Novel Classes on Generating Functions of the Products of (p,q)-Modified Pell Numbers with Several Bivariate Polynomials

Abstract

In this paper, using the symmetrizing operator ${\delta }_{e_1{e}_2}^{2-l}$, we derive new generating functions of the products of $\left(p,q\right)$-modified Pell numbers with various bivariate polynomials, including Mersenne and Mersenne Lucas polynomials, Fibonacci and Lucas polynomials, bivariate Pell and bivariate Pell Lucas polynomials, bivariate Jacobsthal and bivariate Jacobsthal Lucas polynomials, bivariate Vieta–Fibonacci and bivariate Vieta–Lucas polynomials, and bivariate complex Fibonacci and bivariate complex Lucas polynomials.

1. Introduction

A bivariate polynomial is a natural extension of univariate polynomials and plays a significant role in various fields like algebra, calculus, and applied mathematics, especially in multivariate calculus and algebraic geometry [1,2,3].

Modified Pell numbers and their generalizations have been studied for a long time. The recurrence relations of modified Pell numbers is given by $q_n=2q_{n-1}+q_{n-2},$ with the initial conditions $q_0=q_1=1.$ The first generalization of these numbers, called modified k-Pell numbers, ${\left\{q_{k,n}\right\}}_{n\ge 0}$, was defined by Catarino and Vasco in [4] as $q_{k,n}=2q_{k,n-1}+k{q}_{k,n-2},$ with the initial values $q_{k,0}=q_{k,1}=1.$ The $\left(p,q\right)$-modified Pell numbers ${\left\{M{P}_{p,q,n}\right\}}_{n\ge 0}$ [5] represent a second generalization of modified Pell numbers and are defined by the following generating function:

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{z}^n={\displaystyle \frac{1-pz}{1-2pz-q{z}^2}},$$

or by the following recurrence relation:

$$\left\{\begin{array}{c}M{P}_{p,q,0}=1,M{P}_{p,q,1}=p\\ M{P}_{p,q,n}=2pM{P}_{p,q,n-1}+qM{P}_{p,q,n-2}\mathrm{for}n\ge 2\end{array}\right..$$

Binet’s formula is well known and studied in the theory of Fibonacci numbers. This formula can also be carried out on $\left(p,q\right)$-modified Pell numbers. Let $\alpha$ and $\beta$ be the roots of the characteristic equation ${\lambda }^2-2p\lambda -q=0$ of the recurrence relation of $\left(p,q\right)$-modified Pell numbers. In the following proposition, we give Binet’s formula for $\left(p,q\right)$-modified Pell numbers [6,7].

Proposition 1

([8,9,10]). For $n\ge 0,$ Binet’s formula for $\left(p,q\right)$-modified Pell numbers is given by

$$M{P}_{p,q,n}=p\left({\displaystyle \frac{{\alpha }^n+{\beta }^n}{\alpha +\beta }}\right).$$

The Mersenne and Mersenne Lucas sequences are considered interesting classes of numbers with broad applications that offer a wide range of research opportunities (see [11,12]). Similarly, Jacobsthal, Jacobsthal Lucas, Fibonacci, Lucas, Pell and Pell Lucas numbers have been extentively studied (see, for instance, [13,14,15,16,17]). Over the years, numerous articles have appeared in various journals discussing the generalization of these numbers into what know as bivariate polynomials. Alves, in [18], introduced the concept of bivariate Mersenne polynomials ${\left\{M_n(x,y)\right\}}_{n\ge 0}$ as follows:

$$M_n\left(x,y\right)=3y{M}_{n-1}\left(x,y\right)-2x{M}_{n-2}\left(x,y\right),\mathrm{for}n\ge 2,$$

with the initial conditions $M_0\left(x,y\right)=0$ and $M_1\left(x,y\right)=1.$ In addition, the bivariate Mersenne Lucas polynomials ${\left\{m_n(x,y)\right\}}_{n\ge 0}$ are defined in [19,20] using the same recurrence relation but with different initial terms, $m_0\left(x,y\right)=2$ and $m_1\left(x,y\right)=3y$. Zorcelik and Uygun also introduced the bivariate Jacobsthal and bivariate Jacobsthal Lucas polynomials as

$$J_n(x,y)=xy{J}_{n-1}(x,y)+2y{J}_{n-2}(x,y)\mathrm{for}n\ge 2\mathrm{and}{J}_0(x,y)=0,J_1(x,y)=1,$$

and

$$j_n(x,y)=xy{j}_{n-1}(x,y)+2y{j}_{n-2}(x,y)\mathrm{for}n\ge 2\mathrm{and}{j}_0(x,y)=2,j_1(x,y)=xy,$$

respectively, and studied these bivariate polynomials in [21]. The same author (Uygun), in [3], provided the generalized bivariate Jacobsthal and bivariate Jacobsthal Lucas polynomials sequences. In [22], Asci and Gurel introduced the bivariate complex Fibonacci ${\left\{f_n(x,y)\right\}}_{n\ge 0}$ and bivariate complex Lucas ${\left\{l_n(x,y)\right\}}_{n\ge 0}$ polynomials as

$$f_n(x,y)=ix{f}_{n-1}(x,y)+y{f}_{n-2}(x,y),\mathrm{for}n\ge 2\mathrm{and}{f}_0(x,y)=0,f_1(x,y)=1,$$

and

$$l_n(x,y)=ix{l}_{n-1}(x,y)+y{l}_{n-2}(x,y),\mathrm{for}n\ge 2\mathrm{and}{l}_0(x,y)=2,l_1(x,y)=ix.$$

Furthermore, in 2004, Catalani defined the bivariate Fibonacci and bivariate Lucas polynomials ${\left\{F_n(x,y)\right\}}_{n\ge 0}$ and ${\left\{L_n(x,y)\right\}}_{n\ge 0}$ in [23] as follows:

$$\left\{\begin{array}{c}{F}_0(x,y)=0,F_1(x,y)=1\\ F_n(x,y)=x{F}_{n-1}(x,y)+y{F}_{n-2}(x,y)\mathrm{for}n\ge 2\end{array}\right.,$$

(1.1)

and

$$\left\{\begin{array}{c}{L}_0(x,y)=2,L_1(x,y)=x\\ L_n(x,y)=x{L}_{n-1}(x,y)+y{L}_{n-2}(x,y)\mathrm{for}n\ge 2\end{array}\right.,$$

(1.2)

In 2016, Kocer defined the bivariate Vieta–Fibonacci and bivariate Vieta–Lucas polynomials in [8] using the following recurrence relations:

$$\left\{\begin{array}{c}{V}_0(x,y)=0,V_1(x,y)=1\\ V_n(x,y)=x{V}_{n-1}(x,y)-y{V}_{n-2}(x,y),\mathrm{for}n\ge 2\end{array}\right.,$$

(1.3)

and

$$\left\{\begin{array}{c}{v}_0(x,y)=2,v_1(x,y)=x\\ v_n(x,y)=x{v}_{n-1}(x,y)-y{v}_{n-2}(x,y),\mathrm{for}n\ge 2\end{array}\right..$$

(1.4)

respectively. The bivariate Pell and bivariate Pell Lucas polynomials, ${\left\{P_n(x,y)\right\}}_{n\ge 0}$ and ${\left\{Q_n(x,y)\right\}}_{n\ge 0}$, are defined using the following recurrence relations:

$$\left\{\begin{array}{c}{P}_n(x,y)=2xy{P}_{n-1}(x,y)+y{P}_{n-2}(x,y),\mathrm{for}n\ge 2\\ P_0(x,y)=0,P_1(x,y)=1\end{array}\right.,$$

(1.5)

and

$$\left\{\begin{array}{c}{Q}_n(x,y)=2xy{Q}_{n-1}(x,y)+y{Q}_{n-2}(x,y),\mathrm{for}n\ge 2\\ Q_0(x,y)=2,Q_1(x,y)=2xy\end{array}\right..$$

(1.6)

Remark 1.

If we set $y=1$ in Equations (1.1)–(1.6), we obtain the recurrence relations for the Fibonacci, Lucas, Vieta–Fibonacci, Vieta–Lucas, Pell and Pell Lucas polynomials, respectively.

The rest of this paper is organized as follows. In Section 2, the operator ${\delta }_{a_1{a}_2}^k$ and symmetric functions are defined. By utilizing the symmetric functions introduced in the previous section, Section 3 derives new generating functions of the products of $\left(p,q\right)$-modified Pell numbers with bivariate Mersenne and bivariate Mersenne Lucas polynomials, bivariate Fibonacci and bivariate Lucas polynomials, bivariate Pell and bivariate Pell Lucas polynomials, bivariate Jacobsthal and bivariate Jacobsthal Lucas polynomials, bivariate Vieta–Fibonacci and bivariate Vieta–Lucas polynomials, and bivariate complex Fibonacci and bivariate complex Lucas polynomials.

