\), fifth order Jacobsthal-Lucas sequence \(\{j_{n}\}_{n\geq 0}\), adjusted fifth order Jacobsthal sequence \( \{S_{n}\}_{n\geq 0}\) and modified fifth order Jacobsthal-Lucas sequence \( \{R_{n}\}_{n\geq 0}\) are defined, respectively, by the fifth order recurrence relations The sequences \(\{J_{n}\}_{n\geq 0},\) \(\{j_{n}\}_{n\geq 0}\), \( \{S_{n}\}_{n\geq 0}\) and \(\{R_{n}\}_{n\geq 0}\) can be extended to negative subscripts by defining \begin{eqnarray*} J_{-n} &=&-\frac{1}{2}J_{-(n-1)}-\frac{1}{2}J_{-(n-2)}-\frac{1}{2} J_{-(n-3)}-J_{-(n-4)}+\frac{1}{2}J_{-(n-5)}, \\ j_{-n} &=&-\frac{1}{2}j_{-(n-1)}-\frac{1}{2}j_{-(n-2)}-\frac{1}{2}j_{-(n-3)}- \frac{1}{2}j_{-(n-4)}+\frac{1}{2}j_{-(n-5)}, \\ S_{-n} &=&-\frac{1}{2}S_{-(n-1)}-\frac{1}{2}S_{-(n-2)}-\frac{1}{2}S_{-(n-3)}- \frac{1}{2}S_{-(n-4)}+\frac{1}{2}S_{-(n-5)}, \\ R_{-n} &=&-\frac{1}{2}R_{-(n-1)}-\frac{1}{2}R_{-(n-2)}-\frac{1}{2}R_{-(n-3)}- \frac{1}{2}R_{-(n-4)}+\frac{1}{2}R_{-(n-5)}, \end{eqnarray*} for \(n=1,2,3,...\) respectively. Therefore, recurrences (11)-(14) hold for all integer \(n.\)

Next, we present the first few values of the fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers with positive and negative subscripts in the following Table 2:

Table 2. The first few values of the special fifth order numbers with positive and negative subscripts .

\(n\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)	\(12\)	\(13\)
\(J_{n}\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(4\)	\(9\)	\(17\)	\(33\)	\(65\)	\(   132\)	\(265\)	\(529\)	\(  1057\)
\(J_{-n}\)		\(-1\)	\(0\)	\(\frac{1}{2}\)	\(\frac{3}{4}\)	\(-\frac{1}{8}\)	\( -\frac{17}{16}\)	\(-\frac{1}{32}\)	\(\frac{31}{64}\)	\(\frac{95}{128}\)	\(- \frac{33}{256}\)	\(-\frac{545}{512}\)	\(  -\frac{33}{1024}\)	\( \frac{991}{2048}\)
\(j_{n}\)	\(2\)	\(1\)	\(5\)	\(10\)	\(20\)	\(40\)	\(77\)	\(157\)	\(314\)	\(628\)	\(  1256\)	\(  2509\)	\(  5021\)	\(   10042\)
\(j_{-n}\)		\(1\)	\(\frac{1}{2}\)	\(\frac{1}{4}\)	\(-\frac{11}{8}\)	\(\frac{ 13}{16}\)	\(\frac{13}{32}\)	\(\frac{13}{64}\)	\(\frac{13}{128}\)	\(-\frac{371 }{256}\)	\(\frac{397}{512}\)	\(\frac{397}{1024}\)	\(\frac{397}{2048}\)	\(   \frac{397}{4096}\)
\(S_{n}\)	\(  0\)	\(1\)	\(1\)	\(  2\)	\(  4\)	\(   8\)	\(17\)	\(33\)	\(66\)	\(132\)	\(264\)	\(529\)	\(1057\)	\(2114\)
\(S_{-n}\)		\(0\)	\(  0\)	\(0\)	\(\frac{1}{2}\)	\(-\frac{1}{4}\)	\(-\frac{1}{8}\)	\(-\frac{1}{16}\)	\(-\frac{1}{32}\)	\(\frac{31}{64}\)	\(- \frac{33}{128}\)	\(-\frac{33}{256}\)	\(-\frac{33}{512}\)	\(-\frac{33}{1024}\)
\(R_{n}\)	\(  5\)	\(1\)	\(  3\)	\(7\)	\(15\)	\(   36\)	\(  63\)	\(  127\)	\(  255\)	\(  511\)	\(1028\)	\(  2047\)	\(  4095\)	\(   8191\)
\(R_{-n}\)		\(-\frac{1}{2}\)	\(  -\frac{3}{4}\)	\(  - \frac{7}{8}\)	\(-\frac{15}{16}\)	\(  \frac{129}{32}\)	\(-\frac{63}{ 64}\)	\(-\frac{127}{128}\)	\(  -\frac{255}{256}\)	\(-\frac{511}{512 }\)	\(\frac{4097}{1024}\)	\(  -\frac{2047}{2048}\)	\(  - \frac{4095}{4096}\)	\(-\frac{8191}{8192}\)

For all integers \(n,\) Binet formulas of fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers are

\begin{eqnarray*} J_{n} &=&\frac{( \alpha ^{3}-\alpha -2)\alpha ^{n}}{(\alpha -\beta )(\alpha -\gamma )(\alpha -\delta )(\alpha -\lambda )}+\frac{ ( \beta ^{3}-\beta -2)\beta ^{n}}{(\beta -\alpha )(\beta -\gamma )(\beta -\delta )(\beta -\lambda )} \\ &&+\frac{( \gamma ^{3}-\gamma -2)\gamma ^{n}}{(\gamma -\alpha )(\gamma -\beta )(\gamma -\delta )(\gamma -\lambda )}+\frac{(\delta ^{3}-\delta -2)\delta ^{n}}{(\delta -\alpha )(\delta -\beta )(\delta -\gamma )(\delta -\lambda )} \\ &&+\frac{( \lambda ^{3}-\lambda -2)\lambda ^{n}}{(\lambda -\alpha )(\lambda -\beta )(\lambda -\gamma )(\lambda -\delta )}, \end{eqnarray*} \begin{eqnarray*} j_{n} &=&\frac{(\alpha ^{4}+4\alpha ^{3}+4\alpha ^{2}+4\alpha +4)\alpha ^{n-1}}{(\alpha -\beta )(\alpha -\gamma )(\alpha -\delta )(\alpha -\lambda )} +\frac{(\beta ^{4}+4\beta ^{3}+4\beta ^{2}+4\beta +4)\beta ^{n-1}}{(\beta -\alpha )(\beta -\gamma )(\beta -\delta )(\beta -\lambda )} \\ &&+\frac{(\gamma ^{4}+4\gamma ^{3}+4\gamma ^{2}+4\gamma +4)\gamma ^{n-1}}{ (\gamma -\alpha )(\gamma -\beta )(\gamma -\delta )(\gamma -\lambda )}+\frac{ (\delta ^{4}+4\delta ^{3}+4\delta ^{2}+4\delta +4)\delta ^{n-1}}{(\delta -\alpha )(\delta -\beta )(\delta -\gamma )(\delta -\lambda )} \\ &&+\frac{(\lambda ^{4}+4\lambda ^{3}+4\lambda ^{2}+4\lambda +4)\lambda ^{n-1} }{(\lambda -\alpha )(\lambda -\beta )(\lambda -\gamma )(\lambda -\delta )}, \end{eqnarray*} \begin{eqnarray*} S_{n} &=&\frac{\alpha ^{n+3}}{(\alpha -\beta )(\alpha -\gamma )(\alpha -\delta )(\alpha -\lambda )}+\frac{\beta ^{n+3}}{(\beta -\alpha )(\beta -\gamma )(\beta -\delta )(\beta -\lambda )} \\ &&+\frac{\gamma ^{n+3}}{(\gamma -\alpha )(\gamma -\beta )(\gamma -\delta )(\gamma -\lambda )}+\frac{\delta ^{n+3}}{(\delta -\alpha )(\delta -\beta )(\delta -\gamma )(\delta -\lambda )} \\ &&+\frac{\lambda ^{n+3}}{(\lambda -\alpha )(\lambda -\beta )(\lambda -\gamma )(\lambda -\delta )},\\ R_{n}&=&\alpha ^{n}+\beta ^{n}+\gamma ^{n}+\delta ^{n}+\lambda ^{n} \end{eqnarray*} respectively.

Binet formulas of fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers can be given in the following forms:

\begin{eqnarray*} J_{n} &=&\frac{4}{31}\alpha ^{n}-\frac{1}{155}((6\sqrt{5}+5)+2\sqrt{2}\sqrt{ \sqrt{5}+5}(6+\sqrt{5})i)\beta ^{n} \\ &&+\frac{1}{155}(-(6\sqrt{5}+5)+2\sqrt{2}\sqrt{\sqrt{5}+5}(6+\sqrt{5} )i)\gamma ^{n} \\ &&+\frac{1}{155}((6\sqrt{5}-5)+2\sqrt{2}\sqrt{5-\sqrt{5}}(\sqrt{5} -6)i)\delta ^{n} \\ &&+\frac{1}{155}((6\sqrt{5}-5)+2\sqrt{2}\sqrt{5-\sqrt{5}}(-\sqrt{5} +6)i)\lambda ^{n}, \end{eqnarray*} \begin{eqnarray*} j_{n} &=&\frac{38}{31}\alpha ^{n}+\frac{1}{1240}(12(20-7\sqrt{5})+\sqrt{ \sqrt{5}+5}(111\sqrt{2}+3\sqrt{10})i)\beta ^{n} \\ &&+\frac{1}{1240}(12(20-7\sqrt{5})-\sqrt{\sqrt{5}+5}(111\sqrt{2}+3\sqrt{10} )i)\gamma ^{n} \\ &&+\frac{1}{1240}(12(20+7\sqrt{5})+\sqrt{5-\sqrt{5}}(111\sqrt{2}-3\sqrt{10} )i)\delta ^{n} \\ &&+\frac{1}{1240}(12(20+7\sqrt{5})+\sqrt{5-\sqrt{5}}(-111\sqrt{2}+3\sqrt{10} )i)\lambda ^{n}, \end{eqnarray*} \begin{eqnarray*} S_{n} &=&\frac{8}{31}\alpha ^{n}+\frac{1}{1240}(4(-20+7\sqrt{5})-\sqrt{2} \sqrt{\sqrt{5}+5}(37i+i\sqrt{5}))\beta ^{n} \\ &&+\frac{1}{1240}(4(-20+7\sqrt{5})+\sqrt{2}\sqrt{\sqrt{5}+5}(37i+i\sqrt{5} ))\gamma ^{n} \end{eqnarray*}\begin{eqnarray*} &&+\frac{1}{1240}(-4(20+ 7\sqrt{5})+\sqrt{2}\sqrt{\sqrt{5}+5} (21-19\sqrt{5})i)\delta ^{n} \\ &&+\frac{1}{1240}(-4(20+ 7\sqrt{5})+\sqrt{2}\sqrt{\sqrt{5}+5} (-21+19\sqrt{5})i)\lambda ^{n},\\ R_{n}&=&\alpha ^{n}+\beta ^{n}+\gamma ^{n}+\delta ^{n}+\lambda ^{n}\,. \end{eqnarray*} Next, we give the ordinary generating function \(\sum\limits_{n=0}^{\infty }V_{n}x^{n}\) of the sequence \(V_{n}.\)

Lemma 2. Suppose that \(f_{V_{n}}(x)=\sum\limits_{n=0}^{ \infty }V_{n}x^{n}\) is the ordinary generating function of the generalized fifth order Jacobsthal sequence \(\{V_{n}\}_{n\geq 0}.\) Then, \(\sum\limits_{n=0}^{\infty }V_{n}x^{n}\) is given by \begin{equation*} \sum\limits_{n=0}^{\infty }{ V}_{n}{ x}^{n}=\frac{ { V}_{0}{ +(V}_{1}{ -V}_{0} { )x+(V}_{2}{ -V}_{1}{ -V}_{0} { )x}^{2}{ +(V}_{3}{ -V}_{2} { -V}_{1}{ -V}_{0}{ )x}^{3} { +(V}_{4}{ -V}_{3}{ -V}_{2} { -V}_{1}{ -V}_{0}{ )x}^{4}}{ 1-x-x^{2}-x^{3}-x^{4}-2x^{5}}\,. \end{equation*}

The previous Lemma gives the following results as particular examples: generating function of the fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas are \begin{eqnarray*} f_{J_{n}}(x) &=&\sum_{n=0}^{\infty }J_{n}x^{n}=\frac{x-x^{3}-2x^{4}}{ 1-x-x^{2}-x^{3}-x^{4}-2x^{5}}, \\ f_{j_{n}}(x) &=&\sum_{n=0}^{\infty }j_{n}x^{n}=\frac{2-x+2x^{2}+2x^{3}+2x^{4} }{1-x-x^{2}-x^{3}-x^{4}-2x^{5}}, \\ f_{S_{n}}(x) &=&\sum_{n=0}^{\infty }S_{n}x^{n}=\frac{x}{ 1-x-x^{2}-x^{3}-x^{4}-2x^{5}}, \\ f_{R_{n}}(x) &=&\sum_{n=0}^{\infty }R_{n}x^{n}=\frac{5-4x-3x^{2}-2x^{3}-x^{4} }{1-x-x^{2}-x^{3}-x^{4}-2x^{5}}, \end{eqnarray*} respectively.