2. Notations and Some Properties

In this section, we introduce a symmetric function and discuss some of its properties. Additionally, we provide other useful definitions from the literature that will be used in the subsequent sections.

We handle functions on different sets of indeterminates (called alphabets, though we mostly use commutative indeterminates for the moment).

A symmetric function of an alphabet A is a function of the letters that is invariant under permutation of the letters of A. Introducing an additional indeterminate z, we have two fundamental series (see [24,25]):

$${\lambda }_z\left(A\right)=\prod _{a\in A}(1+az),{\sigma }_z\left(A\right)={\displaystyle \frac{1}{{\displaystyle \prod _{a\in A}}(1-az)}}.$$

(2.1)

With (2.1), we have

$${\lambda }_z\left(A\right)={\displaystyle \sum _{n=0}^{\infty }}{\lambda }_n\left(A\right)z^n,{\sigma }_z\left(A\right)={\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(A\right)z^n,$$

where ${\lambda }_n\left(A\right)$ and $S_n\left(A\right)$ are the elementary symmetric and complete functions, respectively.

Definition 1

([24]). Let n be a positive integer and A and B be any two alphabets. Then, $S_n(A-B)$ is defined as

$${\displaystyle \frac{{\displaystyle \prod _{b\in B}}(1-bz)}{{\displaystyle \prod _{a\in A}}(1-az)}}={\displaystyle \sum _{n=0}^{\infty }}{S}_n(A-B)z^n={\sigma }_z(A-B),$$

(2.2)

with the condition that $S_n(A-B)=0$ for $n<0$.

Corollary 1.

Taking $A=\left\{0\right\}$ in (2.2) gives

$$\prod _{b\in B}(1-bz)={\displaystyle \sum _{n=0}^{\infty }}{S}_n(-B)z^n={\lambda }_z(-B).$$

(2.3)

Further, in the case where $A=\left\{0\right\}$ or $B=\left\{0\right\},$ we have

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n(A-B)z^n={\sigma }_z\left(A\right)\times {\lambda }_z(-B).$$

(2.4)

Thus, we obtain (see [24])

$$S_n(A-B)={\displaystyle \sum _{k=0}^n}{S}_{n-k}\left(A\right)S_k(-B),$$

Definition 2

([19]). Let n be a positive integer and $A=\left\{a_1,a_2\right\}$ be a set of given variables. Then, the nth symmetric function $S_n(a_1+a_2)$ is defined as

$$S_n\left(A\right)=S_n(a_1+a_2)={\displaystyle \frac{a_1^{n+1}-a_2^{n+1}}{a_1-a_2}},$$

with

$$\begin{array}{ccc}\hfill S_0\left(A\right)& =& S_0(a_1+a_2)=1,\hfill \\ \hfill S_1\left(A\right)& =& S_1(a_1+a_2)=a_1+a_2,\hfill \\ \hfill S_2\left(A\right)& =& S_2(a_1+a_2)=a_1^2+a_1{a}_2+a_2^2,\hfill \\ & & ⋮\hfill \end{array}$$

Definition 3

([19]). The symmetrizing operator ${\delta }_{e_1{e}_2}^k$ is defined as

$${\delta }_{e_1{e}_2}^{k}f\left(e_1\right)={\displaystyle \frac{e_1^{k}f\left(e_1\right)-e_2^{k}f\left(e_2\right)}{e_1-e_2}},\mathit{for}\mathit{all}k\in {\mathbb{N}}_0.$$

(2.5)

Proposition 2

([20,26,27]). For $n\in \mathbb{N},$ the symmetric functions of bivariate Fibonacci and Lucas polynomials, bivariate Pell and Pell Lucas polynomials, bivariate Jacobsthal and Jacobsthal Lucas polynomials, bivariate Vieta–Fibonacci and Lucas polynomials, bivariate complex Fibonacci and Lucas polynomials, bivariate Mersenne and Mersenne Lucas polynomials, and $\left(p,q\right)$-modified Pell numbers are, respectively, given by

$$\begin{array}{cc}\hfill F_n(x,y)& =S_{n-1}\left(a_1+\left[-a_2\right]\right)\mathit{and}{L}_n(x,y)=2S_n\left(a_1+\left[-a_2\right]\right)-x{S}_{n-1}\left(a_1+\left[-a_2\right]\right),\hfill \\ & \mathit{with}{a}_{1,2}={\displaystyle \frac{x\pm \sqrt{x^2+4y}}{2}}.\hfill \\ \hfill P_n(x,y)& =S_{n-1}\left(a_1+\left[-a_2\right]\right)\mathit{and}{Q}_n(x,y)=2S_n\left(a_1+\left[-a_2\right]\right)-2xy{S}_{n-1}\left(a_1+\left[-a_2\right]\right),\hfill \\ & \mathit{with}{a}_{1,2}=xy\pm \sqrt{x^2{y}^2+y}.\hfill \\ \hfill J_n(x,y)& =S_{n-1}\left(a_1+\left[-a_2\right]\right)\mathit{and}{j}_n(x,y)=2S_n\left(a_1+\left[-a_2\right]\right)-xy{S}_{n-1}\left(a_1+\left[-a_2\right]\right),\hfill \\ & \mathit{with}{a}_{1,2}={\displaystyle \frac{xy\pm \sqrt{x^2{y}^2+8y}}{2}}.\hfill \\ \hfill V_n(x,y)& =S_{n-1}\left(a_1+\left[-a_2\right]\right)\mathit{and}{v}_n(x,y)=2S_n\left(a_1+\left[-a_2\right]\right)-x{S}_{n-1}\left(a_1+\left[-a_2\right]\right),\hfill \\ & \mathit{with}{a}_{1,2}={\displaystyle \frac{x\pm \sqrt{x^2-4y}}{2}}.\hfill \\ \hfill f_n(x,y)& =S_{n-1}\left(a_1+\left[-a_2\right]\right)\mathit{and}{l}_n(x,y)=2S_n\left(a_1+\left[-a_2\right]\right)-ix{S}_{n-1}\left(a_1+\left[-a_2\right]\right),\hfill \\ & \mathit{with}{a}_{1,2}={\displaystyle \frac{ix\pm \sqrt{-x^2+4y}}{2}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill M(x,y)& =S_{n-1}\left(a_1+\left[-a_2\right]\right)\mathit{and}{m}_n(x,y)=2S_n\left(a_1+\left[-a_2\right]\right)-3y{S}_{n-1}\left(a_1+\left[-a_2\right]\right),\hfill \\ & \mathit{with}{a}_{1,2}={\displaystyle \frac{3y\pm \sqrt{9y^2-8x}}{2}}.\hfill \\ \hfill M{P}_{p,q,n}& =S_n\left(a_1+\left[-a_2\right]\right)-p{S}_{n-1}\left(a_1+\left[-a_2\right]\right),\hfill \\ & \mathit{with}{a}_{1,2}=p+\sqrt{p^2+q}.\hfill \end{array}$$

3. A New Class of Ordinary Generating Functions of Binary Products of $\left(\mathit{p},\mathit{q}\right)$-Modified Pell Numbers with Bivariate Polynomials and Some Special Cases

The following theorem is one of the key tools of the proof of our main result. Although it was proved in [16], we include its proof here for the completeness of this paper.

Theorem 1.

Given two alphabets, $A=\left\{a_1,a_2,\cdots ,a_k\right\}$ and $E=\left\{e_1,e_2\right\},$ we have

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(A\right)S_{n+1-l}\left(E\right)z^n=$$

$${\displaystyle \frac{S_{1-l}\left(E\right)+S_1\left(-A\right)e_1{e}_2{S}_{-l}\left(E\right)z-e_1^{2-l}{e}_2^{2-l}{z}^{3-l}{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-l+3}\left(-A\right)S_n\left(E\right)z^n}{\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_1^n{z}^n\right)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_2^n{z}^n\right)}},$$

(3.1)

for all $n,$$k\in {\mathbb{N}}_0$ and $l\in \left\{0,1,2\right\}$.

Proof. 

By applying the operator ${\delta }_{e_1{e}_2}^{2-l}$ to the series $f\left(e_{1}z\right)={\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(A\right)e_1^n{z}^n$, we obtain

$$\begin{array}{ccc}\hfill {\delta }_{e_1{e}_2}^{2-l}f\left(e_{1}z\right)& =& {\displaystyle \frac{e_1^{2-l}{\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(A\right)e_1^n{z}^n-e_2^{2-l}{\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(A\right)e_2^n{z}^n}{e_1-e_2}}\hfill \\ & =& \sum _{n=0}^{\infty }{S}_n\left(A\right)\left({\displaystyle \frac{e_1^{n-l+2}-e_2^{n-l+2}}{e_1-e_2}}\right)z^n\hfill \\ & =& \sum _{n=0}^{\infty }{S}_n\left(A\right)S_{n+1-l}\left(E\right)z^n.\hfill \end{array}$$

On the other hand, by applying the operator ${\delta }_{e_1{e}_2}^{2-l}$ to the series $f\left(e_{1}z\right)={\displaystyle \frac{1}{{\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_1^n{z}^n}},$ we obtain

$$\begin{array}{ccc}\hfill {\delta }_{e_1{e}_2}^{2-l}f\left(e_{1}z\right)& =& {\displaystyle \frac{\frac{e_1^{2-l}}{{\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_1^n{z}^n}-\frac{e_2^{2-l}}{{\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_2^n{z}^n}}{e_1-e_2}}\hfill \\ & =& {\displaystyle \frac{e_1^{2-l}{\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_2^n{z}^n-e_2^{2-l}{\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_1^n{z}^n}{(e_1-e_2)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_1^n{z}^n\right)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_2^n{z}^n\right)}}\hfill \\ & =& {\displaystyle \frac{{\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_1^n{e}_2^n\frac{e_1^{2-l-n}-e_2^{2-l-n}}{e_1-e_2}{z}^n}{\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_1^n{z}^n\right)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_2^n{z}^n\right)}}.\hfill \end{array}$$

Equivalently,

$$\begin{array}{ccc}\hfill {\delta }_{e_1{e}_2}^{2-l}f\left(e_{1}z\right)& =& {\displaystyle \frac{{\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_1^n{e}_2^n{S}_{1-n-l}\left(E\right)z^n}{\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_1^n{z}^n\right)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_2^n{z}^n\right)}}\hfill \\ & =& {\displaystyle \frac{{\displaystyle \sum _{n=0}^{1-l}}{S}_n\left(-A\right)e_1^n{e}_2^n{S}_{1-n-l}\left(E\right)z^n+{\displaystyle \sum _{n=3-l}^{\infty }}{S}_n\left(-A\right)e_1^n{e}_2^n{S}_{1-n-l}\left(E\right)z^n}{\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_1^n{z}^n\right)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_2^n{z}^n\right)}}\hfill \\ & =& {\displaystyle \frac{{\displaystyle \sum _{n=0}^{1-l}}{S}_n\left(-A\right)e_1^n{e}_2^n{S}_{1-n-l}\left(E\right)z^n-{\displaystyle \sum _{n=3-l}^{\infty }}{S}_n\left(-A\right)e_1^{2-l}{e}_2^{2-l}\left(\frac{e_1^{n+l-2}-e_2^{n+l-2}}{e_1-e_2}\right)z^n}{\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_1^n{z}^n\right)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_2^n{z}^n\right)}},\hfill \end{array}$$

which also gives

$$\begin{array}{ccc}\hfill {\delta }_{e_1{e}_2}^{2-l}f\left(e_{1}z\right)& =& {\displaystyle \frac{S_{1-l}\left(E\right)+S_1\left(-A\right)e_1{e}_2{S}_{-l}\left(E\right)z-e_1^{2-l}{e}_2^{2-l}{\displaystyle \sum _{n=3-l}^{\infty }}{S}_n\left(-A\right)S_{n+l-3}\left(E\right)z^n}{\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_1^n{z}^n\right)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_2^n{z}^n\right)}}\hfill \\ & =& {\displaystyle \frac{S_{1-l}\left(E\right)+S_1\left(-A\right)e_1{e}_2{S}_{-l}\left(E\right)z-e_1^{2-l}{e}_2^{2-l}{z}^{3-l}{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-l+3}\left(-A\right)S_n\left(E\right)z^n}{\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_1^n{z}^n\right)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(-A\right)e_2^n{z}^n\right)}}.\hfill \end{array}$$

Hence, we obtain the desired result. □

• For $A=\left\{a_1,a_2\right\},$$E=\left\{e_1,e_2\right\}$ and $l\in \left\{0,1,2\right\}$, Theorem 1 allows us to deduce the following lemmas.

Lemma 1.

Given two alphabets, $E=\left\{e_1,e_2\right\}$ and $A=\left\{a_1,a_2\right\},$ then we have

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(A\right)S_{n+1}\left(E\right)z^n={\displaystyle \frac{(e_1+e_2)z^2-e_1{e}_2(a_1+a_2)z^3}{\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n(-A)e_1^n{z}^n\right)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n(-A)e_2^n{z}^n\right)}}.$$

(3.2)

Note that, based on relationship (3.2), we obtain

$${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}\left(A\right)S_n\left(E\right)z^n={\displaystyle \frac{(e_1+e_2)z-e_1{e}_2(a_1+a_2)z^2}{\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n(-A)e_1^n{z}^n\right)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n(-A)e_2^n{z}^n\right)}}.$$

(3.3)

Lemma 2.

Given two alphabets, $E=\left\{e_1,e_2\right\}$ and $A=\left\{a_1,a_2\right\},$ we have

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(A\right)S_n\left(E\right)z^n={\displaystyle \frac{1-a_1{a}_2{e}_1{e}_2{z}^2}{\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n(-A)e_1^n{z}^n\right)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n(-A)e_2^n{z}^n\right)}}.$$

(3.4)

Based on relationship (3.4), we obtain

$${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}\left(A\right)S_{n-1}\left(E\right)z^n={\displaystyle \frac{z-a_1{a}_2{e}_1{e}_2{z}^3}{\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n(-A)e_1^n{z}^n\right)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n(-A)e_2^n{z}^n\right)}}.$$

(3.5)

Lemma 3.

Given two alphabets, $E=\left\{e_1,e_2\right\}$ and $A=\left\{a_1,a_2\right\},$ we have

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(A\right)S_{n-1}\left(E\right)z^n={\displaystyle \frac{(a_1+a_2)z-a_1{a}_2(e_1+e_2)z^2}{\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n(-A)e_1^n{z}^n\right)\left({\displaystyle \sum _{n=0}^{\infty }}{S}_n(-A)e_2^n{z}^n\right)}}.$$

(3.6)

In this part, we derive new generating functions of the products of $\left(p,q\right)$-modified Pell numbers with bivariate Fibonacci and bivariate Lucas polynomials, bivariatePell and bivariate Pell Lucas polynomials, bivariate Jacobsthal and bivariate Jacobsthal Lucas polynomials, bivariate Vieta–Fibonacci and bivariate Vieta–Lucas polynomials, and bivariate complex Fibonacci and bivariate complex Lucas polynomials. Furthermore, we present the special cases of these generating functions.

• For the case of $A=\left\{a_1,-a_2\right\}$ and $E=\left\{e_1,-e_2\right\}$, by replacing $a_2$ with $\left(-a_2\right)$ and $e_2$ with $\left(-e_2\right)$ in Equations (3.3)–(3.6) we have

  $${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+\left[-a_2\right])S_n(e_1+\left[-e_2\right])z^n={\displaystyle \frac{(e_1-e_2)z+e_1{e}_2(a_1-a_2)z^2}{(1-a_1{e}_{1}z)(1+a_2{e}_{1}z)(1+a_1{e}_{2}z)(1-a_2{e}_{2}z)}},$$

  (3.7)

  $${\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(a_1+\left[-a_2\right]\right)S_n\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{1-a_1{a}_2{e}_1{e}_2{z}^2}{(1-a_1{e}_{1}z)(1+a_2{e}_{1}z)(1+a_1{e}_{2}z)(1-a_2{e}_{2}z)}},$$

  (3.8)

  $${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}\left(a_1+\left[-a_2\right]\right)S_{n-1}\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{z-a_1{a}_2{e}_1{e}_2{z}^3}{(1-a_1{e}_{1}z)(1+a_2{e}_{1}z)(1+a_1{e}_{2}z)(1-a_2{e}_{2}z)}},$$

  (3.9)

  $${\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+\left[-a_2\right])S_{n-1}(e_1+\left[-e_2\right])z^n={\displaystyle \frac{(a_1-a_2)z+a_1{a}_2(e_1-e_2)z^2}{(1-a_1{e}_{1}z)(1+a_2{e}_{1}z)(1+a_1{e}_{2}z)(1-a_2{e}_{2}z)}},$$

  (3.10)

  respectively.
  This case consists of six related parts.
  Firstly, the following substitutions,

  $$\left\{\begin{array}{c}\hfill a_1-a_2=2p\hfill \\ \hfill a_1{a}_2=q\hfill \end{array}\right.\mathrm{and}\left\{\begin{array}{c}\hfill e_1-e_2=x\hfill \\ \hfill e_1{e}_2=y\hfill \end{array}\right.,$$

  in (3.7)–(3.10) would give

  $${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}\left(a_1+\left[-a_2\right]\right)S_n\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{xz+2py{z}^2}{1-2pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-2pqxy{z}^3+q^2{y}^2{z}^4}},$$

  (3.11)

  $${\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(a_1+\left[-a_2\right]\right)S_n\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{1-qy{z}^2}{1-2pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-2pqxy{z}^3+q^2{y}^2{z}^4}},$$

  (3.12)

  $${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}\left(a_1+\left[-a_2\right]\right)S_{n-1}\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{z-qy{z}^3}{1-2pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-2pqxy{z}^3+q^2{y}^2{z}^4}},$$

  (3.13)

  $${\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(a_1+\left[-a_2\right]\right)S_{n-1}\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{2pz+qx{z}^2}{1-2pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-2pqxy{z}^3+q^2{y}^2{z}^4}},$$

  (3.14)

  respectively, and we have the following results.