2. Binomial transform of the generalized fifth order Jacobsthal sequence \(V_{n}\)

In [8], p. 137, Knuth introduced the idea of the binomial transform. Given a sequence of numbers \((a_{n})\), its binomial transform \(( \hat{a}_{n})\) may be defined by the rule \begin{equation*} \hat{a}_{n}=\sum\limits_{i=0}^{n}\binom{n}{i}a_{i},\text{ with inversion } a_{n}=\sum\limits_{i=0}^{n}\binom{n}{i}(-1)^{n-i}\hat{a}_{i}, \end{equation*} or, in the symmetric version \begin{equation*} \hat{a}_{n}=\sum\limits_{i=0}^{n}\binom{n}{i}(-1)^{i+1}a_{i},\text{ with inversion }a_{n}=\sum\limits_{i=0}^{n}\binom{n}{i}(-1)^{i+1}\hat{a}_{i}. \end{equation*} For more information on binomial transform, see, for example, [9,10,11,12] and references therein. For recent works on binomial transform of well-known sequences, see for example, [13,14,15,16,17,18,19,20,21,22,23,24,25].

In this section, we define the binomial transform of the generalized fifth order Jacobsthal sequence \(V_{n}\) and as special cases the binomial transform of the fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas sequences will be introduced.

Definition 1. The binomial transform of the generalized fifth order Jacobsthal sequence \( V_{n}\) is defined by \begin{equation*} b_{n}=\widehat{V}_{n}=\sum\limits_{i=0}^{n}\binom{n}{i}V_{i}. \end{equation*}

The few terms of \(b_{n}\) are \begin{eqnarray*} b_{0} &=&\sum\limits_{i=0}^{0}\binom{0}{i}V_{i}=V_{0}, \\ b_{1} &=&\sum\limits_{i=0}^{1}\binom{1}{i}V_{i}=V_{0}+V_{1}, \\ b_{2} &=&\sum\limits_{i=0}^{2}\binom{2}{i}V_{i}=V_{0}+2V_{1}+V_{2}, \\ b_{3} &=&\sum\limits_{i=0}^{3}\binom{3}{i}V_{i}=V_{0}+3V_{1}+3V_{2}+V_{3}, \\ b_{4} &=&\sum\limits_{i=0}^{4}\binom{4}{i} V_{i}=V_{0}+4V_{1}+6V_{2}+4V_{3}+V_{4}. \end{eqnarray*} Translated to matrix language, \(b_{n}\) has the nice (lower-triangular matrix) form \begin{equation*} \left( \begin{array}{c} b_{0} \\ b_{1} \\ b_{2} \\ b_{3} \\ b_{4} \\ \vdots \end{array} \right) =\left( \begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & \cdots \\ 1 & 1 & 0 & 0 & 0 & \cdots \\ 1 & 2 & 1 & 0 & 0 & \cdots \\ 1 & 3 & 3 & 1 & 0 & \cdots \\ 1 & 4 & 6 & 4 & 1 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array} \right) \left( \begin{array}{c} V_{0} \\ V_{1} \\ V_{2} \\ V_{3} \\ V_{4} \\ \vdots \end{array} \right) . \end{equation*} As special cases of \(b_{n}=\widehat{V}_{n}\), the binomial transforms of the fifth order Jacobsthal and fifth order Jacobsthal-Lucas sequences are defined as follows: The binomial transform of the fifth order Jacobsthal sequence \(J_{n}\) is \begin{equation*} \widehat{J}_{n}=\sum\limits_{i=0}^{n}\binom{n}{i}J_{i}, \end{equation*} and the binomial transform of the fifth order Jacobsthal-Lucas sequence \( j_{n}\) is \begin{equation*} \widehat{j}_{n}=\sum\limits_{i=0}^{n}\binom{n}{i}j_{i}, \end{equation*} The binomial transform of the adjusted fifth order Jacobsthal sequence \( S_{n} \) is \begin{equation*} \widehat{S}_{n}=\sum\limits_{i=0}^{n}\binom{n}{i}S_{i}, \end{equation*} and the binomial transform of the modified fifth order Jacobsthal-Lucas sequence \(R_{n}\) is \begin{equation*} \widehat{R}_{n}=\sum\limits_{i=0}^{n}\binom{n}{i}R_{i}. \end{equation*}

Lemma 3. For \(n\geq 0,\) the binomial transform of the generalized fifth order Jacobsthal sequence \(V_{n}\) satisfies the following relation: \begin{equation*} b_{n+1}=\sum\limits_{i=0}^{n}\binom{n}{i}(V_{i}+V_{i+1}). \end{equation*}

Proof. We use the following well-known identity: \begin{equation*} \binom{n+1}{i}=\binom{n}{i}+\binom{n}{i-1}. \end{equation*} Note also that \begin{equation*} \binom{n+1}{0}=\binom{n}{0}=1\text{ and }\binom{n}{n+1}=0. \end{equation*} Then \begin{eqnarray*} b_{n+1} &=&V_{0}+\sum\limits_{i=1}^{n+1}\binom{n+1}{i}V_{i} \\ &=&V_{0}+\sum\limits_{i=1}^{n+1}\binom{n}{i}V_{i}+\sum\limits_{i=1}^{n+1} \binom{n}{i-1}V_{i} \\ &=&V_{0}+\sum\limits_{i=1}^{n}\binom{n}{i}V_{i}+\sum\limits_{i=0}^{n}\binom{n }{i}V_{i+1} \\ &=&\sum\limits_{i=0}^{n}\binom{n}{i}V_{i}+\sum\limits_{i=0}^{n}\binom{n}{i} V_{i+1} \\ &=&\sum\limits_{i=0}^{n}\binom{n}{i}(V_{i}+V_{i+1}). \end{eqnarray*} This completes the proof.

Remark 1. From the Lemma 3, we see that \begin{equation*} b_{n+1}=b_{n}+\sum\limits_{i=0}^{n}\binom{n}{i}V_{i+1}. \end{equation*}

The following theorem gives recurrent relations of the binomial transform of the generalized fifth order Jacobsthal sequence.

Theorem 4. For \(n\geq 0,\) the binomial transform of the generalized fifth order Jacobsthal sequence \(V_{n}\) satisfies the following recurrence relation:

Proof. To show (15), writing \begin{equation*} b_{n+5}=r_{1}\times b_{n+4}+s_{1}\times b_{n+3}+t_{1}\times b_{n+2}+u_{1}\times b_{n+1}+v_{1}\times b_{n} \end{equation*} and taking the values \(n=0,1,2,3,4\) and then solving the system of equations \begin{eqnarray*} b_{5} &=&r_{1}\times b_{4}+s_{1}\times b_{3}+t_{1}\times b_{2}+u_{1}\times b_{1}+v_{1}\times b_{0} \\ b_{6} &=&r_{1}\times b_{5}+s_{1}\times b_{4}+t_{1}\times b_{3}+u_{1}\times b_{2}+v_{1}\times b_{1} \\ b_{7} &=&r_{1}\times b_{6}+s_{1}\times b_{5}+t_{1}\times b_{4}+u_{1}\times b_{3}+v_{1}\times b_{2} \\ b_{8} &=&r_{1}\times b_{7}+s_{1}\times b_{6}+t_{1}\times b_{5}+u_{1}\times b_{4}+v_{1}\times b_{3} \\ b_{9} &=&r_{1}\times b_{8}+s_{1}\times b_{7}+t_{1}\times b_{6}+u_{1}\times b_{5}+v_{1}\times b_{4} \end{eqnarray*} we find that \(r_{1}=6,s_{1}=-13,t_{1}=14,u_{1}=-7,v_{1}=3.\)

The sequence \(\{b_{n}\}_{n\geq 0}\) can be extended to negative subscripts by defining \begin{equation*} b_{-n}=\frac{7}{3}b_{-(n-1)}-\frac{14}{3}b_{-(n-2)}+\frac{13}{3} b_{-(n-3)}-2b_{-(n-4)}+\frac{1}{3}b_{-(n-5)} \end{equation*} for \(n=1,2,3,...\). Therefore, recurrence (15) holds for all integer \(n.\)

Note that the recurence relation (15) is independent from initial values. So,

\begin{eqnarray*} \widehat{J}_{n+5} &=&6\widehat{J}_{n+4}-13\widehat{J}_{n+3}+14\widehat{J} _{n+2}-7\widehat{J}_{n+1}+3\widehat{J}_{n}, \\ \widehat{j}_{n+5} &=&6\widehat{j}_{n+4}-13\widehat{j}_{n+3}+14\widehat{j} _{n+2}-7\widehat{j}_{n+1}+3\widehat{j}_{n}, \\ \widehat{S}_{n+5} &=&6\widehat{S}_{n+4}-13\widehat{S}_{n+3}+14\widehat{S} _{n+2}-7\widehat{S}_{n+1}+3\widehat{S}_{n}, \end{eqnarray*}\begin{eqnarray*} \widehat{R}_{n+5} &=&6\widehat{R}_{n+4}-13\widehat{R}_{n+3}+14\widehat{R} _{n+2}-7\widehat{R}_{n+1}+3\widehat{R}_{n}. \end{eqnarray*}

The first few terms of the binomial transform of the generalized fifth order Jacobsthal sequence with positive subscript and negative subscript are given in the following Table 3.

Table 3. A few binomial transform (terms) of the generalized fifth order Jacobsthal sequence.

\(n\)	\(b_{n}\)	\(b_{-n}\)
\(0\)	\(V_{0}\)	\(V_{0}\)
\(1\)	\(V_{0}+V_{1}\)	\(\frac{1}{3}\left(V_{0}-2V_{1}+V_{2}-2V_{3}+V_{4}\right) \)
\(2\)	\(V_{0}+2V_{1}+V_{2}\)	\(-\frac{1}{9}\left(11V_{0}+2V_{1}+2V_{2}+11V_{3}-7V_{4}\right) \)
\(3\)	\(V_{0}+3V_{1}+3V_{2}+V_{3}\)	\(-\frac{1}{27}\left(47V_{0}-34V_{1}+47V_{2}-7V_{3}-7V_{4}\right) \)
\(4\)	\(  V_{0}+4V_{1}+6V_{2}+4V_{3}+V_{4}\)	\(\frac{1}{81}\left(115V_{0}+115V_{1}-128V_{2}+277V_{3}-128V_{4}\right) \)
\(5\)	\(  3V_{0}+6V_{1}+11V_{2}+11V_{3}+6V_{4}\)	\(\frac{1}{243} \left( 1411V_{0}-533V_{1}+682V_{2}+682V_{3}-533V_{4}\right) \)
\(6\)	\(  15V_{0}+15V_{1}+23V_{2}+28V_{3}+23V_{4}\)	\(\frac{1}{729} \left( 1411V_{0}-4421V_{1}+5056V_{2}-4421V_{3}+1411V_{4}\right) \)
\(7\)	\(  61V_{0}+53V_{1}+61V_{2}+74V_{3}+74V_{4}\)	\(-\frac{1}{2187 }\left( 29207V_{0}+776V_{1}+776V_{2}+29207V_{3}-16720V_{4}\right) \)

The first few terms of the binomial transform numbers of the fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas sequences with positive subscript and negative subscript are given in the following Table 4.

Table 4. A few binomial transform (terms).

\(n\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\widehat{J}_{n}\)	\(0\)	\(1\)	\(3\)	\(7\)	\(  15\)	\(34\)	\(89\)	\(262\)	\(807\)	\(2489\)	\(7590\)	\(22914\)
\(\widehat{J}_{-n}\)		\(-\frac{2}{3}\)	\(-\frac{8}{9}\)	\(\frac{1}{27}\)	\( \frac{136}{81}\)	\(\frac{298}{243}\)	\(-\frac{2375}{729}\)	\(-\frac{14\,039}{ 2187}\)	\(\frac{14\,392}{6561}\)	\(\frac{375\,247}{19\,683}\)	\(\frac{ 788\,590}{59\,049}\)	\(-\frac{6474\,437}{177\,147}\)
\(\widehat{j}_{n}\)	\(2\)	\(3\)	\(9\)	\(30\)	\(96\)	\(297\)	\(900\)	\(   2700\)	\(8076\)	\(24165\)	\(72393\)	\(217077\)
\(\widehat{j}_{-n}\)		\(\frac{5}{3}\)	\(-\frac{4}{9}\)	\(-\frac{85}{27}\)	\( -\frac{85}{81}\)	\(\frac{1859}{243}\)	\(\frac{7691}{729}\)	\(-\frac{20740}{ 2187}\)	\(-\frac{243814}{6561}\)	\(-\frac{243814}{19683}\)	\(\frac{5011547}{ 59049}\)	\(\frac{20777630}{177147}\)
\(\widehat{S}_{n}\)	\(0\)	\(1\)	\(3\)	\(8\)	\(22\)	\(63\)	\(186\)	\(   558\)	\(1682\)	\(5067\)	\(15235\)	\(45739\)
\(\widehat{S}_{-n}\)		\(-\frac{1}{3}\)	\(\frac{2}{9}\)	\(\frac{29}{27}\)	\( \frac{29}{81}\)	\(-\frac{619}{243}\)	\(-\frac{2563}{729}\)	\(\frac{6914}{2187 }\)	\(\frac{81272}{6561}\)	\(\frac{81272}{19683}\)	\(-\frac{1670515}{59049}\)	\(-\frac{6925876}{177147}\)
\(\widehat{R}_{n}\)	\(5\)	\(6\)	\(10\)	\(24\)	\(70\)	\(221\)	\( 700 \)	\(  2169\)	\(6590\)	\(19806\)	\(59295\)	\(177469\)
\(\widehat{R}_{-n}\)		\(\frac{7}{3}\)	\(-\frac{35}{9}\)	\(-\frac{188}{27}\)	\(\frac{325}{81}\)	\(\frac{5347}{243}\)	\(\frac{8020}{729}\)	\(-\frac{102788 }{2187}\)	\(-\frac{498635}{6561}\)	\(\frac{925102}{19683}\)	\(\frac{14526055 }{59049}\)	\(\frac{21789082}{177147}\)