Theorem 2.

For $n\in \mathbb{N},$ the new generating function of the product of $\left(p,q\right)$-modified Pell numbers with bivariate Fibonacci polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{F}_n\left(x,y\right)z^n={\displaystyle \frac{pz+qx{z}^2+pqy{z}^3}{1-2pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-2pqxy{z}^3+q^2{y}^2{z}^4}}.$$

(3.15)

Proof. 

Referring to [28], we have

$$F_n\left(x,y\right)=S_{n-1}(e_1+[-e_2]).$$

We see that

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{F}_n\left(x,y\right)z^n& =& {\displaystyle \sum _{n=0}^{\infty }}\left(S_n(a_1+[-a_2])-p{S}_{n-1}(a_1+[-a_2])\right)S_{n-1}(e_1+[-e_2])z^n\hfill \\ & =& {\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & & -p{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & =& {\displaystyle \frac{2pz+qx{z}^2}{1-2pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-2pqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{p\left(z-qy{z}^3\right)}{1-2pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-2pqxy{z}^3+q^2{y}^2{z}^4}},\hfill \end{array}$$

after a simple calculation, we have

$$M{P}_{p,q,n}{F}_n\left(x,y\right)={\displaystyle \frac{pz+qx{z}^2+pqy{z}^3}{1-2pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-2pqxy{z}^3+q^2{y}^2{z}^4}}.$$

So, the proof is completed. □

Theorem 3.

For $n\in \mathbb{N},$ the new generating function of the product of $\left(p,q\right)$-modified Pell numbers with bivariate Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{L}_n\left(x,y\right)z^n={\displaystyle \frac{2-3pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-pqxy{z}^3}{1-2pxz-\left(q{x}^2+2y\left(2p^2+q\right)\right)z^2-2pqxy{z}^3+q^2{y}^2{z}^4}}.$$

(3.16)

Proof. 

We know that (see [28])

$$L_n\left(x,y\right)=2S_n(e_1+[-e_2])-x{S}_{n-1}(e_1+[-e_2]),$$

We see that

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{L}_n\left(x,y\right)z^n& =& {\displaystyle \sum _{n=0}^{\infty }}\left(\begin{array}{c}\left(S_n(a_1+[-a_2])-p{S}_{n-1}(a_1+[-a_2])\right)\\ \times (2S_n(e_1+[-e_2])-x{S}_{n-1}(e_1+[-e_2]))\end{array}\right)z^n\hfill \\ & =& 2{\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_n(e_1+[-e_2])z^n\hfill \\ & & -x{\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & & -2p{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_n(e_1+[-e_2])z^n\hfill \\ & & +px{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n.\hfill \end{array}$$

According to relationships (3.11)–(3.14), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{L}_n\left(x,y\right)z^n& =& {\displaystyle \frac{2\left(1-qy{z}^2\right)}{1-2pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-2pqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{x\left(2pz+qx{z}^2\right)}{1-2pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-2pqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{2p\left(xz+2py{z}^2\right)}{1-2pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-2pqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & +{\displaystyle \frac{px\left(z-qy{z}^3\right)}{1-2pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-2pqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & =& {\displaystyle \frac{2-3pxz-(q{x}^2+2y\left(2p^2+q\right))z^2-pqxy{z}^3}{1-2pxz-\left(q{x}^2+2y\left(2p^2+q\right)\right)z^2-2pqxy{z}^3+q^2{y}^2{z}^4}}.\hfill \end{array}$$

Thus, this completes the proof. □

Corollary 2.

By substituting $(p=1$, $q=k)$ and $(p=q=1)$ into Equations (3.15) and (3.16), we obtain the following new generating functions:

(1) The new generating function of the product of modified k-Pell numbers with bivariate Fibonacci polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_{k,n}{F}_n\left(x,y\right)z^n={\displaystyle \frac{z+kx{z}^2+ky{z}^3}{1-2xz-(k{x}^2+2y\left(2+k\right))z^2-2kxy{z}^3+k^2{y}^2{z}^4}}.$$

(2) The new generating function of the product of modified k-Pell numbers with bivariate Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_{k,n}{L}_n\left(x,y\right)z^n={\displaystyle \frac{2-3xz-(k{x}^2+2y\left(2+k\right))z^2-kxy{z}^3}{1-2xz-\left(k{x}^2+2y\left(2+k\right)\right)z^2-2kxy{z}^3+k^2{y}^2{z}^4}}.$$

(3) The new generating function of the product of modified Pell numbers with bivariate Fibonacci polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_n{F}_n\left(x,y\right)z^n={\displaystyle \frac{z+x{z}^2+y{z}^3}{1-2xz-(x^2+6y)z^2-2xy{z}^3+y^2{z}^4}}.$$

(4) The new generating function of the product of modified Pell numbers with bivariate Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_n{L}_n\left(x,y\right)z^n={\displaystyle \frac{2-3xz-\left(6y+x^2\right)z^2-xy{z}^3}{1-2xz-(x^2+6y)z^2-2xy{z}^3+y^2{z}^4}}.$$

• Putting $\left(y=1\right)$ and $\left(x=y=1\right)$ in Theorems 2 and 3 and Corollary 2, we obtain the following table (Table 1):

Table 1. New generating functions of the products of some numbers and polynomials.

Coefficient of  ${\mathit{z}}^{\mathit{n}}$	Generating Function	$\mathbf{Coefficient}\mathbf{of}{\mathit{z}}^{\mathit{n}}$	Generating Function
$M{P}_{p,q,n}{F}_n\left(x\right)$	${\displaystyle \frac{pz+qx{z}^2+pq{z}^3}{1-2pxz-\left(q{x}^2+2\left(2p^2+q\right)\right)z^2-2pqx{z}^3+q^2{z}^4}}$	$M{P}_{p,q,n}{F}_n$	${\displaystyle \frac{pz+q{z}^2+pq{z}^3}{1-2pz-\left(4p^2+3q\right)z^2-2pq{z}^3+q^2{z}^4}}$
$M{P}_{p,q,n}{L}_n\left(x\right)$	${\displaystyle \frac{2-3pxz-(q{x}^2+2\left(2p^2+q\right))z^2-pqx{z}^3}{1-2pxz-\left(q{x}^2+2\left(2p^2+q\right)\right)z^2-2pqx{z}^3+q^2{z}^4}}$	$M{P}_{p,q,n}{L}_n$	${\displaystyle \frac{2-3pz-(4p^2+3q)z^2-pq{z}^3}{1-2pz-(4p^2+3q)z^2-2pq{z}^3+q^2{z}^4}}$
$q_{k,n}{F}_n\left(x\right)$	${\displaystyle \frac{z+kx{z}^2+k{z}^3}{1-2xz-\left(k{x}^2+2\left(2+k\right)\right)z^2-2kx{z}^3+k^2{z}^4}}$	$q_{k,n}{F}_n$	${\displaystyle \frac{z+k{z}^2+k{z}^3}{1-2z-\left(4+3k\right)z^2-2k{z}^3+k^2{z}^4}}$
$q_{k,n}{L}_n\left(x\right)$	${\displaystyle \frac{2-3xz-(k{x}^2+2\left(2+k\right))z^2-kx{z}^3}{1-2xz-\left(k{x}^2+2\left(2+k\right)\right)z^2-2kx{z}^3+k^2{z}^4}}$	$q_{k,n}{L}_n$	${\displaystyle \frac{2-3z-(4+3k)z^2-k{z}^3}{1-2z-(4+3k)z^2-2k{z}^3+k^2{z}^4}}$
$q_n{F}_n\left(x\right)$	${\displaystyle \frac{z+x{z}^2+z^3}{1-2xz-\left(x^2+6\right)z^2-2x{z}^3+z^4}}$	$q_n{F}_n$	${\displaystyle \frac{z+z^2+z^3}{1-2z-7z^2-2z^3+z^4}}$
$q_n{L}_n\left(x\right)$	${\displaystyle \frac{2-3xz-(x^2+6)z^2-x{z}^3}{1-2xz-\left(x^2+6\right)z^2-2x{z}^3+z^4}}$	$q_n{L}_n$	${\displaystyle \frac{2-3z-7z^2-z^3}{1-2z-7z^2-2z^3+z^4}}$

Table 1. New generating functions of the products of some numbers and polynomials.

Coefficient of  ${\mathit{z}}^{\mathit{n}}$	Generating Function	$\mathbf{Coefficient}\mathbf{of}{\mathit{z}}^{\mathit{n}}$	Generating Function
$M{P}_{p,q,n}{F}_n\left(x\right)$	${\displaystyle \frac{pz+qx{z}^2+pq{z}^3}{1-2pxz-\left(q{x}^2+2\left(2p^2+q\right)\right)z^2-2pqx{z}^3+q^2{z}^4}}$	$M{P}_{p,q,n}{F}_n$	${\displaystyle \frac{pz+q{z}^2+pq{z}^3}{1-2pz-\left(4p^2+3q\right)z^2-2pq{z}^3+q^2{z}^4}}$
$M{P}_{p,q,n}{L}_n\left(x\right)$	${\displaystyle \frac{2-3pxz-(q{x}^2+2\left(2p^2+q\right))z^2-pqx{z}^3}{1-2pxz-\left(q{x}^2+2\left(2p^2+q\right)\right)z^2-2pqx{z}^3+q^2{z}^4}}$	$M{P}_{p,q,n}{L}_n$	${\displaystyle \frac{2-3pz-(4p^2+3q)z^2-pq{z}^3}{1-2pz-(4p^2+3q)z^2-2pq{z}^3+q^2{z}^4}}$
$q_{k,n}{F}_n\left(x\right)$	${\displaystyle \frac{z+kx{z}^2+k{z}^3}{1-2xz-\left(k{x}^2+2\left(2+k\right)\right)z^2-2kx{z}^3+k^2{z}^4}}$	$q_{k,n}{F}_n$	${\displaystyle \frac{z+k{z}^2+k{z}^3}{1-2z-\left(4+3k\right)z^2-2k{z}^3+k^2{z}^4}}$
$q_{k,n}{L}_n\left(x\right)$	${\displaystyle \frac{2-3xz-(k{x}^2+2\left(2+k\right))z^2-kx{z}^3}{1-2xz-\left(k{x}^2+2\left(2+k\right)\right)z^2-2kx{z}^3+k^2{z}^4}}$	$q_{k,n}{L}_n$	${\displaystyle \frac{2-3z-(4+3k)z^2-k{z}^3}{1-2z-(4+3k)z^2-2k{z}^3+k^2{z}^4}}$
$q_n{F}_n\left(x\right)$	${\displaystyle \frac{z+x{z}^2+z^3}{1-2xz-\left(x^2+6\right)z^2-2x{z}^3+z^4}}$	$q_n{F}_n$	${\displaystyle \frac{z+z^2+z^3}{1-2z-7z^2-2z^3+z^4}}$
$q_n{L}_n\left(x\right)$	${\displaystyle \frac{2-3xz-(x^2+6)z^2-x{z}^3}{1-2xz-\left(x^2+6\right)z^2-2x{z}^3+z^4}}$	$q_n{L}_n$	${\displaystyle \frac{2-3z-7z^2-z^3}{1-2z-7z^2-2z^3+z^4}}$

Secondly, the following substitutions,

$$\left\{\begin{array}{c}{a}_1-a_2=2p\\ a_1{a}_2=q\end{array}\right.\mathrm{and}\left\{\begin{array}{c}{e}_1-e_2=xy\\ e_1{e}_2=2y\end{array}\right.,$$

in (3.7)–(3.10) would give

$${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_n(e_1+[-e_2])z^n={\displaystyle \frac{xyz+4py{z}^2}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}},$$

(3.17)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_n(e_1+[-e_2])z^n={\displaystyle \frac{1-2qy{z}^2}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}},$$

(3.18)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n={\displaystyle \frac{z-2qy{z}^3}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}},$$

(3.19)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n={\displaystyle \frac{2pz+qxy{z}^2}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}},$$

(3.20)

respectively, and we have the following theorems.

Theorem 4.

For $n\in \mathbb{N},$ the new generating function of the product of $\left(p,q\right)$-modified Pell numbers with bivariate Jacobsthal polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{J}_n\left(x,y\right)z^n={\displaystyle \frac{pz+qxy{z}^2+2pqy{z}^3}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}}.$$

(3.21)

Proof. 

Using [28], we have $J_n\left(x,y\right)=S_{n-1}(e_1+[-e_2])$. Then, we can see that

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{J}_n\left(x,y\right)z^n& =& \hfill {\displaystyle \sum _{n=0}^{\infty }}\left(S_n(a_1+[-a_2])-p{S}_{n-1}(a_1+[-a_2])\right)S_{n-1}(e_1+[-e_2]))z^n\hfill \\ & =& {\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & & -p{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & =& \hfill {\displaystyle \frac{2pz+qxy{z}^2}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}}\hfill \\ & & \hfill -{\displaystyle \frac{p\left(z-2qy{z}^3\right)}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}},\hfill \end{array}$$

after a simple calculation, we have

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{J}_n\left(x,y\right)z^n={\displaystyle \frac{pz+qxy{z}^2+2pqy{z}^3}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}}.$$

This completes the proof. □

Theorem 5.

For $n\in \mathbb{N},$ the new generating function of the product of $\left(p,q\right)$-modified Pell numbers with bivariate Jacobsthal Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{j}_n\left(x,y\right)z^n={\displaystyle \frac{2-3pxyz-\left(q{x}^2{y}^2+4y\left(2p^2+q\right)\right)z^2-2pqx{y}^2{z}^3}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}}.$$

(3.22)

Proof. 

We know that (see [28])

$$j_n\left(x,y\right)=2S_n(e_1+[-e_2])-xy{S}_{n-1}(e_1+[-e_2]),$$

We see that

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{j}_n\left(x,y\right)z^n& =& {\displaystyle \sum _{n=0}^{\infty }}\left(\begin{array}{c}\left(S_n(a_1+[-a_2])-p{S}_{n-1}(a_1+[-a_2])\right)\\ \times (2S_n(e_1+[-e_2])-xy{S}_{n-1}(e_1+[-e_2]))\end{array}\right)z^n\hfill \\ & =& 2{\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_n(e_1+[-e_2])z^n\hfill \\ & & -xy{\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & & -2p{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_n(e_1+[-e_2])z^n\hfill \\ & & +pxy{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n,\hfill \end{array}$$

Using relationships (3.17)–(3.20), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{j}_n\left(x,y\right)z^n& =& \hfill {\displaystyle \frac{2\left(1-2qy{z}^2\right)}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}}\hfill \\ & & \hfill -{\displaystyle \frac{xy\left(2pz+qxy{z}^2\right)}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}}\hfill \\ & & \hfill -{\displaystyle \frac{2p\left(xyz+4py{z}^2\right)}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}}\hfill \\ & & \hfill +{\displaystyle \frac{pxy\left(z-2qy{z}^3\right)}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}}\hfill \\ & =& \hfill {\displaystyle \frac{2-3pxyz-\left(q{x}^2{y}^2+4y\left(2p^2+q\right)\right)z^2-2pqx{y}^2{z}^3}{1-2pxyz-(q{x}^2{y}^2+4y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+4q^2{y}^2{z}^4}}.\hfill \end{array}$$

Thus, this completes the proof. □

Corollary 3.