Eq. (3) can be used to obtain Binet's formula of the binomial transform of generalized fifth order Jacobsthal numbers. Binet's formula of the binomial transform of generalized fifth order Jacobsthal numbers can be given as

where \begin{eqnarray*} { C}_{1} &{ =}&{ b}_{4}{ -(\theta _{2}+\theta _{3}+\theta _{4}+\theta _{5})b}_{3}{ +(\theta _{2}\theta _{5}+\theta _{2}\theta _{3}+\theta _{5}\theta _{3}+\theta _{2} \theta _{4}+\theta _{5}\theta _{4}+\theta _{3}\theta _{4})b}_{2} \\ &&{ -(\theta _{2}\theta _{5}\theta _{3}+\theta _{2}\theta _{5}\theta _{4}+\theta _{2} \theta _{3}\theta _{4}+\theta _{5}\theta _{3}\theta _{4})b}_{1}{ +(\theta _{2}\theta _{5}\theta _{3}\theta _{4})b}_{0}{ ,} \\ { C}_{2} &{ =}&{ b}_{4}{ -(\theta _{1}+\theta _{3}+\theta _{4}+\theta _{5})b}_{3}{ +( \theta _{1}\theta _{5}+\theta _{1}\theta _{3}+\theta _{1}\theta _{4}+\theta _{5}\theta _{3}+\theta _{5}\theta _{4}+\theta _{3}\theta _{4})b}_{2} \\ &&{ -(\theta _{1}\theta _{5}\theta _{3}+\theta _{1}\theta _{5}\theta _{4}+\theta _{1}\theta _{3}\theta _{4}+\theta _{5}\theta _{3}\theta _{4})b}_{1}{ +(\theta _{1}\theta _{5}\theta _{3} \theta _{4})b}_{0}{ ,} \end{eqnarray*}\begin{eqnarray*} { C}_{3} &{ =}&{ b}_{4}{ -(\theta _{1}+\theta _{2}+\theta _{4}+\theta _{5})b}_{3}{ +(\theta _{1}\theta _{2}+ \theta _{1}\theta _{5}+\theta _{2}\theta _{5}+\theta _{1}\theta _{4}+\theta _{2} \theta _{4}+\theta _{5}\theta _{4})b}_{2} \\ &&{ -(\theta _{1}\theta _{2}\theta _{5}+ \theta _{1}\theta _{2}\theta _{4}+\theta _{1}\theta _{5}\theta _{4}+\theta _{2}\theta _{5}\theta _{4})b}_{1}{ +(\theta _{1}\theta _{2}\theta _{5}\theta _{4})b}_{0}{ ,} \\ { C}_{4} &{ =}&{ b}_{4}{ -(\theta _{1}+\theta _{2}+\theta _{3}+\theta _{5})b}_{3}{ +(\theta _{1}\theta _{2}+ \theta _{1}\theta _{5}+\theta _{1}\theta _{3}+\theta _{2}\theta _{5}+\theta _{2}\theta _{3}+\theta _{5}\theta _{3})b}_{2} \\ &&{ -(\theta _{1}\theta _{2}\theta _{5}+\theta _{1}\theta _{2}\theta _{3}+\theta _{1}\theta _{5}\theta _{3}+\theta _{2}\theta _{5}\theta _{3})b}_{1}{ +(\theta _{1}\theta _{2}\theta _{5}\theta _{3})b}_{0}{ ,} \\ { C}_{5} &{ =}&{ b}_{4}{ -(\theta _{1}+\theta _{2}+\theta _{3}+\theta _{4})b}_{3}{ +(\theta _{1}\theta _{2}+\theta _{1}\theta _{3}+\theta _{1}\theta _{4}+\theta _{2}\theta _{3}+\theta _{2} \theta _{4}+\theta _{3}\theta _{4})b}_{2} \\ &&{ -(\theta _{1}\theta _{2}\theta _{3}+ \theta _{1}\theta _{2}\theta _{4}+\theta _{1}\theta _{3}\theta _{4}+\theta _{2} \theta _{3}\theta _{4})b}_{1}{ +(\theta _{1}\theta _{2}\theta _{3}\theta _{4})b}_{0}{ .} \end{eqnarray*} Here, \(\theta _{1},\theta _{2},\theta _{3},\theta _{4}\) and \(\theta _{5}\) are the roots of the equation \begin{equation*} x^{5}-6x^{4}+13x^{3}-14x^{2}+7x-3=\left( x-3\right) \left( x^{4}-3x^{3}+4x^{2}-2x+1\right) =0. \end{equation*} Moreover, the approximate value of \(\theta _{1},\theta _{2},\theta _{3},\theta _{4}\) and \(\theta _{5}\) are given by \begin{eqnarray*} \theta _{1} &=&3, \\ \theta _{2} &=&1.30901699437495+0.951056516295154i, \\ \theta _{3} &=&1.30901699437495-0.951056516295154i, \\ \theta _{4} &=&0.190983005625053\,+0.587785252292473i, \\ \theta _{5} &=&0.190983005625053\,-0.587785252292473i. \end{eqnarray*} Note that \begin{eqnarray*} \theta _{1}+\theta _{2}+\theta _{3}+\theta _{4}+\theta _{5} &=&6, \\ \theta _{1}\theta _{2}+ \theta _{1}\theta _{5}+\theta _{1}\theta _{3}+\theta _{2}\theta _{5}+\theta _{1}\theta _{4}+\theta _{2}\theta _{3}+\theta _{5}\theta _{3}+\theta _{2} \theta _{4}+\theta _{5}\theta _{4}+\theta _{3}\theta _{4} &=&13 ,\\ \theta _{1}\theta _{2}\theta _{5}+\theta _{1}\theta _{2}\theta _{3}+\theta _{1}\theta _{5}\theta _{3}+ \theta _{1}\theta _{2}\theta _{4}+\theta _{1}\theta _{5}\theta _{4}+\theta _{2}\theta _{5}\theta _{3}+\theta _{1}\theta _{3}\theta _{4}+\theta _{2}\theta _{5}\theta _{4}+\theta _{2} \theta _{3}\theta _{4}+\theta _{5}\theta _{3}\theta _{4} &=&14 ,\\ \theta _{1}\theta _{2}\theta _{5}\theta _{3}+\theta _{1}\theta _{2}\theta _{5}\theta _{4}+\theta _{1}\theta _{2}\theta _{3}\theta _{4}+\theta _{1}\theta _{5}\theta _{3} \theta _{4}+\theta _{2}\theta _{5}\theta _{3}\theta _{4} &=&7, \\ \theta _{1}\theta _{2}\theta _{3}\theta _{4}\theta _{5} &=&3. \end{eqnarray*} For all integers \(n,\) (Binet's formulas of) binomial transforms of fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers (using initial conditions in (16)) can be expressed using Binet's formulas as \begin{eqnarray*} \widehat{J}_{n} &=&\frac{(\theta _{1}^{3}-3\theta _{1}^{2}+2\theta _{1}-2)\theta _{1}^{n}}{(\theta _{1}-\theta _{2})(\theta _{1}-\theta _{3})(\theta _{1}-\theta _{4})(\theta _{1}-\theta _{5})}+\frac{(\theta _{2}^{3}-3\theta _{2}^{2}+2\theta _{2}-2)\theta _{2}^{n}}{(\theta _{2}-\theta _{1})(\theta _{2}-\theta _{3})(\theta _{2}-\theta _{4})(\theta _{2}-\theta _{5})} \\ &&+\frac{(\theta _{3}^{3}-3\theta _{3}^{2}+2\theta _{3}-2)\theta _{3}^{n}}{ (\theta _{3}-\theta _{1})(\theta _{3}-\theta _{2})(\theta _{3}-\theta _{4})(\theta _{3}-\theta _{5})}+\frac{(\theta _{4}^{3}-3\theta _{4}^{2}+2\theta _{4}-2)\theta _{4}^{n}}{(\theta _{4}-\theta _{1})(\theta _{4}-\theta _{2})(\theta _{4}-\theta _{3})(\theta _{4}-\theta _{5})} \\ &&+\frac{(\theta _{5}^{3}-3\theta _{5}^{2}+2\theta _{5}-2)\theta _{5}^{n}}{ (\theta _{5}-\theta _{1})(\theta _{5}-\theta _{2})(\theta _{5}-\theta _{3})(\theta _{5}-\theta _{4})}, \end{eqnarray*} \begin{eqnarray*} \widehat{j}_{n} &=&\frac{(2\theta _{1}^{4}-9\theta _{1}^{3}+17\theta _{1}^{2}-13\theta _{1}+5)\theta _{1}^{n}}{(\theta _{1}-\theta _{2})(\theta _{1}-\theta _{3})(\theta _{1}-\theta _{4})(\theta _{1}-\theta _{5})}+\frac{ (2\theta _{2}^{4}-9\theta _{2}^{3}+17\theta _{2}^{2}-13\theta _{2}+5)\theta _{2}^{n}}{(\theta _{2}-\theta _{1})(\theta _{2}-\theta _{3})(\theta _{2}-\theta _{4})(\theta _{2}-\theta _{5})} \\ &&+\frac{(2\theta _{3}^{4}-9\theta _{3}^{3}+17\theta _{3}^{2}-13\theta _{3}+5)\theta _{3}^{n}}{(\theta _{3}-\theta _{1})(\theta _{3}-\theta _{2})(\theta _{3}-\theta _{4})(\theta _{3}-\theta _{5})}+\frac{(2\theta _{4}^{4}-9\theta _{4}^{3}+17\theta _{4}^{2}-13\theta _{4}+5)\theta _{4}^{n}}{ (\theta _{4}-\theta _{1})(\theta _{4}-\theta _{2})(\theta _{4}-\theta _{3})(\theta _{4}-\theta _{5})} \\ &&+\frac{(2\theta _{5}^{4}-9\theta _{5}^{3}+17\theta _{5}^{2}-13\theta _{5}+5)\theta _{5}^{n}}{(\theta _{5}-\theta _{1})(\theta _{5}-\theta _{2})(\theta _{5}-\theta _{3})(\theta _{5}-\theta _{4})}, \end{eqnarray*} \begin{eqnarray*} \widehat{S}_{n} &=&\frac{(\theta _{1}-1)^{3}\theta _{1}^{n}}{(\theta _{1}-\theta _{2})(\theta _{1}-\theta _{3})(\theta _{1}-\theta _{4})(\theta _{1}-\theta _{5})}{ +}\frac{(\theta _{2}-1)^{3}\theta _{2}^{n}}{ (\theta _{2}-\theta _{1})(\theta _{2}-\theta _{3})(\theta _{2}-\theta _{4})(\theta _{2}-\theta _{5})} \\ &&{ +}\frac{(\theta _{3}-1)^{3}\theta _{3}^{n}}{(\theta _{3}-\theta _{1})(\theta _{3}-\theta _{2})(\theta _{3}-\theta _{4})(\theta _{3}-\theta _{5})}{ +}\frac{(\theta _{4}-1)^{3}\theta _{4}^{n}}{ (\theta _{4}-\theta _{1})(\theta _{4}-\theta _{2})(\theta _{4}-\theta _{3})(\theta _{4}-\theta _{5})} \end{eqnarray*}\begin{eqnarray*} &&{ +}\frac{(\theta _{5}-1)^{3}\theta _{5}^{n}}{(\theta _{5}-\theta _{1})(\theta _{5}-\theta _{2})(\theta _{5}-\theta _{3})(\theta _{5}-\theta _{4})},\\ \widehat{R}_{n}&=&\theta _{1}^{n}+\theta _{2}^{n}+\theta _{3}^{n}+\theta _{4}^{n}+\theta _{5}^{n}, \end{eqnarray*} respectively.

3. Generating functions and obtaining Binet formula of binomial transform from generating function

The generating function of the binomial transform of the generalized fifth order Jacobsthal sequence \(V_{n}\) is a power series centered at the origin whose coefficients are the binomial transform of the generalized fifth order Jacobsthal sequence.