By putting $(p=1$ and $q=k)$ and $(p=q=1)$ in Equations (3.21) and (3.22), we obtain the following new generating functions:

(1) The new generating function of the product of modified k-Pell numbers with bivariate Jacobsthal polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_{k,n}{J}_n\left(x,y\right)z^n={\displaystyle \frac{z+kxy{z}^2+2ky{z}^3}{1-2xyz-(k{x}^2{y}^2+4y\left(2+k\right))z^2-4kx{y}^2{z}^3+4k^2{y}^2{z}^4}}.$$

(2) The new generating function of the product of modified k-Pell numbers with bivariate Jacobsthal Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_{k,n}{j}_n\left(x,y\right)z^n={\displaystyle \frac{2-3xyz-\left(k{x}^2{y}^2+4y\left(2+k\right)\right)z^2-2kx{y}^2{z}^3}{1-2xyz-(k{x}^2{y}^2+4y\left(2+k\right))z^2-4kx{y}^2{z}^3+4k^2{y}^2{z}^4}}.$$

(3) The new generating function of the product of modified Pell numbers with bivariate Jacobsthal polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_n{J}_n\left(x,y\right)z^n={\displaystyle \frac{z+xy{z}^2+2y{z}^3}{1-2xyz-(x^2{y}^2+12y)z^2-4x{y}^2{z}^3+4y^2{z}^4}}.$$

(4) The new generating function of the product of modified Pell numbers with bivariate Jacobsthal Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_n{j}_n\left(x,y\right)z^n={\displaystyle \frac{2-3xyz-\left(x^2{y}^2+12y\right)z^2-2x{y}^2{z}^3}{1-2xyz-(x^2{y}^2+12y)z^2-4x{y}^2{z}^3+4y^2{z}^4}}.$$

• Putting $x=y=1$ in Theorems 4 and 5 and Corollary 3, we obtain the following table (Table 2):

Table 2. New generating functions of the products of some numbers and polynomials.

Coefficient of ${\mathit{z}}^{\mathit{n}}$	Generating Function
$M{P}_{p,q,n}{J}_n$	${\displaystyle \frac{pz+q{z}^2+2pq{z}^3}{1-2pz-\left(8p^2+5q\right)z^2-4pq{z}^3+4q^2{z}^4}}$
$M{P}_{p,q,n}{j}_n$	${\displaystyle \frac{2-3pz-\left(8p^2+5q\right)z^2-2pq{z}^3}{1-2pz-\left(8p^2+5q\right)z^2-4pq{z}^3+4q^2{z}^4}}$
$q_{k,n}{J}_n$	${\displaystyle \frac{z+k{z}^2+2k{z}^3}{1-2z-\left(8+5k\right)z^2-4k{z}^3+4k^2{z}^4}}$
$q_{k,n}{j}_n$	${\displaystyle \frac{2-3z-\left(8+5k\right)z^2-2k{z}^3}{1-2z-\left(8+5k\right)z^2-4k{z}^3+4k^2{z}^4}}$
$q_n{J}_n$	${\displaystyle \frac{z+z^2+2z^3}{1-2z-13z^2-4z^3+4z^4}}$
$q_n{j}_n$	${\displaystyle \frac{2-3z-13z^2-2z^3}{1-2z-13z^2-4z^3+4z^4}}$

Table 2. New generating functions of the products of some numbers and polynomials.

Coefficient of ${\mathit{z}}^{\mathit{n}}$	Generating Function
$M{P}_{p,q,n}{J}_n$	${\displaystyle \frac{pz+q{z}^2+2pq{z}^3}{1-2pz-\left(8p^2+5q\right)z^2-4pq{z}^3+4q^2{z}^4}}$
$M{P}_{p,q,n}{j}_n$	${\displaystyle \frac{2-3pz-\left(8p^2+5q\right)z^2-2pq{z}^3}{1-2pz-\left(8p^2+5q\right)z^2-4pq{z}^3+4q^2{z}^4}}$
$q_{k,n}{J}_n$	${\displaystyle \frac{z+k{z}^2+2k{z}^3}{1-2z-\left(8+5k\right)z^2-4k{z}^3+4k^2{z}^4}}$
$q_{k,n}{j}_n$	${\displaystyle \frac{2-3z-\left(8+5k\right)z^2-2k{z}^3}{1-2z-\left(8+5k\right)z^2-4k{z}^3+4k^2{z}^4}}$
$q_n{J}_n$	${\displaystyle \frac{z+z^2+2z^3}{1-2z-13z^2-4z^3+4z^4}}$
$q_n{j}_n$	${\displaystyle \frac{2-3z-13z^2-2z^3}{1-2z-13z^2-4z^3+4z^4}}$

Thirdly, the following substitutions,

$$\left\{\begin{array}{c}{a}_1-a_2=2p\\ a_1{a}_2=q\end{array}\right.\mathrm{and}\left\{\begin{array}{c}{e}_1-e_2=2xy\\ e_1{e}_2=y\end{array}\right.,$$

in (3.7)–(3.10) would give

$${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_n(e_1+[-e_2])z^n={\displaystyle \frac{2xyz+2py{z}^2}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}},$$

(3.23)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_n(e_1+[-e_2])z^n={\displaystyle \frac{1-qy{z}^2}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}},$$

(3.24)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n={\displaystyle \frac{z-qy{z}^3}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}},$$

(3.25)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n={\displaystyle \frac{2pz+2qxy{z}^2}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}},$$

(3.26)

respectively; thus, we obtain the following theorems.

Theorem 6.

For $n\in \mathbb{N},$ the new generating function of the product of $\left(p,q\right)$-modified Pell numbers with bivariate Pell polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{P}_n\left(x,y\right)z^n={\displaystyle \frac{pz+2qxy{z}^2+pqy{z}^3}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}}.$$

(3.27)

Proof. 

From [28], we have $P_n\left(x,y\right)=S_{n-1}(e_1+[-e_2])$; then,

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{P}_n\left(x,y\right)z^n& =& {\displaystyle \sum _{n=0}^{\infty }}\left(S_n(a_1+[-a_2])-p{S}_{n-1}(a_1+[-a_2])\right)S_{n-1}(e_1+[-e_2])z^n\hfill \\ & =& {\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & & -p{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & =& {\displaystyle \frac{2pz+2qxy{z}^2}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{p\left(z-qy{z}^3\right)}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}},\hfill \end{array}$$

After a simple calculation, we have

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{P}_n\left(x,y\right)z^n={\displaystyle \frac{pz+2qxy{z}^2+pqy{z}^3}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}}.$$

This completes the proof. □

Theorem 7.

For $n\in \mathbb{N},$ the new generating function of the product of $\left(p,q\right)$-modified Pell numbers with bivariate Pell Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{Q}_n\left(x,y\right)z^n={\displaystyle \frac{2-6pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-2pqx{y}^2{z}^3}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}}.$$

(3.28)

Proof. 

We know that (see [28])

$$Q_n\left(x,y\right)=2S_n(e_1+[-e_2])-2xy{S}_{n-1}(e_1+[-e_2]),$$

Then,

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{Q}_n\left(x,y\right)z^n& =& {\displaystyle \sum _{n=0}^{\infty }}\left(\begin{array}{c}\left(S_n(a_1+[-a_2])-p{S}_{n-1}(a_1+[-a_2])\right)\\ \times (2S_n(e_1+[-e_2])-2xy{S}_{n-1}(e_1+[-e_2]))\end{array}\right)z^n\hfill \\ & =& 2{\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_n(e_1+[-e_2])z^n\hfill \\ & & -2xy{\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & & -2p{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_n(e_1+[-e_2])z^n\hfill \\ & & +2pxy{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n.\hfill \end{array}$$

Using relationships (3.23)–(3.26), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{Q}_n\left(x,y\right)z^n& =& {\displaystyle \frac{2\left(1-qy{z}^2\right)}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{2xy\left(2pz+2qxy{z}^2\right)}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{2p\left(2xyz+2py{z}^2\right)}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & +{\displaystyle \frac{2pxy\left(z-qy{z}^3\right)}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}}\hfill \\ & =& {\displaystyle \frac{2-6pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-2pqx{y}^2{z}^3}{1-4pxyz-(4q{x}^2{y}^2+2y\left(2p^2+q\right))z^2-4pqx{y}^2{z}^3+q^2{y}^2{z}^4}}.\hfill \end{array}$$

This completes the proof. □

Corollary 4.

By putting $(p=1$ and $q=k)$ and $(p=q=1)$ in Equations (3.27) and (3.28), we obtain the following new generating functions:

(1) The new generating function of the product of modified k-Pell numbers with bivariate Pell polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_{k,n}{P}_n\left(x,y\right)z^n={\displaystyle \frac{z+2kxy{z}^2+ky{z}^3}{1-4xyz-(4k{x}^2{y}^2+2y\left(2+k\right))z^2-4kx{y}^2{z}^3+k^2{y}^2{z}^4}}.$$

(2) The new generating function of the product of modified k-Pell numbers with bivariate Pell Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_{k,n}{Q}_n\left(x,y\right)z^n={\displaystyle \frac{2-6xyz-(4k{x}^2{y}^2+2y\left(2+k\right))z^2-2kx{y}^2{z}^3}{1-4xyz-(4k{x}^2{y}^2+2y\left(2+k\right))z^2-4kx{y}^2{z}^3+k^2{y}^2{z}^4}}.$$

(3) The new generating function of the product of modified Pell numbers with bivariate Pell polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_n{P}_n\left(x,y\right)z^n={\displaystyle \frac{z+2xy{z}^2+y{z}^3}{1-4xyz-(4x^2{y}^2+6y)z^2-4x{y}^2{z}^3+y^2{z}^4}}.$$

(4) The new generating function of the product of modified Pell numbers with bivariate Pell Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_n{Q}_n\left(x,y\right)z^n={\displaystyle \frac{2-6xyz-\left(4x^2{y}^2+6y\right)z^2-2x{y}^2{z}^3}{1-4xyz-(4x^2{y}^2+6y)z^2-4x{y}^2{z}^3+y^2{z}^4}}.$$

• Putting $\left(y=1\right)$ and $\left(x=y=1\right)$ in Theorems 6 and 7 and Corollary 4, we obtain the following table (Table 3):

Table 3. New generating functions of the products of some numbers and polynomials.

Coefficient of ${\mathit{z}}^{\mathit{n}}$	Generating Function	Coefficient of ${\mathit{z}}^{\mathit{n}}$	Generating Function
$M{P}_{p,q,n}{P}_n\left(x\right)$	${\displaystyle \frac{pz+2qx{z}^2+pq{z}^3}{1-4pxz-\left(4q{x}^2+2\left(2p^2+q\right)\right)z^2-4pqx{z}^3+q^2{z}^4}}$	$M{P}_{p,q,n}{P}_n$	${\displaystyle \frac{pz+2q{z}^2+pq{z}^3}{1-4pz-\left(4p^2+6q\right)z^2-4pq{z}^3+q^2{z}^4}}$
$M{P}_{p,q,n}{Q}_n\left(x\right)$	${\displaystyle \frac{2-6pxz-\left(4q{x}^2+2\left(2p^2+q\right)\right)z^2-2pqx{z}^3}{1-4pxz-\left(4q{x}^2+2\left(2p^2+q\right)\right)z^2-4pqx{z}^3+q^2{z}^4}}$	$M{P}_{p,q,n}{Q}_n$	${\displaystyle \frac{2-6pz-\left(4p^2+6q\right)z^2-2pq{z}^3}{1-4pz-\left(4p^2+6q\right)z^2-4pq{z}^3+q^2{z}^4}}$
$q_{k,n}{P}_n\left(x\right)$	${\displaystyle \frac{z+2kx{z}^2+k{z}^3}{1-4xz-\left(4k{x}^2+2\left(2+k\right)\right)z^2-4kx{z}^3+k^2{z}^4}}$	$q_{k,n}{P}_n$	${\displaystyle \frac{z+2k{z}^2+k{z}^3}{1-4z-\left(4+6k\right)z^2-4k{z}^3+k^2{z}^4}}$
$q_{k,n}{Q}_n\left(x\right)$	${\displaystyle \frac{2-6xz-\left(4k{x}^2+2\left(2+k\right)\right)z^2-2kx{z}^3}{1-4xz-\left(4k{x}^2+2\left(2+k\right)\right)z^2-4kx{z}^3+k^2{z}^4}}$	$q_{k,n}{Q}_n$	${\displaystyle \frac{2-6z-\left(4+6k\right)z^2-2k{z}^3}{1-4z-\left(4+6k\right)z^2-4k{z}^3+k^2{z}^4}}$
$q_n{P}_n\left(x\right)$	${\displaystyle \frac{z+2x{z}^2+z^3}{1-4xz-\left(4x^2+6\right)z^2-4x{z}^3+z^4}}$	$q_n{P}_n$	${\displaystyle \frac{z+2z^2+z^3}{1-4z-10z^2-4z^3+z^4}}$
$q_n{Q}_n\left(x\right)$	${\displaystyle \frac{2-6xz-\left(4x^2+6\right)z^2-2x{z}^3}{1-4xz-\left(4x^2+6\right)z^2-4x{z}^3+z^4}}$	$q_n{Q}_n$	${\displaystyle \frac{2-6z-10z^2-2z^3}{1-4z-10z^2-4z^3+z^4}}$

Table 3. New generating functions of the products of some numbers and polynomials.

Coefficient of ${\mathit{z}}^{\mathit{n}}$	Generating Function	Coefficient of ${\mathit{z}}^{\mathit{n}}$	Generating Function
$M{P}_{p,q,n}{P}_n\left(x\right)$	${\displaystyle \frac{pz+2qx{z}^2+pq{z}^3}{1-4pxz-\left(4q{x}^2+2\left(2p^2+q\right)\right)z^2-4pqx{z}^3+q^2{z}^4}}$	$M{P}_{p,q,n}{P}_n$	${\displaystyle \frac{pz+2q{z}^2+pq{z}^3}{1-4pz-\left(4p^2+6q\right)z^2-4pq{z}^3+q^2{z}^4}}$
$M{P}_{p,q,n}{Q}_n\left(x\right)$	${\displaystyle \frac{2-6pxz-\left(4q{x}^2+2\left(2p^2+q\right)\right)z^2-2pqx{z}^3}{1-4pxz-\left(4q{x}^2+2\left(2p^2+q\right)\right)z^2-4pqx{z}^3+q^2{z}^4}}$	$M{P}_{p,q,n}{Q}_n$	${\displaystyle \frac{2-6pz-\left(4p^2+6q\right)z^2-2pq{z}^3}{1-4pz-\left(4p^2+6q\right)z^2-4pq{z}^3+q^2{z}^4}}$
$q_{k,n}{P}_n\left(x\right)$	${\displaystyle \frac{z+2kx{z}^2+k{z}^3}{1-4xz-\left(4k{x}^2+2\left(2+k\right)\right)z^2-4kx{z}^3+k^2{z}^4}}$	$q_{k,n}{P}_n$	${\displaystyle \frac{z+2k{z}^2+k{z}^3}{1-4z-\left(4+6k\right)z^2-4k{z}^3+k^2{z}^4}}$
$q_{k,n}{Q}_n\left(x\right)$	${\displaystyle \frac{2-6xz-\left(4k{x}^2+2\left(2+k\right)\right)z^2-2kx{z}^3}{1-4xz-\left(4k{x}^2+2\left(2+k\right)\right)z^2-4kx{z}^3+k^2{z}^4}}$	$q_{k,n}{Q}_n$	${\displaystyle \frac{2-6z-\left(4+6k\right)z^2-2k{z}^3}{1-4z-\left(4+6k\right)z^2-4k{z}^3+k^2{z}^4}}$
$q_n{P}_n\left(x\right)$	${\displaystyle \frac{z+2x{z}^2+z^3}{1-4xz-\left(4x^2+6\right)z^2-4x{z}^3+z^4}}$	$q_n{P}_n$	${\displaystyle \frac{z+2z^2+z^3}{1-4z-10z^2-4z^3+z^4}}$
$q_n{Q}_n\left(x\right)$	${\displaystyle \frac{2-6xz-\left(4x^2+6\right)z^2-2x{z}^3}{1-4xz-\left(4x^2+6\right)z^2-4x{z}^3+z^4}}$	$q_n{Q}_n$	${\displaystyle \frac{2-6z-10z^2-2z^3}{1-4z-10z^2-4z^3+z^4}}$

Fourthly, the following substitutions,

$$\left\{\begin{array}{c}\hfill a_1-a_2=2p\hfill \\ \hfill a_1{a}_2=q\hfill \end{array}\right.\mathrm{and}\left\{\begin{array}{c}\hfill e_1-e_2=x\hfill \\ \hfill e_1{e}_2=-y\hfill \end{array}\right.,$$

in (3.7)–(3.10) would give

$${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}\left(a_1+\left[-a_2\right]\right)S_n\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{xz-2py{z}^2}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}},$$

(3.29)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(a_1+\left[-a_2\right]\right)S_n\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{1+qy{z}^2}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}},$$

(3.30)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}\left(a_1+\left[-a_2\right]\right)S_{n-1}\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{z+qy{z}^3}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}},$$

(3.31)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(a_1+\left[-a_2\right]\right)S_{n-1}\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{2pz+qx{z}^2}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}},$$

(3.32)

respectively, and we have the following results.

Theorem 8.

For $n\in \mathbb{N},$ the new generating function of the product of $\left(p,q\right)$-modified Pell numbers with bivariate Vieta–Fibonacci polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{V}_n\left(x,y\right)z^n={\displaystyle \frac{pz+qx{z}^2-pqy{z}^3}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}}.$$

(3.33)

Proof. 

Referring to [27], we have

$$V_n\left(x,y\right)=S_{n-1}(e_1+[-e_2]).$$

We see that

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{V}_n\left(x,y\right)z^n& =& {\displaystyle \sum _{n=0}^{\infty }}\left(S_n(a_1+[-a_2])-p{S}_{n-1}(a_1+[-a_2])\right)S_{n-1}(e_1+[-e_2])z^n\hfill \\ & =& {\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & & -p{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & =& {\displaystyle \frac{2pz+qx{z}^2}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{p\left(z+qy{z}^3\right)}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}},\hfill \end{array}$$

After a simple calculation, we have

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{V}_n\left(x,y\right)z^n={\displaystyle \frac{pz+qx{z}^2-pqy{z}^3}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}}.$$

So, the proof is completed. □

Theorem 9.