Next, we give the ordinary generating function \(f_{b_{n}}(x)=\sum \limits_{n=0}^{\infty }b_{n}x^{n}\) of the sequence \(b_{n}.\)

Lemma 4. Suppose that \(f_{b_{n}}(x)=\sum\limits_{n=0}^{\infty }b_{n}x^{n}\) is the ordinary generating function of the binomial transform of the generalized fifth order Jacobsthal sequence \(\{V_{n}\}_{n\geq 0}.\) Then, \(f_{b_{n}}(x)\) is given by

Proof. Using Lemma 1, we obtain \begin{eqnarray*} f_{b_{n}}(x) &=&\frac{{ b}_{0}{ +(b}_{1} { -6b}_{0}{ )x+(b}_{2}{ -6b}_{1}+13 { b}_{0}{ )x}^{2}{ +(b}_{3} { -6b}_{2}+13{ b}_{1}{ -14b}_{0} { )x}^{3}{ +(b}_{4}{ -6b}_{3}+13 { b}_{2}{ -14b}_{1}+7{ b}_{0} { )x}^{4}}{1-6x+13x^{2}-14x^{3}+7x^{4}-3x^{5}} \\ &=&\frac{V_{0}{ +(V_{1}-5V_{0})x+(8V_{0}-4V_{1}+V_{2})x}^{2} { +(4V_{1}-6V_{0}-3V_{2}+V_{3})x}^{3}{ +(V_{0}-2V_{1}+V_{2}-2V_{3}+V_{4})x}^{4}}{1-6x+13x^{2}-14x^{3}+7x^{4}-3x^{5}}\,, \end{eqnarray*} where \begin{eqnarray*} b_{0} &=&V_{0}, \\ b_{1} &=&V_{0}+V_{1}, \\ b_{2} &=&V_{0}+2V_{1}+V_{2}, \\ b_{3} &=&V_{0}+3V_{1}+3V_{2}+V_{3}, \\ b_{4} &=&V_{0}+4V_{1}+6V_{2}+4V_{3}+V_{4}. \end{eqnarray*}

Note that P. Barry shows in [26] that if \(A(x)\) is the generating function of the sequence \(\{a_{n}\},\) then \begin{equation*} S(x)=\frac{1}{1-x}A\left(\frac{x}{1-x}\right) \end{equation*} is the generating function of the sequence \(\{b_{n}\}\) with \( b_{n}=\sum\limits_{i=0}^{n}\binom{n}{i}a_{i}.\) In our case, since \begin{equation*} A(x)=\frac{{ V}_{0}{ +(V}_{1}{ -V}_{0} { )x+(V}_{2}{ -V}_{1}{ -V}_{0} { )x}^{2}{ +(V}_{3}{ -V}_{2} { -V}_{1}{ -V}_{0}{ )x}^{3} { +(V}_{4}{ -V}_{3}{ -V}_{2} { -V}_{1}{ -V}_{0}{ )x}^{4}}{ 1-x-x^{2}-x^{3}-x^{4}-2x^{5}},\text{ see Lemma 2,} \end{equation*} we obtain \begin{eqnarray*} S(x) &=&\frac{1}{1-x}A(\frac{x}{1-x}) \\ &=&\frac{V_{0}{ +(V_{1}-5V_{0})x+(8V_{0}-4V_{1}+V_{2})x}^{2} { +(4V_{1}-6V_{0}-3V_{2}+V_{3})x}^{3}{ +(V_{0}-2V_{1}+V_{2}-2V_{3}+V_{4})x}^{4}}{1-6x+13x^{2}-14x^{3}+7x^{4}-3x^{5}}\,. \end{eqnarray*} The Lemma 4 gives the following results as particular examples.

Corollary 3. Generating functions of the binomial transform of the fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers are \begin{eqnarray*} \sum_{n=0}^{\infty }\widehat{J}_{n}x^{n} &=&\frac{x-3x^{2}+2x^{3}-2x^{4}}{ 1-6x+13x^{2}-14x^{3}+7x^{4}-3x^{5}}\,, \\ \sum_{n=0}^{\infty }\widehat{j}_{n}x^{n} &=&\frac{2-9x+17x^{2}-13x^{3}+5x^{4} }{1-6x+13x^{2}-14x^{3}+7x^{4}-3x^{5}} \,,\\ \sum_{n=0}^{\infty }\widehat{S}_{n}x^{n} &=&\frac{x-3x^{2}+3x^{3}-x^{4}}{ 1-6x+13x^{2}-14x^{3}+7x^{4}-3x^{5}} \,,\\ \sum_{n=0}^{\infty }\widehat{R}_{n}x^{n} &=&\frac{ 5-24x+39x^{2}-28x^{3}+7x^{4}}{1-6x+13x^{2}-14x^{3}+7x^{4}-3x^{5}}\,. \end{eqnarray*} respectively.

4. Simson formulas

There is a well-known Simson Identity (formula) for Fibonacci sequence \( \{F_{n}\}\), namely, \begin{equation*} F_{n+1}F_{n-1}-F_{n}^{2}=(-1)^{n}\,, \end{equation*} which was derived first by R. Simson in 1753 and it is now called as Cassini Identity (formula) as well. This can be written in the form \begin{equation*} \left\vert \begin{array}{cc} F_{n+1} & F_{n} \\ F_{n} & F_{n-1} \end{array} \right\vert =(-1)^{n}. \end{equation*} The following theorem gives generalization of this result to the generalized Pentanacci sequence \(\{W_{n}\}.\)

Theorem 5.(Simson formula of generalized Pentanacci numbers) For all integers \(n,\) we have

Proof. Eq. (18) is given in [27], Theorem 3.1.

Taking \(\{W_{n}\}=\{b_{n}\}\) in the above theorem and considering \( b_{n+5}=6b_{n+4}-13b_{n+3}+14b_{n+2}-7b_{n+1}+3b_{n},\) \( r=6,s=-13,t=14,u=-7,v=3,\) we have the following proposition.

Proposition 1. For all integers \(n,\) Simson formula of binomial transforms of generalized fifth order Jacobsthal numbers is given as \begin{equation*} \left\vert \begin{array}{ccccc} b_{n+4} & b_{n+3} & b_{n+2} & b_{n+1} & b_{n} \\ b_{n+3} & b_{n+2} & b_{n+1} & b_{n} & b_{n-1} \\ b_{n+2} & b_{n+1} & b_{n} & b_{n-1} & b_{n-2} \\ b_{n+1} & b_{n} & b_{n-1} & b_{n-2} & b_{n-3} \\ b_{n} & b_{n-1} & b_{n-2} & b_{n-3} & b_{n-4} \end{array} \right\vert =3^{n}\left\vert \begin{array}{ccccc} b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \\ b_{3} & b_{2} & b_{1} & b_{0} & b_{-1} \\ b_{2} & b_{1} & b_{0} & b_{-1} & b_{-2} \\ b_{1} & b_{0} & b_{-1} & b_{-2} & b_{-3} \\ b_{0} & b_{-1} & b_{-2} & b_{-3} & b_{-4} \end{array} \right\vert . \end{equation*}

The Proposition 1 gives the following results as particular examples.

Corollary 4. For all integers \(n,\) Simson formula of binomial transforms of the fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers are given as \begin{eqnarray*} \left\vert \begin{array}{ccccc} \widehat{J}_{n+4} & \widehat{J}_{n+3} & \widehat{J}_{n+2} & \widehat{J}_{n+1} & \widehat{J}_{n} \\ \widehat{J}_{n+3} & \widehat{J}_{n+2} & \widehat{J}_{n+1} & \widehat{J}_{n} & \widehat{J}_{n-1} \\ \widehat{J}_{n+2} & \widehat{J}_{n+1} & \widehat{J}_{n} & \widehat{J}_{n-1} & \widehat{J}_{n-2} \\ \widehat{J}_{n+1} & \widehat{J}_{n} & \widehat{J}_{n-1} & \widehat{J}_{n-2} & \widehat{J}_{n-3} \\ \widehat{J}_{n} & \widehat{J}_{n-1} & \widehat{J}_{n-2} & \widehat{J}_{n-3} & \widehat{J}_{n-4} \end{array} \right\vert &=&44\times 3^{n-4}, \\ \left\vert \begin{array}{ccccc} \widehat{j}_{n+4} & \widehat{j}_{n+3} & \widehat{j}_{n+2} & \widehat{j}_{n+1} & \widehat{j}_{n} \\ \widehat{j}_{n+3} & \widehat{j}_{n+2} & \widehat{j}_{n+1} & \widehat{j}_{n} & \widehat{j}_{n-1} \\ \widehat{j}_{n+2} & \widehat{j}_{n+1} & \widehat{j}_{n} & \widehat{j}_{n-1} & \widehat{j}_{n-2} \\ \widehat{j}_{n+1} & \widehat{j}_{n} & \widehat{j}_{n-1} & \widehat{j}_{n-2} & \widehat{j}_{n-3} \\ \widehat{j}_{n} & \widehat{j}_{n-1} & \widehat{j}_{n-2} & \widehat{j}_{n-3} & \widehat{j}_{n-4} \end{array} \right\vert &=&38\times 3^{n}, \\ \left\vert \begin{array}{ccccc} \widehat{S}_{n+4} & \widehat{S}_{n+3} & \widehat{S}_{n+2} & \widehat{S}_{n+1} & \widehat{S}_{n} \\ \widehat{S}_{n+3} & \widehat{S}_{n+2} & \widehat{S}_{n+1} & \widehat{S}_{n} & \widehat{S}_{n-1} \\ \widehat{S}_{n+2} & \widehat{S}_{n+1} & \widehat{S}_{n} & \widehat{S}_{n-1} & \widehat{S}_{n-2} \\ \widehat{S}_{n+1} & \widehat{S}_{n} & \widehat{S}_{n-1} & \widehat{S}_{n-2} & \widehat{S}_{n-3} \\ \widehat{S}_{n} & \widehat{S}_{n-1} & \widehat{S}_{n-2} & \widehat{S}_{n-3} & \widehat{S}_{n-4} \end{array} \right\vert &=&8\times 3^{n-4}, \\ \left\vert \begin{array}{ccccc} \widehat{R}_{n+4} & \widehat{R}_{n+3} & \widehat{R}_{n+2} & \widehat{R}_{n+1} & \widehat{R}_{n} \\ \widehat{R}_{n+3} & \widehat{R}_{n+2} & \widehat{R}_{n+1} & \widehat{R}_{n} & \widehat{R}_{n-1} \\ \widehat{R}_{n+2} & \widehat{R}_{n+1} & \widehat{R}_{n} & \widehat{R}_{n-1} & \widehat{R}_{n-2} \\ \widehat{R}_{n+1} & \widehat{R}_{n} & \widehat{R}_{n-1} & \widehat{R}_{n-2} & \widehat{R}_{n-3} \\ \widehat{R}_{n} & \widehat{R}_{n-1} & \widehat{R}_{n-2} & \widehat{R}_{n-3} & \widehat{R}_{n-4} \end{array} \right\vert &=&120125\times 3^{n-4}, \end{eqnarray*} respectively.

5. Some identities

In this section, we obtain some identities of binomial transforms of fifth order Jacobsthal, fifth order Jacobsthal-Lucas, adjusted fifth order Jacobsthal and modified fifth order Jacobsthal-Lucas numbers. First, we present a few basic relations between \(\{\widehat{J}_{n}\}\) and \(\{\widehat{j }_{n}\}\).

Lemma 5. The following equalities are true: \begin{eqnarray*} 171\widehat{J}_{n} &=&-22\widehat{j}_{n+6}+123\widehat{j}_{n+5}-259\widehat{j }_{n+4}+329\widehat{j}_{n+3}-193\widehat{j}_{n+2}, \\ 57\widehat{J}_{n} &=&-3\widehat{j}_{n+5}+9\widehat{j}_{n+4}+7\widehat{j} _{n+3}-13\widehat{j}_{n+2}-22\widehat{j}_{n+1}, \\ 57\widehat{J}_{n} &=&-9\widehat{j}_{n+4}+46\widehat{j}_{n+3}-55\widehat{j} _{n+2}-\widehat{j}_{n+1}-9\widehat{j}_{n}, \\ 57\widehat{J}_{n} &=&-8\widehat{j}_{n+3}+62\widehat{j}_{n+2}-127\widehat{j} _{n+1}+54\widehat{j}_{n}-27\widehat{j}_{n-1}, \\ 57\widehat{J}_{n} &=&14\widehat{j}_{n+2}-23\widehat{j}_{n+1}-58\widehat{j} _{n}+29\widehat{j}_{n-1}-24\widehat{j}_{n-2}, \end{eqnarray*} and \begin{eqnarray*} 198\widehat{j}_{n} &=&5\widehat{J}_{n+6}-285\widehat{J}_{n+5}+1532\widehat{J} _{n+4}-2764\widehat{J}_{n+3}+2003\widehat{J}_{n+2}, \\ 66\widehat{j}_{n} &=&-85\widehat{J}_{n+5}+489\widehat{J}_{n+4}-898\widehat{J} _{n+3}+656\widehat{J}_{n+2}+5\widehat{J}_{n+1}, \\ 22\widehat{j}_{n} &=&-7\widehat{J}_{n+4}+69\widehat{J}_{n+3}-178\widehat{J} _{n+2}+200\widehat{J}_{n+1}-85\widehat{J}_{n}, \\ 22\widehat{j}_{n} &=&27\widehat{J}_{n+3}-87\widehat{J}_{n+2}+102\widehat{J} _{n+1}-36\widehat{J}_{n}-21\widehat{J}_{n-1}, \\ 22\widehat{j}_{n} &=&75\widehat{J}_{n+2}-249\widehat{J}_{n+1}+342\widehat{J} _{n}-210\widehat{J}_{n-1}+81\widehat{J}_{n-2}. \end{eqnarray*}

Proof. Writing \begin{equation*} \widehat{J}_{n}=a\times \widehat{j}_{n+6}+b\times \widehat{j}_{n+5}+c\times \widehat{j}_{n+4}+d\times \widehat{j}_{n+3}+e\times \widehat{j}_{n+2} \end{equation*} and solving the system of equations \begin{eqnarray*} \widehat{J}_{0} &=&a\times \widehat{j}_{6}+b\times \widehat{j}_{5}+c\times \widehat{j}_{4}+d\times \widehat{j}_{3}+e\times \widehat{j}_{2} \\ \widehat{J}_{1} &=&a\times \widehat{j}_{7}+b\times \widehat{j}_{6}+c\times \widehat{j}_{5}+d\times \widehat{j}_{4}+e\times \widehat{j}_{3} \\ \widehat{J}_{2} &=&a\times \widehat{j}_{8}+b\times \widehat{j}_{7}+c\times \widehat{j}_{6}+d\times \widehat{j}_{5}+e\times \widehat{j}_{4} \\ \widehat{J}_{3} &=&a\times \widehat{j}_{9}+b\times \widehat{j}_{8}+c\times \widehat{j}_{7}+d\times \widehat{j}_{6}+e\times \widehat{j}_{5} \\ \widehat{J}_{4} &=&a\times \widehat{j}_{10}+b\times \widehat{j}_{9}+c\times \widehat{j}_{8}+d\times \widehat{j}_{7}+e\times \widehat{j}_{6} \end{eqnarray*} we find that \(a=-\frac{22}{171},b=\frac{41}{57},c=-\frac{259}{171},d=\frac{ 329}{171},e=-\frac{193}{171}.\) The other equalities can be proved similarly.

Now, we give a few basic relations between \(\{\widehat{J}_{n}\}\) and \(\{ \widehat{S}_{n}\}\).

Lemma 6. The following equalities are true: \begin{eqnarray*} 18\widehat{J}_{n} &=&7\widehat{S}_{n+6}-39\widehat{S}_{n+5}+82\widehat{S} _{n+4}-104\widehat{S}_{n+3}+61\widehat{S}_{n+2}, \\ 6\widehat{J}_{n} &=&\widehat{S}_{n+5}-3\widehat{S}_{n+4}-2\widehat{S}_{n+3}+4 \widehat{S}_{n+2}+7\widehat{S}_{n+1}, \\ 2\widehat{J}_{n} &=&\widehat{S}_{n+4}-5\widehat{S}_{n+3}+6\widehat{S}_{n+2}+ \widehat{S}_{n}, \\ 2\widehat{J}_{n} &=&\widehat{S}_{n+3}-7\widehat{S}_{n+2}+14\widehat{S} _{n+1}-6\widehat{S}_{n}+3\widehat{S}_{n-1}, \\ 2\widehat{J}_{n} &=&-\widehat{S}_{n+2}+\widehat{S}_{n+1}+8\widehat{S}_{n}-4 \widehat{S}_{n-1}+3\widehat{S}_{n-2}, \end{eqnarray*} and \begin{eqnarray*} 99\widehat{S}_{n} &=&\widehat{J}_{n+6}+42\widehat{J}_{n+5}-248\widehat{J} _{n+4}+457\widehat{J}_{n+3}-332\widehat{J}_{n+2}, \\ 33\widehat{S}_{n} &=&16\widehat{J}_{n+5}-87\widehat{J}_{n+4}+157\widehat{J} _{n+3}-113\widehat{J}_{n+2}+\widehat{J}_{n+1}, \\ 11\widehat{S}_{n} &=&3\widehat{J}_{n+4}-17\widehat{J}_{n+3}+37\widehat{J} _{n+2}-37\widehat{J}_{n+1}+16\widehat{J}_{n}, \\ 11\widehat{S}_{n} &=&\widehat{J}_{n+3}-2\widehat{J}_{n+2}+5\widehat{J} _{n+1}-5\widehat{J}_{n}+9\widehat{J}_{n-1}, \\ 11\widehat{S}_{n} &=&4\widehat{J}_{n+2}-8\widehat{J}_{n+1}+9\widehat{J}_{n}+2 \widehat{J}_{n-1}+3\widehat{J}_{n-2}. \end{eqnarray*}

Next, we present a few basic relations between \(\{\widehat{J}_{n}\}\) and \(\{ \widehat{R}_{n}\}\).

Lemma 7. The following equalities are true: \begin{eqnarray*} 43245\widehat{J}_{n} &=&164\widehat{R}_{n+6}+1182\widehat{R}_{n+5}-7993 \widehat{R}_{n+4}+15017\widehat{R}_{n+3}-17692\widehat{R}_{n+2}, \\ 14415\widehat{J}_{n} &=&722\widehat{R}_{n+5}-3375\widehat{R}_{n+4}+5771 \widehat{R}_{n+3}-6280\widehat{R}_{n+2}+164\widehat{R}_{n+1}, \\ 4805\widehat{J}_{n} &=&319\widehat{R}_{n+4}-1205\widehat{R}_{n+3}+1276 \widehat{R}_{n+2}-1630\widehat{R}_{n+1}+722\widehat{R}_{n}, \\ 4805\widehat{J}_{n} &=&709\widehat{R}_{n+3}-2871\widehat{R}_{n+2}+2836 \widehat{R}_{n+1}-1511\widehat{R}_{n}+957\widehat{R}_{n-1}, \\ 4805\widehat{J}_{n} &=&1383\widehat{R}_{n+2}-6381\widehat{R}_{n+1}+8415 \widehat{R}_{n}-4006\widehat{R}_{n-1}+2127\widehat{R}_{n-2}, \end{eqnarray*} and \begin{eqnarray*} 396\widehat{R}_{n} &=&1163\widehat{J}_{n+6}-7881\widehat{J}_{n+5}+19664 \widehat{J}_{n+4}-22810\widehat{J}_{n+3}+10379\widehat{J}_{n+2}, \\ 132\widehat{R}_{n} &=&-301\widehat{J}_{n+5}+1515\widehat{J}_{n+4}-2176 \widehat{J}_{n+3}+746\widehat{J}_{n+2}+1163\widehat{J}_{n+1}, \\ 44\widehat{R}_{n} &=&-97\widehat{J}_{n+4}+579\widehat{J}_{n+3}-1156\widehat{J }_{n+2}+1090\widehat{J}_{n+1}-301\widehat{J}_{n}, \\ 44\widehat{R}_{n} &=&-3\widehat{J}_{n+3}+105\widehat{J}_{n+2}-268\widehat{J} _{n+1}+378\widehat{J}_{n}-291\widehat{J}_{n-1}, \\ 44\widehat{R}_{n} &=&87\widehat{J}_{n+2}-229\widehat{J}_{n+1}+336\widehat{J} _{n}-270\widehat{J}_{n-1}-9\widehat{J}_{n-2}. \end{eqnarray*}

Now, we give a few basic relations between \(\{\widehat{j}_{n}\}\) and \(\{ \widehat{S}_{n}\}\).

Lemma 8. The following equalities are true: \begin{eqnarray*} 36\widehat{j}_{n} &=&-83\widehat{S}_{n+6}+465\widehat{S}_{n+5}-872\widehat{S} _{n+4}+706\widehat{S}_{n+3}-83\widehat{S}_{n+2}, \end{eqnarray*}\begin{eqnarray*} 12\widehat{j}_{n} &=&-11\widehat{S}_{n+5}+69\widehat{S}_{n+4}-152\widehat{S} _{n+3}+166\widehat{S}_{n+2}-83\widehat{S}_{n+1}, \\ 4\widehat{j}_{n} &=&\widehat{S}_{n+4}-3\widehat{S}_{n+3}+4\widehat{S}_{n+2}-2 \widehat{S}_{n+1}-11\widehat{S}_{n}, \\ 4\widehat{j}_{n} &=&3\widehat{S}_{n+3}-9\widehat{S}_{n+2}+12\widehat{S} _{n+1}-18\widehat{S}_{n}+3\widehat{S}_{n-1}, \\ 4\widehat{j}_{n} &=&9\widehat{S}_{n+2}-27\widehat{S}_{n+1}+24\widehat{S} _{n}-18\widehat{S}_{n-1}+9\widehat{S}_{n-2}, \end{eqnarray*} and \begin{eqnarray*} 171\widehat{S}_{n} &=&-44\widehat{j}_{n+6}+246\widehat{j}_{n+5}-461\widehat{j }_{n+4}+373\widehat{j}_{n+3}-44\widehat{j}_{n+2}, \\ 57\widehat{S}_{n} &=&-6\widehat{j}_{n+5}+37\widehat{j}_{n+4}-81\widehat{j} _{n+3}+88\widehat{j}_{n+2}-44\widehat{j}_{n+1}, \\ 57\widehat{S}_{n} &=&\widehat{j}_{n+4}-3\widehat{j}_{n+3}+4\widehat{j} _{n+2}-2\widehat{j}_{n+1}-18\widehat{j}_{n}, \\ 57\widehat{S}_{n} &=&3\widehat{j}_{n+3}-9\widehat{j}_{n+2}+12\widehat{j} _{n+1}-25\widehat{j}_{n}+3\widehat{j}_{n-1}, \\ 57\widehat{S}_{n} &=&9\widehat{j}_{n+2}-27\widehat{j}_{n+1}+17\widehat{j} _{n}-18\widehat{j}_{n-1}+9\widehat{j}_{n-2}. \end{eqnarray*}

Next, we present a few basic relations between \(\{\widehat{j}_{n}\}\) and \(\{ \widehat{R}_{n}\}\).

Lemma 9. The following equalities are true: \begin{eqnarray*} 43245\widehat{j}_{n} &=&11881\widehat{R}_{n+6}-71634\widehat{R}_{n+5}+151312 \widehat{R}_{n+4}-141779\widehat{R}_{n+3}+41176\widehat{R}_{n+2}, \\ 14415\widehat{j}_{n} &=&-116\widehat{R}_{n+5}-1047\widehat{R}_{n+4}+8185 \widehat{R}_{n+3}-13997\widehat{R}_{n+2}+11881\widehat{R}_{n+1}, \\ 4805\widehat{j}_{n} &=&-581\widehat{R}_{n+4}+3231\widehat{R}_{n+3}-5207 \widehat{R}_{n+2}+4231\widehat{R}_{n+1}-116\widehat{R}_{n}, \\ 4805\widehat{j}_{n} &=&-255\widehat{R}_{n+3}+2346\widehat{R}_{n+2}-3903 \widehat{R}_{n+1}+3951\widehat{R}_{n}-1743\widehat{R}_{n-1}, \\ 4805\widehat{j}_{n} &=&816\widehat{R}_{n+2}-588\widehat{R}_{n+1}+381\widehat{ R}_{n}+42\widehat{R}_{n-1}-765\widehat{R}_{n-2}, \end{eqnarray*} and \begin{eqnarray*} 342\widehat{R}_{n} &=&457\widehat{j}_{n+6}-2397\widehat{j}_{n+5}+3994 \widehat{j}_{n+4}-2396\widehat{j}_{n+3}-1025\widehat{j}_{n+2}, \\ 114\widehat{R}_{n} &=&115\widehat{j}_{n+5}-649\widehat{j}_{n+4}+1334\widehat{ j}_{n+3}-1408\widehat{j}_{n+2}+457\widehat{j}_{n+1}, \\ 114\widehat{R}_{n} &=&41\widehat{j}_{n+4}-161\widehat{j}_{n+3}+202\widehat{j} _{n+2}-348\widehat{j}_{n+1}+345\widehat{j}_{n}, \\ 114\widehat{R}_{n} &=&85\widehat{j}_{n+3}-331\widehat{j}_{n+2}+226\widehat{j} _{n+1}+58\widehat{j}_{n}+123\widehat{j}_{n-1}, \\ 114\widehat{R}_{n} &=&179\widehat{j}_{n+2}-879\widehat{j}_{n+1}+1248\widehat{ j}_{n}-472\widehat{j}_{n-1}+255\widehat{j}_{n-2}. \end{eqnarray*}

Now, we give a few basic relations between \(\{\widehat{S}_{n}\}\) and \(\{ \widehat{R}_{n}\}\).

Lemma 10. The following equalities are true: \begin{eqnarray*} 43245\widehat{S}_{n} &=&-3857\widehat{R}_{n+6}+23568\widehat{R}_{n+5}-50024 \widehat{R}_{n+4}+47053\widehat{R}_{n+3}-13622\widehat{R}_{n+2}, \\ 14415\widehat{S}_{n} &=&142\widehat{R}_{n+5}+39\widehat{R}_{n+4}-2315 \widehat{R}_{n+3}+4459\widehat{R}_{n+2}-3857\widehat{R}_{n+1}, \\ 4805\widehat{S}_{n} &=&297\widehat{R}_{n+4}-1387\widehat{R}_{n+3}+2149 \widehat{R}_{n+2}-1617\widehat{R}_{n+1}+142\widehat{R}_{n}, \\ 4805\widehat{S}_{n} &=&395\widehat{R}_{n+3}-1712\widehat{R}_{n+2}+2541 \widehat{R}_{n+1}-1937\widehat{R}_{n}+891\widehat{R}_{n-1}, \\ 4805\widehat{S}_{n} &=&658\widehat{R}_{n+2}-2594\widehat{R}_{n+1}+3593 \widehat{R}_{n}-1874\widehat{R}_{n-1}+1185\widehat{R}_{n-2}, \end{eqnarray*} and \begin{eqnarray*} 72\widehat{R}_{n} &=&-287\widehat{S}_{n+6}+1509\widehat{S}_{n+5}-2516 \widehat{S}_{n+4}+1510\widehat{S}_{n+3}+649\widehat{S}_{n+2}, \\ 24\widehat{R}_{n} &=&-71\widehat{S}_{n+5}+405\widehat{S}_{n+4}-836\widehat{S} _{n+3}+886\widehat{S}_{n+2}-287\widehat{S}_{n+1}, \\ 8\widehat{R}_{n} &=&-7\widehat{S}_{n+4}+29\widehat{S}_{n+3}-36\widehat{S} _{n+2}+70\widehat{S}_{n+1}-71\widehat{S}_{n}, \\ 8\widehat{R}_{n} &=&-13\widehat{S}_{n+3}+55\widehat{S}_{n+2}-28\widehat{S} _{n+1}-22\widehat{S}_{n}-21\widehat{S}_{n-1}, \\ 8\widehat{R}_{n} &=&-23\widehat{S}_{n+2}+141\widehat{S}_{n+1}-204\widehat{S} _{n}+70\widehat{S}_{n-1}-39\widehat{S}_{n-2}. \end{eqnarray*}

6. On the recurrence properties of binomial transform of the generalized fifth order Jacobsthal sequence

Taking \(r_{1}=6,s_{1}=-13,t_{1}=14,u_{1}=-7,v_{1}=3\) and \(H_{n}=\widehat{R} _{n}\) in Theorem 3, we obtain the following Proposition.

Proposition 2. For \(n\in \mathbb{Z},\) binomial Transform of the generalized fifth order Jacobsthal sequence have the following identity: \begin{eqnarray*} b_{-n}&=&\frac{1}{24} 3^{-n}(b_{0}\widehat{R}_{n}^{4}-4b_{n} \widehat{R}_{n}^{3}+3b_{0}\widehat{R}_{2n}^{2}+12\widehat{R} _{n}^{2}b_{2n}-6b_{0}\widehat{R}_{n}^{2}\widehat{R}_{2n}-6b_{0}\widehat{R} _{4n}-8b_{n}\widehat{R}_{3n}-12\widehat{R}_{2n}b_{2n}-24\widehat{R} _{n}b_{3n}+24b_{4n}\\ &&+8b_{0}\widehat{R}_{n}\widehat{R}_{3n}+12b_{n}\widehat{R} _{n}\widehat{R}_{2n})\\ &=&3^{-n}(b_{4n}-\widehat{R}_{n}b_{3n}+\frac{1}{2}(\widehat{R}_{n}^{2}- \widehat{R}_{2n})b_{2n}-\frac{1}{6}(\widehat{R}_{n}^{3}+2\widehat{R}_{3n}-3 \widehat{R}_{2n}\widehat{R}_{n})b_{n}\\ &&+\frac{1}{24}(\widehat{R}_{n}^{4}+3 \widehat{R}_{2n}^{2}-6\widehat{R}_{n}^{2}\widehat{R}_{2n}-6\widehat{R}_{4n}+8 \widehat{R}_{3n}\widehat{R}_{n})b_{0}). \end{eqnarray*}

Using Proposition 2 (and Corollary 2), we obtain the following corollary which gives the connection between the special cases of binomial transform of generalized fifth order Jacobsthal sequence at the positive index and the negative index: for binomial transform of fifth order Jacobsthal, fifth order Jacobsthal-Lucas numbers: take \(b_{n}=\widehat{J}_{n}\) with \(\widehat{J}_{0}=0,\widehat{J}_{1}=1, \widehat{J}_{2}=3,\widehat{J}_{3}=7,\widehat{J}_{4}=15,\) take \(b_{n}= \widehat{j}_{n}\) with \(\widehat{j}_{0}=2,\widehat{j}_{1}=3,\widehat{j}_{2}=9, \widehat{j}_{3}=30,\widehat{j}_{4}=96,\) take \(b_{n}=\widehat{S}_{n}\) with \( \widehat{S}_{0}=0,\widehat{S}_{1}=1,\widehat{S}_{2}=3,\widehat{S}_{3}=8, \widehat{S}_{4}=22,\) take \(b_{n}=\widehat{R}_{n}\) with \(\widehat{R}_{0}=5, \widehat{R}_{1}=6,\widehat{R}_{2}=10,\widehat{R}_{3}=24,\widehat{R}_{4}=70,\) respectively. Note that in this case we have \(H_{n}=\widehat{R}_{n}\). Note also that \(G_{n}\neq \widehat{S}_{n}\).

Corollary 5. For \(n\in \mathbb{Z},\) we have the following recurrence relations:

• (a) Recurrence relations of binomial transforms of fifth order Jacobsthal numbers (take \(b_{n}=\widehat{J}_{n}\) in Proposition 2): \begin{equation*} \widehat{J}_{-n}=3^{-n}(\widehat{J}_{4n}-\widehat{R}_{n}\widehat{J}_{3n}+ \frac{1}{2}(\widehat{R}_{n}^{2}-\widehat{R}_{2n})\widehat{J}_{2n}-\frac{1}{6} (\widehat{R}_{n}^{3}+2\widehat{R}_{3n}-3\widehat{R}_{2n}\widehat{R}_{n}) \widehat{J}_{n}). \end{equation*}
• (b) Recurrence relations of binomial transforms of fifth order Jacobsthal-Lucas numbers (take \(b_{n}=\widehat{j}_{n}\) in Proposition 2): \begin{equation*} \widehat{j}_{-n}=3^{-n}(\widehat{j}_{4n}-\widehat{R}_{n}\widehat{j}_{3n}+ \frac{1}{2}(\widehat{R}_{n}^{2}-\widehat{R}_{2n})\widehat{j}_{2n}-\frac{1}{6} (\widehat{R}_{n}^{3}+2\widehat{R}_{3n}-3\widehat{R}_{2n}\widehat{R}_{n}) \widehat{j}_{n}+\frac{1}{12}(\widehat{R}_{n}^{4}+3\widehat{R}_{2n}^{2}-6 \widehat{R}_{n}^{2}\widehat{R}_{2n}-6\widehat{R}_{4n}+8\widehat{R}_{3n} \widehat{R}_{n})). \end{equation*}
• (c) Recurrence relations of binomial transforms of adjusted fifth order Jacobsthal numbers (take \(b_{n}=\widehat{S}_{n}\) in Proposition 2): \begin{equation*} \widehat{S}_{-n}=3^{-n}(\widehat{S}_{4n}-\widehat{R}_{n}\widehat{S}_{3n}+ \frac{1}{2}(\widehat{R}_{n}^{2}-\widehat{R}_{2n})\widehat{S}_{2n}-\frac{1}{6} (\widehat{R}_{n}^{3}+2\widehat{R}_{3n}-3\widehat{R}_{2n}\widehat{R}_{n}) \widehat{S}_{n}). \end{equation*}
• (d) Recurrence relations of binomial transforms of modified fifth order Jacobsthal-Lucas numbers (take \(b_{n}=\widehat{R}_{n}\) in Proposition 2 or take \(H_{n}=\widehat{R}_{n}\) in Corollary 2): \begin{equation*} \widehat{R}_{-n}=\frac{1}{24} 3^{-n}(\widehat{R}_{n}^{4}+3 \widehat{R}_{2n}^{2}-6\widehat{R}_{n}^{2}\widehat{R}_{2n}-6\widehat{R}_{4n}+8 \widehat{R}_{3n}\widehat{R}_{n}). \end{equation*}

7. Sum formulas

7.1. Sums of terms with positive subscripts

The following proposition presents some formulas of binomial transform of generalized fifth order Jacobsthal numbers with positive subscripts.

Proposition 3. If \(r=6,s=-13,t=14,u=-7,v=3,\) then for \(n\geq 0\) we have the following formulas:

• (a) \(\sum_{k=0}^{n}b_{k}=\frac{1}{2}(b_{n+5}-5b_{n+4}+8 b_{n+3}-6b_{n+2}+b_{n+1}-b_{4}+5b_{3}-8b_{2}+6b_{1}-b_{0}).\)
• (b) \(\sum_{k=0}^{n}b_{2k}=\frac{1}{88} (21b_{2n+2}-103b_{2n+1}+244b_{2n}-98b_{2n-1}+69b_{2n-2}-21b_{4}+103b_{3}-156b_{2}+98b_{1}+19b_{0}). \)
• (c) \(\sum_{k=0}^{n}b_{2k+1}=\frac{1}{88}(23b_{2n+2}-29b_{2n+1}+196 b_{2n}-78b_{2n-1}+63b_{2n-2}-23b_{4}+117b_{3}-196b_{2}+166b_{1}-63b_{0} ).\)

Proof. Take \(r=6,s=-13,t=14,u=-7,v=3,\) in Theorem 2.1 in [28].

From the last proposition, we have the following corollary which gives sum formulas of binomial transform of fifth order Jacobsthal numbers (take \( b_{n}=\widehat{J}_{n}\) with \(\widehat{J}_{0}=0,\widehat{J}_{1}=1,\widehat{J} _{2}=3,\widehat{J}_{3}=7,\widehat{J}_{4}= 15\)).

Corollary 6. For \(n\geq 0\) we have the following formulas:

• (a) \(\sum_{k=0}^{n}\widehat{J}_{k}=\frac{1}{2}(\widehat{J}_{n+5}-5 \widehat{J}_{n+4}+8 \widehat{J}_{n+3}-6\widehat{J}_{n+2}+\widehat{ J}_{n+1}+2).\)
• (b) \(\sum_{k=0}^{n}\widehat{J}_{2k}=\frac{1}{88}(21\widehat{J} _{2n+2}-103\widehat{J}_{2n+1}+244\widehat{J}_{2n}-98\widehat{J}_{2n-1}+69 \widehat{J}_{2n-2}+36).\)
• (c) \(\sum_{k=0}^{n}\widehat{J}_{2k+1}=\frac{1}{88}(23\widehat{J} _{2n+2}-29\widehat{J}_{2n+1}+196 \widehat{J}_{2n}-78\widehat{J} _{2n-1}+63\widehat{J}_{2n-2}+52 ).\)

Taking \(b_{n}=\widehat{j}_{n}\) with \(\widehat{j}_{0}=2,\widehat{j}_{1}=3, \widehat{j}_{2}=9,\widehat{j}_{3}=30,\widehat{j}_{4}=96\) in the last proposition, we have the following corollary which presents sum formulas of binomial transform of fifth order Jacobsthal-Lucas numbers.

Corollary 7. For \(n\geq 0\) we have the following formulas:

• (a) \(\sum_{k=0}^{n}\widehat{j}_{k}=\frac{1}{2}(\widehat{j}_{n+5}-5 \widehat{j}_{n+4}+8 \widehat{j}_{n+3}-6\widehat{j}_{n+2}+\widehat{ j}_{n+1}-2).\)
• (b) \(\sum_{k=0}^{n}\widehat{j}_{2k}=\frac{1}{88}(21\widehat{j} _{2n+2}-103\widehat{j}_{2n+1}+244\widehat{j}_{2n}-98\widehat{j}_{2n-1}+69 \widehat{j}_{2n-2}+2).\)
• (c) \(\sum_{k=0}^{n}\widehat{j}_{2k+1}=\frac{1}{88}(23\widehat{j} _{2n+2}-29\widehat{j}_{2n+1}+196 \widehat{j}_{2n}-78\widehat{j} _{2n-1}+63\widehat{j}_{2n-2}-90).\)

From the last proposition, we have the following corollary which gives sum formulas of binomial transform of adjusted fifth order Jacobsthal numbers (take \(b_{n}=\widehat{S}_{n}\) with \(\widehat{S}_{0}=0,\widehat{S}_{1}=1, \widehat{S}_{2}=3,\widehat{S}_{3}=8,\widehat{S}_{4}=22\)).

Corollary 8. For \(n\geq 0\) we have the following formulas:

• (a) \(\sum_{k=0}^{n}\widehat{S}_{k}=\frac{1}{2}(\widehat{S}_{n+5}-5 \widehat{S}_{n+4}+8 \widehat{S}_{n+3}-6\widehat{S}_{n+2}+\widehat{ S}_{n+1}).\)
• (b) \(\sum_{k=0}^{n}\widehat{S}_{2k}=\frac{1}{88}(21\widehat{S} _{2n+2}-103\widehat{S}_{2n+1}+244\widehat{S}_{2n}-98\widehat{S}_{2n-1}+69 \widehat{S}_{2n-2}-8).\)
• (c) \(\sum_{k=0}^{n}\widehat{S}_{2k+1}=\frac{1}{88}(23\widehat{S} _{2n+2}-29\widehat{S}_{2n+1}+196 \widehat{S}_{2n}-78\widehat{S} _{2n-1}+63\widehat{S}_{2n-2}+8).\)

Taking \(b_{n}=\widehat{R}_{n}\) with \(\widehat{R}_{0}=5,\widehat{R}_{1}=6, \widehat{R}_{2}=10,\widehat{R}_{3}=24,\widehat{R}_{4}=70\) in the last proposition, we have the following corollary which presents sum formulas of binomial transform of modified fifth order Jacobsthal-Lucas numbers.

Corollary 9. For \(n\geq 0\) we have the following formulas:

• (a) \(\sum_{k=0}^{n}\widehat{R}_{k}=\frac{1}{2}(\widehat{R}_{n+5}-5 \widehat{R}_{n+4}+8 \widehat{R}_{n+3}-6\widehat{R}_{n+2}+\widehat{ R}_{n+1}+1).\)
• (b) \(\sum_{k=0}^{n}\widehat{R}_{2k}=\frac{1}{88}(21\widehat{R} _{2n+2}-103\widehat{R}_{2n+1}+244\widehat{R}_{2n}-98\widehat{R}_{2n-1}+69 \widehat{R}_{2n-2}+125).\)
• (c) \(\sum_{k=0}^{n}\widehat{R}_{2k+1}=\frac{1}{88}(23\widehat{R} _{2n+2}-29\widehat{R}_{2n+1}+196 \widehat{R}_{2n}-78\widehat{R} _{2n-1}+63\widehat{R}_{2n-2}-81 ).\)

7.2. Sums of terms with negative subscripts

The following proposition presents some formulas of binomial transform of generalized fifth order Jacobsthal numbers with negative subscripts.

Proposition 4. If \(r=6,s=-13,t=14,u=-7,v=3,\) then for \(n\geq 1\) we have the following formulas:

• (a) \(\sum_{k=1}^{n}b_{-k}=\frac{1}{2} (-b_{-n+4}+5b_{-n+3}-8b_{-n+2}+6b_{-n+1}-b_{-n}+b_{4}-5b_{3}+8b_{2}-6b_{1}+b_{0}). \)
• (b) \(\sum_{k=1}^{n}b_{-2k}=\frac{1}{88}(- 23b_{-2n+3}+117b_{-2n+2}-196b_{-2n+1}+166 b_{-2n}-63b_{-2n-1}+21b_{4}-103b_{3}+156b_{2}-98b_{1}-19b_{0}).\)
• (c) \(\sum_{k=1}^{n}b_{-2k+1}=\frac{1}{88} (-21b_{-2n+3}+103b_{-2n+2}-156b_{-2n+1}+98b_{-2n}-69b_{-2n-1}+23b_{4}-117b_{3}+196b_{2}-166b_{1}+63b_{0}). \)

Proof. Take \(r=6,s=-13,t=14,u=-7,v=3,\) in Theorem 3.1 in [28].

From the last proposition, we have the following corollary which gives sum formulas of binomial transform of fifth order Jacobsthal numbers (take \( b_{n}=\widehat{J}_{n}\) with \(\widehat{J}_{0}=0,\widehat{J}_{1}=1,\widehat{J} _{2}=3,\widehat{J}_{3}=7,\widehat{J}_{4}= 15).\)

Corollary 10. For \(n\geq 1,\) binomial transform of fifth order Jacobsthal numbers have the following properties.

• (a) \(\sum_{k=1}^{n}\widehat{J}_{-k}= \frac{1}{2}(-\widehat{ J}_{-n+4}+5\widehat{J}_{-n+3}-8\widehat{J}_{-n+2}+6\widehat{J}_{-n+1}- \widehat{J}_{-n}-2).\)
• (b) \(\sum_{k=1}^{n}\widehat{J}_{-2k}=\frac{1}{88}(- 23 \widehat{J}_{-2n+3}+117\widehat{J}_{-2n+2}-196\widehat{J}_{-2n+1}+166 \widehat{J}_{-2n}-63\widehat{J}_{-2n-1}-36).\)
• (c) \(\sum_{k=1}^{n}\widehat{J}_{-2k+1}= \frac{1}{88}(-21 \widehat{J}_{-2n+3}+103\widehat{J}_{-2n+2}-156\widehat{J}_{-2n+1}+98\widehat{ J}_{-2n}-69\widehat{J}_{-2n-1}-52).\)

Taking \(b_{n}=\widehat{j}_{n}\) with \(\widehat{j}_{0}=2,\widehat{j}_{1}=3, \widehat{j}_{2}=9,\widehat{j}_{3}=30,\widehat{j}_{4}=96\) in the last proposition, we have the following corollary which presents sum formulas of binomial transform of fifth order Jacobsthal-Lucas numbers.

Corollary 11. For \(n\geq 1,\) binomial transform of fifth order Jacobsthal-Lucas numbers have the following properties.

• (a) \(\sum_{k=1}^{n}\widehat{j}_{-k}=\frac{1}{2}(-\widehat{j}_{-n+4}+5 \widehat{j}_{-n+3}-8\widehat{j}_{-n+2}+6\widehat{j}_{-n+1}-\widehat{j} _{-n}+2).\)
• (b) \(\sum_{k=1}^{n}\widehat{j}_{-2k}= \frac{1}{88} (- 23\widehat{j}_{-2n+3}+117\widehat{j}_{-2n+2}-196\widehat{j} _{-2n+1}+166 \widehat{j}_{-2n}-63\widehat{j}_{-2n-1}-2).\)
• (c) \(\sum_{k=1}^{n}\widehat{j}_{-2k+1}= \frac{1}{88}(-21 \widehat{j}_{-2n+3}+103\widehat{j}_{-2n+2}-156\widehat{j}_{-2n+1}+98\widehat{ j}_{-2n}-69\widehat{j}_{-2n-1}+90).\)

From the last proposition, we have the following corollary which gives sum formulas of binomial transform of adjusted fifth order Jacobsthal numbers (take \(b_{n}=\widehat{S}_{n}\) with \(\widehat{S}_{0}=0,\widehat{S}_{1}=1, \widehat{S}_{2}=3,\widehat{S}_{3}=8,\widehat{S}_{4}=22\)).

Corollary 12. For \(n\geq 1,\) binomial transform of adjusted fifth order Jacobsthal numbers have the following properties.

• (a) \(\sum_{k=1}^{n}\widehat{S}_{-k}=\frac{1}{2}(-\widehat{S}_{-n+4}+5 \widehat{S}_{-n+3}-8\widehat{S}_{-n+2}+6\widehat{S}_{-n+1}-\widehat{S} _{-n}). \)
• (b) \(\sum_{k=1}^{n}\widehat{S}_{-2k}=\frac{1}{88}(- 23 \widehat{S}_{-2n+3}+117\widehat{S}_{-2n+2}-196\widehat{S}_{-2n+1}+166 \widehat{S}_{-2n}-63\widehat{S}_{-2n-1}+8).\)
• (c) \(\sum_{k=1}^{n}\widehat{S}_{-2k+1}=\frac{1}{88}(-21\widehat{S} _{-2n+3}+103\widehat{S}_{-2n+2}-156\widehat{S}_{-2n+1}+98\widehat{S}_{-2n}-69 \widehat{S}_{-2n-1}-8).\)

Taking \(b_{n}=\widehat{R}_{n}\) with \(\widehat{R}_{0}=5,\widehat{R}_{1}=6, \widehat{R}_{2}=10,\widehat{R}_{3}=24,\widehat{R}_{4}=70\) in the last proposition, we have the following corollary which presents sum formulas of binomial transform of modified fifth order Jacobsthal-Lucas numbers.

Corollary 13. For \(n\geq 1,\) binomial transform of modified fifth order Jacobsthal-Lucas numbers have the following properties.

• (a) \(\ \sum_{k=1}^{n}\widehat{R}_{-k}=\frac{1}{2}(-\widehat{R} _{-n+4}+5\widehat{R}_{-n+3}-8\widehat{R}_{-n+2}+6\widehat{R}_{-n+1}-\widehat{ R}_{-n}-1).\)
• (b) \(\ \sum_{k=1}^{n}\widehat{R}_{-2k}=\frac{1}{88}(- 23 \widehat{R}_{-2n+3}+117\widehat{R}_{-2n+2}-196\widehat{R}_{-2n+1}+166 \widehat{R}_{-2n}-63\widehat{R}_{-2n-1}-125).\)
• (c) \(\ \sum_{k=1}^{n}\widehat{R}_{-2k+1}=\frac{1}{88}(-21\widehat{R} _{-2n+3}+103\widehat{R}_{-2n+2}-156\widehat{R}_{-2n+1}+98\widehat{R}_{-2n}-69 \widehat{R}_{-2n-1}+81).\)

8. Matrices related with binomial transform of generalized fifth order Jacobsthal numbers

We define the square matrix \(A\) of order \(5\) as: \begin{equation*} A=\left( \begin{array}{ccccc} 6 & -13 & 14 & -7 & 3 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array} \right)\,, \end{equation*} such that \(\det A=3.\) From (1) we have and from (6) (or using (19) and induction) we have \begin{equation*} \left( \begin{array}{c} b_{n+4} \\ b_{n+3} \\ b_{n+2} \\ b_{n+1} \\ b_{n} \end{array} \right) =\left( \begin{array}{ccccc} 6 & -13 & 14 & -7 & 3 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array} \right) ^{n}\left( \begin{array}{c} b_{4} \\ b_{3} \\ b_{2} \\ b_{1} \\ b_{0} \end{array} \right) . \end{equation*} If we take \(b_{n}=\widehat{S}_{n}\) in (19) we have We also, for \(n\geq 0,\) define \begin{equation*} B_{n}=\left( \begin{array}{ccccc} \sum_{k=0}^{n+1}\sum_{l=k}^{n+1}\sum_{p=l}^{n+1}\widehat{S}_{k} & E_{1} & E_{6} & E_{11} & 3\sum_{k=0}^{n}\sum_{l=k}^{n}\sum_{p=l}^{n}\widehat{S}_{k} \\ \sum_{k=0}^{n}\sum_{l=k}^{n}\sum_{p=l}^{n}\widehat{S}_{k} & E_{2} & E_{7} & E_{12} & 3\sum_{k=0}^{n-1}\sum_{l=k}^{n-1}\sum_{p=l}^{n-1}\widehat{S}_{k} \\ \sum_{k=0}^{n-1}\sum_{l=k}^{n-1}\sum_{p=l}^{n-1}\widehat{S}_{k} & E_{3} & E_{8} & E_{13} & 3\sum_{k=0}^{n-2}\sum_{l=k}^{n-2}\sum_{p=l}^{n-2}\widehat{S} _{k} \\ \sum_{k=0}^{n-2}\sum_{l=k}^{n-2}\sum_{p=l}^{n-2}\widehat{S}_{k} & E_{4} & E_{9} & E_{14} & 3\sum_{k=0}^{n-3}\sum_{l=k}^{n-3}\sum_{p=l}^{n-3}\widehat{S} _{k} \\ \sum_{k=0}^{n-3}\sum_{l=k}^{n-3}\sum_{p=l}^{n-3}\widehat{S}_{k} & E_{5} & E_{10} & E_{15} & 3\sum_{k=0}^{n-4}\sum_{l=k}^{n-4}\sum_{p=l}^{n-4}\widehat{S }_{k} \end{array} \right)\,, \end{equation*} and \begin{equation*} C_{n}=\left( \begin{array}{c} \begin{array}{ccccc} b_{n+1} & -13b_{n}+14b_{n-1}-7b_{n-2}+3b_{n-3} & 14b_{n}-7b_{n-1}+3b_{n-2} & -7b_{n}+3b_{n-1} & 3b_{n} \\ b_{n} & -13b_{n-1}+14b_{n-2}-7b_{n-3}+3b_{n-4} & 14b_{n-1}-7b_{n-2}+3b_{n-3} & -7b_{n-1}+3b_{n-2} & 3b_{n-1} \\ b_{n-1} & -13b_{n-2}+14b_{n-3}-7b_{n-4}+3b_{n-5} & 14b_{n-2}-7b_{n-3}+3b_{n-4} & -7b_{n-2}+3b_{n-3} & 3b_{n-2} \\ b_{n-2} & -13b_{n-3}+14b_{n-4}-7b_{n-5}+3b_{n-6} & 14b_{n-3}-7b_{n-4}+3b_{n-5} & -7b_{n-3}+3b_{n-4} & 3b_{n-3} \\ b_{n-3} & -13b_{n-4}+14b_{n-5}-7b_{n-6}+3b_{n-7} & 14b_{n-4}-7b_{n-5}+3b_{n-6} & -7b_{n-4}+3b_{n-5} & 3b_{n-4} \end{array} \end{array} \right)\,, \end{equation*} where \begin{equation*} \left\{\begin{array}{c} E_{1} \\ E_{2} \\ E_{3} \\ E_{4} \\ E_{5}\\ E_{6} \\ E_{7} \\ E_{8} \\ E_{9} \\ E_{10}\\ E_{11} \\ E_{12} \\ E_{13} \\ E_{14} \\ E_{15} \end{array}\right\} =\left\{ \begin{array}{c} -13\sum_{k=0}^{n}\sum_{l=k}^{n}\sum_{p=l}^{n}\widehat{S}_{k}+14 \sum_{k=0}^{n-1}\sum_{l=k}^{n-1}\sum_{p=l}^{n-1}\widehat{S} _{k}-7\sum_{k=0}^{n-2}\sum_{l=k}^{n-2}\sum_{p=l}^{n-2}\widehat{S} _{k}+3\sum_{k=0}^{n-3}\sum_{l=k}^{n-3}\sum_{p=l}^{n-3}\widehat{S}_{k} \\ -13\sum_{k=0}^{n-1}\sum_{l=k}^{n-1}\sum_{p=l}^{n-1}\widehat{S} _{k}+14\sum_{k=0}^{n-2}\sum_{l=k}^{n-2}\sum_{p=l}^{n-2}\widehat{S} _{k}-7\sum_{k=0}^{n-3}\sum_{l=k}^{n-3}\sum_{p=l}^{n-3}\widehat{S} _{k}+3\sum_{k=0}^{n-4}\sum_{l=k}^{n-4}\sum_{p=l}^{n-4}\widehat{S}_{k} \\ -13\sum_{k=0}^{n-2}\sum_{l=k}^{n-2}\sum_{p=l}^{n-2}\widehat{S} _{k}+14\sum_{k=0}^{n-3}\sum_{l=k}^{n-3}\sum_{p=l}^{n-3}\widehat{S} _{k}-7\sum_{k=0}^{n-4}\sum_{l=k}^{n-4}\sum_{p=l}^{n-4}\widehat{S} _{k}+3\sum_{k=0}^{n-5}\sum_{l=k}^{n-5}\sum_{p=l}^{n-5}\widehat{S}_{k} \\ -13\sum_{k=0}^{n-3}\sum_{l=k}^{n-3}\sum_{p=l}^{n-3}\widehat{S} _{k}+14\sum_{k=0}^{n-4}\sum_{l=k}^{n-4}\sum_{p=l}^{n-4}\widehat{S} _{k}-7\sum_{k=0}^{n-5}\sum_{l=k}^{n-5}\sum_{p=l}^{n-5}\widehat{S} _{k}+3\sum_{k=0}^{n-6}\sum_{l=k}^{n-6}\sum_{p=l}^{n-6}\widehat{S}_{k} \\ -13\sum_{k=0}^{n-4}\sum_{l=k}^{n-4}\sum_{p=l}^{n-4}\widehat{S} _{k}+14\sum_{k=0}^{n-5}\sum_{l=k}^{n-5}\sum_{p=l}^{n-5}\widehat{S} _{k}-7\sum_{k=0}^{n-6}\sum_{l=k}^{n-6}\sum_{p=l}^{n-6}\widehat{S} _{k}+3\sum_{k=0}^{n-7}\sum_{l=k}^{n-7}\sum_{p=l}^{n-7}\widehat{S}_{k}\\ 14\sum_{k=0}^{n}\sum_{l=k}^{n}\sum_{p=l}^{n}\widehat{S}_{k}-7 \sum_{k=0}^{n-1}\sum_{l=k}^{n-1}\sum_{p=l}^{n-1}\widehat{S} _{k}+3\sum_{k=0}^{n-2}\sum_{l=k}^{n-2}\sum_{p=l}^{n-2}\widehat{S}_{k} \\ 14\sum_{k=0}^{n-1}\sum_{l=k}^{n-1}\sum_{p=l}^{n-1}\widehat{S} _{k}-7\sum_{k=0}^{n-2}\sum_{l=k}^{n-2}\sum_{p=l}^{n-2}\widehat{S} _{k}+3\sum_{k=0}^{n-3}\sum_{l=k}^{n-3}\sum_{p=l}^{n-3}\widehat{S}_{k} \\ 14\sum_{k=0}^{n-2}\sum_{l=k}^{n-2}\sum_{p=l}^{n-2}\widehat{S} _{k}-7\sum_{k=0}^{n-3}\sum_{l=k}^{n-3}\sum_{p=l}^{n-3}\widehat{S} _{k}+3\sum_{k=0}^{n-4}\sum_{l=k}^{n-4}\sum_{p=l}^{n-4}\widehat{S}_{k} \\ 14\sum_{k=0}^{n-3}\sum_{l=k}^{n-3}\sum_{p=l}^{n-3}\widehat{S} _{k}-7\sum_{k=0}^{n-4}\sum_{l=k}^{n-4}\sum_{p=l}^{n-4}\widehat{S} _{k}+3\sum_{k=0}^{n-5}\sum_{l=k}^{n-5}\sum_{p=l}^{n-5}\widehat{S}_{k} \\ 14\sum_{k=0}^{n-4}\sum_{l=k}^{n-4}\sum_{p=l}^{n-4}\widehat{S} _{k}-7\sum_{k=0}^{n-5}\sum_{l=k}^{n-5}\sum_{p=l}^{n-5}\widehat{S} _{k}+3\sum_{k=0}^{n-6}\sum_{l=k}^{n-6}\sum_{p=l}^{n-6}\widehat{S}_{k}\\ -7\sum_{k=0}^{n}\sum_{l=k}^{n}\sum_{p=l}^{n}\widehat{S}_{k}+3 \sum_{k=0}^{n-1}\sum_{l=k}^{n-1}\sum_{p=l}^{n-1}\widehat{S}_{k} \\ -7\sum_{k=0}^{n-1}\sum_{l=k}^{n-1}\sum_{p=l}^{n-1}\widehat{S} _{k}+3\sum_{k=0}^{n-2}\sum_{l=k}^{n-2}\sum_{p=l}^{n-2}\widehat{S}_{k} \\ -7\sum_{k=0}^{n-2}\sum_{l=k}^{n-2}\sum_{p=l}^{n-2}\widehat{S} _{k}+3\sum_{k=0}^{n-3}\sum_{l=k}^{n-3}\sum_{p=l}^{n-3}\widehat{S}_{k} \\ -7\sum_{k=0}^{n-3}\sum_{l=k}^{n-3}\sum_{p=l}^{n-3}\widehat{S} _{k}+3\sum_{k=0}^{n-4}\sum_{l=k}^{n-4}\sum_{p=l}^{n-4}\widehat{S}_{k} \\ -7\sum_{k=0}^{n-4}\sum_{l=k}^{n-4}\sum_{p=l}^{n-4}\widehat{S} _{k}+3\sum_{k=0}^{n-5}\sum_{l=k}^{n-5}\sum_{p=l}^{n-5}\widehat{S}_{k} \end{array}\right\}\,. \end{equation*} By convention, we assume that \begin{eqnarray*} \sum_{k=0}^{0}\sum_{l=k}^{0}\sum_{p=l}^{0}\widehat{S}_{k} =0,\quad \sum_{k=0}^{-1}\sum_{l=k}^{-1}\sum_{p=l}^{-1}\widehat{S} _{k}=0,\quad\sum_{k=0}^{-2}\sum_{l=k}^{-2}\sum_{p=l}^{-2}\widehat{S}_{k}=0,\quad \sum_{k=0}^{-3}\sum_{l=k}^{-3}\sum_{p=l}^{-3}\widehat{S}_{k} =0,\\ \sum_{k=0}^{-4}\sum_{l=k}^{-4}\sum_{p=l}^{-4}\widehat{S}_{k}=\frac{1}{3} ,\quad\sum_{k=0}^{-5}\sum_{l=k}^{-5}\sum_{p=l}^{-5}\widehat{S}_{k}=\frac{7}{9}, \quad\sum_{k=0}^{-6}\sum_{l=k}^{-6}\sum_{p=l}^{-6}\widehat{S}_{k} =\frac{7}{27} ,\quad\sum_{k=0}^{-7}\sum_{l=k}^{-7}\sum_{p=l}^{-7}\widehat{S}_{k}=-\frac{128}{81} . \end{eqnarray*}

Theorem 6. For all integers \(m,n\geq 0,\) we have

• (a) \(B_{n}=A^{n}.\)
• (b) \(C_{1}A^{n}=A^{n}C_{1}.\)
• (c) \(C_{n+m}=C_{n}B_{m}=B_{m}C_{n}.\)

Proof.

• (a) Proof can be done by mathematical induction on \(n.\)
• (b) After matrix multiplication, (b) follows.
• (c) We have \(C_{n}=AC_{n-1}.\) From the last equation, using induction, we obtain \(C_{n}=A^{n-1}C_{1}.\) Now \begin{equation*} C_{n+m}=A^{n+m-1}C_{1}=A^{n-1}A^{m}C_{1}=A^{n-1}C_{1}A^{m}=C_{n}B_{m} \end{equation*} and similarly \begin{equation*} C_{n+m}=B_{m}C_{n}. \end{equation*}

Theorem 7. For \(m,n\geq 0,\) we have \begin{eqnarray*} b_{n+m} &=&b_{n}\sum_{k=0}^{m+1}\sum_{l=k}^{m+1}\sum_{p=l}^{m+1}\widehat{S} _{k} \\ &&+b_{n-1}\left( -13\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m}\widehat{S} _{k}+14\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}-7\sum_{k=0}^{m-2}\sum_{l=k}^{m-2}\sum_{p=l}^{m-2}\widehat{S} _{k} +3\sum_{k=0}^{m-3}\sum_{l=k}^{m-3}\sum_{p=l}^{m-3}\widehat{S} _{k} \right) \\ &&+b_{n-2}\left( 14\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m}\widehat{S} _{k}-7\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}+3\sum_{k=0}^{m-2}\sum_{l=k}^{m-2}\sum_{p=l}^{m-2}\widehat{S} _{k} \right) \\ &&+b_{n-3}\left( -7\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m}\widehat{S} _{k}+3\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S}_{k}\right) + 3b_{n-4}\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m}\widehat{S} _{k}. \end{eqnarray*}

Proof. From the equation \(C_{n+m}=C_{n}B_{m}=B_{m}C_{n},\) we see that an element of \(C_{n+m}\) is the product of row \(C_{n}\) and a column \( B_{m}.\) From the last equation, we say that an element of \(C_{n+m}\) is the product of a row \(C_{n}\) and column \(B_{m}.\) We just compare the linear combination of the 2nd row and 1st column entries of the matrices \(C_{n+m}\) and \(C_{n}B_{m}\). This completes the proof.

Corollary 14. For \(m,n\geq 0,\) we have \begin{eqnarray*} \widehat{J}_{n+m} &=&\widehat{J}_{n}\sum_{k=0}^{m+1}\sum_{l=k}^{m+1} \sum_{p=l}^{m+1}\widehat{S}_{k} +\widehat{J}_{n-1}\left( -13\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m} \widehat{S}_{k}+14\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}-7\sum_{k=0}^{m-2}\sum_{l=k}^{m-2}\sum_{p=l}^{m-2}\widehat{S} _{k} +3\sum_{k=0}^{m-3}\sum_{l=k}^{m-3}\sum_{p=l}^{m-3}\widehat{S} _{k} \right) \\ &&+\widehat{J}_{n-2}\left( 14\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m} \widehat{S}_{k}-7\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}+3\sum_{k=0}^{m-2}\sum_{l=k}^{m-2}\sum_{p=l}^{m-2}\widehat{S} _{k} \right) \\ &&+\widehat{J}_{n-3}\left( -7\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m} \widehat{S}_{k}+3\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}\right) + 3\widehat{J}_{n-4}\sum_{k=0}^{m}\sum_{l=k}^{m} \sum_{p=l}^{m}\widehat{S}_{k}, \end{eqnarray*} \begin{eqnarray*} \widehat{j}_{n+m} &=&\widehat{j}_{n}\sum_{k=0}^{m+1}\sum_{l=k}^{m+1} \sum_{p=l}^{m+1}\widehat{S}_{k} \\ &&+\widehat{j}_{n-1}\left( -13\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m} \widehat{S}_{k}+14\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}-7\sum_{k=0}^{m-2}\sum_{l=k}^{m-2}\sum_{p=l}^{m-2}\widehat{S} _{k} +3\sum_{k=0}^{m-3}\sum_{l=k}^{m-3}\sum_{p=l}^{m-3}\widehat{S} _{k} \right) \\ &&+\widehat{j}_{n-2}\left( 14\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m} \widehat{S}_{k}-7\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}+3\sum_{k=0}^{m-2}\sum_{l=k}^{m-2}\sum_{p=l}^{m-2}\widehat{S} _{k} \right) \\ &&+\widehat{j}_{n-3}\left( -7\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m} \widehat{S}_{k}+3\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}\right) + 3\widehat{j}_{n-4}\sum_{k=0}^{m}\sum_{l=k}^{m} \sum_{p=l}^{m}\widehat{S}_{k}, \end{eqnarray*} \begin{eqnarray*} \widehat{S}_{n+m} &=&\widehat{S}_{n}\sum_{k=0}^{m+1}\sum_{l=k}^{m+1} \sum_{p=l}^{m+1}\widehat{S}_{k} \\ &&+\widehat{S}_{n-1}\left( -13\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m} \widehat{S}_{k}+14\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}-7\sum_{k=0}^{m-2}\sum_{l=k}^{m-2}\sum_{p=l}^{m-2}\widehat{S} _{k} +3\sum_{k=0}^{m-3}\sum_{l=k}^{m-3}\sum_{p=l}^{m-3}\widehat{S} _{k} \right) \\ &&+\widehat{S}_{n-2}\left( 14\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m} \widehat{S}_{k}-7\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}+3\sum_{k=0}^{m-2}\sum_{l=k}^{m-2}\sum_{p=l}^{m-2}\widehat{S} _{k} \right) \\ &&+\widehat{S}_{n-3}\left( -7\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m} \widehat{S}_{k}+3\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}\right) + 3\widehat{S}_{n-4}\sum_{k=0}^{m}\sum_{l=k}^{m} \sum_{p=l}^{m}\widehat{S}_{k}, \end{eqnarray*} and \begin{eqnarray*} \widehat{R}_{n+m} &=&\widehat{R}_{n}\sum_{k=0}^{m+1}\sum_{l=k}^{m+1} \sum_{p=l}^{m+1}\widehat{S}_{k} \\ &&+\widehat{R}_{n-1}\left( -13\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m} \widehat{S}_{k}+14\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}-7\sum_{k=0}^{m-2}\sum_{l=k}^{m-2}\sum_{p=l}^{m-2}\widehat{S} _{k} +3\sum_{k=0}^{m-3}\sum_{l=k}^{m-3}\sum_{p=l}^{m-3}\widehat{S} _{k}\right) \\ &&+\widehat{R}_{n-2}\left( 14\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m} \widehat{S}_{k}-7\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}+3\sum_{k=0}^{m-2}\sum_{l=k}^{m-2}\sum_{p=l}^{m-2}\widehat{S} _{k} \right) \\ &&+\widehat{R}_{n-3}\left( -7\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m} \widehat{S}_{k}+3\sum_{k=0}^{m-1}\sum_{l=k}^{m-1}\sum_{p=l}^{m-1}\widehat{S} _{k}\right) +3\widehat{R}_{n-4}\sum_{k=0}^{m}\sum_{l=k}^{m}\sum_{p=l}^{m} \widehat{S}_{k}. \end{eqnarray*}

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.