For $n\in \mathbb{N},$ the new generating function of the product of $\left(p,q\right)$-modified Pell numbers with bivariate Vieta–Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{v}_n\left(x,y\right)z^n={\displaystyle \frac{2-3pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+pqxy{z}^3}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}}.$$

(3.34)

Proof. 

We know that (see [27])

$$v_n\left(x,y\right)=2S_n(e_1+[-e_2])-x{S}_{n-1}(e_1+[-e_2]),$$

We see that

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{v}_n\left(x,y\right)z^n& =& {\displaystyle \sum _{n=0}^{\infty }}\left(\begin{array}{c}\left(S_n(a_1+[-a_2])-p{S}_{n-1}(a_1+[-a_2])\right)\\ \times (2S_n(e_1+[-e_2])-x{S}_{n-1}(e_1+[-e_2]))\end{array}\right)z^n\hfill \\ & =& 2{\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_n(e_1+[-e_2])z^n\hfill \\ & & -x{\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & & -2p{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_n(e_1+[-e_2])z^n\hfill \\ & & +px{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n.\hfill \end{array}$$

According to relationships (3.29)–(3.32), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{v}_n\left(x,y\right)z^n& =& {\displaystyle \frac{2\left(1+qy{z}^2\right)}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{x\left(2pz+qx{z}^2\right)}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{2p\left(xz-2py{z}^2\right)}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & +{\displaystyle \frac{px\left(z+qy{z}^3\right)}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & =& {\displaystyle \frac{2-3pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+pqxy{z}^3}{1-2pxz-(q{x}^2-2y\left(2p^2+q\right))z^2+2pqxy{z}^3+q^2{y}^2{z}^4}}.\hfill \end{array}$$

Thus, this completes the proof. □

Corollary 5.

By putting $(p=1$ and $q=k)$ and $(p=q=1)$ in Equations (3.33) and (3.34), we obtain the following new generating functions:

(1) The new generating function of the product of modified k-Pell numbers with bivariate Vieta–Fibonacci polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_{k,n}{V}_n\left(x,y\right)z^n={\displaystyle \frac{z+kx{z}^2-ky{z}^3}{1-2xz-(k{x}^2-2y\left(2+k\right))z^2+2kxy{z}^3+k^2{y}^2{z}^4}}.$$

(2) The new generating function of the product of modified k-Pell numbers with bivariate Vieta–Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_{k,n}{v}_n\left(x,y\right)z^n={\displaystyle \frac{2-3xz-(k{x}^2-2y\left(2+k\right))z^2+kxy{z}^3}{1-2xz-(k{x}^2-2y\left(2+k\right))z^2+2kxy{z}^3+k^2{y}^2{z}^4}}.$$

(3) The new generating function of the product of modified Pell numbers with bivariate Vieta–Fibonacci polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_n{V}_n\left(x,y\right)z^n={\displaystyle \frac{z+x{z}^2-y{z}^3}{1-2xz-(x^2-6y)z^2+2xy{z}^3+y^2{z}^4}}.$$

(4) The new generating function of the product of modified Pell numbers with bivariate Vieta–Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_n{v}_n\left(x,y\right)z^n={\displaystyle \frac{2-3xz-\left(x^2-6y\right)z^2+xy{z}^3}{1-2xz-(x^2-6y)z^2+2xy{z}^3+y^2{z}^4}}.$$

• Putting $y=1$ in Theorems 8 and 9 and Corollary 5, we obtain the following table (Table 4):

Table 4. New generating functions of the products of some numbers and polynomials.

Coefficient of ${\mathit{z}}^{\mathit{n}}$	Generating Function
$M{P}_{p,q,n}{V}_n\left(x\right)$	${\displaystyle \frac{pz+qx{z}^2-pq{z}^3}{1-2pxz-(q{x}^2-2\left(2p^2+q\right))z^2+2pqx{z}^3+q^2{z}^4}}$
$M{P}_{p,q,n}{v}_n\left(x\right)$	${\displaystyle \frac{2-3pxz-(q{x}^2-2\left(2p^2+q\right))z^2+pqx{z}^3}{1-2pxz-(q{x}^2-2\left(2p^2+q\right))z^2+2pqx{z}^3+q^2{z}^4}}$
$q_{k,n}{V}_n\left(x\right)$	${\displaystyle \frac{z+kx{z}^2-k{z}^3}{1-2xz-(k{x}^2-2\left(2+k\right))z^2+2kx{z}^3+k^2{z}^4}}$
$q_{k,n}{v}_n\left(x\right)$	${\displaystyle \frac{2-3xz-(k{x}^2-2\left(2+k\right))z^2+kx{z}^3}{1-2xz-(k{x}^2-2\left(2+k\right))z^2+2kx{z}^3+k^2{z}^4}}$
$q_n{V}_n\left(x\right)$	${\displaystyle \frac{z+x{z}^2-z^3}{1-2xz-(x^2-6)z^2+2x{z}^3+z^4}}$
$q_n{v}_n\left(x\right)$	${\displaystyle \frac{2-3xz-(x^2-6)z^2+x{z}^3}{1-2xz-(x^2-6)z^2+2x{z}^3+z^4}}$

Table 4. New generating functions of the products of some numbers and polynomials.

Coefficient of ${\mathit{z}}^{\mathit{n}}$	Generating Function
$M{P}_{p,q,n}{V}_n\left(x\right)$	${\displaystyle \frac{pz+qx{z}^2-pq{z}^3}{1-2pxz-(q{x}^2-2\left(2p^2+q\right))z^2+2pqx{z}^3+q^2{z}^4}}$
$M{P}_{p,q,n}{v}_n\left(x\right)$	${\displaystyle \frac{2-3pxz-(q{x}^2-2\left(2p^2+q\right))z^2+pqx{z}^3}{1-2pxz-(q{x}^2-2\left(2p^2+q\right))z^2+2pqx{z}^3+q^2{z}^4}}$
$q_{k,n}{V}_n\left(x\right)$	${\displaystyle \frac{z+kx{z}^2-k{z}^3}{1-2xz-(k{x}^2-2\left(2+k\right))z^2+2kx{z}^3+k^2{z}^4}}$
$q_{k,n}{v}_n\left(x\right)$	${\displaystyle \frac{2-3xz-(k{x}^2-2\left(2+k\right))z^2+kx{z}^3}{1-2xz-(k{x}^2-2\left(2+k\right))z^2+2kx{z}^3+k^2{z}^4}}$
$q_n{V}_n\left(x\right)$	${\displaystyle \frac{z+x{z}^2-z^3}{1-2xz-(x^2-6)z^2+2x{z}^3+z^4}}$
$q_n{v}_n\left(x\right)$	${\displaystyle \frac{2-3xz-(x^2-6)z^2+x{z}^3}{1-2xz-(x^2-6)z^2+2x{z}^3+z^4}}$

Fifthly, the following substitutions,

$$\left\{\begin{array}{c}\hfill a_1-a_2=2p\hfill \\ \hfill a_1{a}_2=q\hfill \end{array}\right.\mathrm{and}\left\{\begin{array}{c}\hfill e_1-e_2=ix\hfill \\ \hfill e_1{e}_2=y\hfill \end{array}\right.,$$

in (3.7)–(3.10) would give

$${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}\left(a_1+\left[-a_2\right]\right)S_n\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{ixz+2py{z}^2}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}},$$

(3.35)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(a_1+\left[-a_2\right]\right)S_n\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{1-qy{z}^2}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}},$$

(3.36)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}\left(a_1+\left[-a_2\right]\right)S_{n-1}\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{z-qy{z}^3}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}},$$

(3.37)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(a_1+\left[-a_2\right]\right)S_{n-1}\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{2pz+iqx{z}^2}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}},$$

(3.38)

respectively; thus, we obtain the following theorems.

Theorem 10.

For $n\in \mathbb{N},$ the new generating function of the product of $\left(p,q\right)$-modified Pell numbers with bivariate complex Fibonacci polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{f}_n\left(x,y\right)z^n={\displaystyle \frac{pz+iqx{z}^2+pqy{z}^3}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}}.$$

(3.39)

Proof. 

Referring to [26], we have

$$f_n\left(x,y\right)=S_{n-1}(e_1+[-e_2]).$$

We see that

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{f}_n\left(x,y\right)z^n& =& {\displaystyle \sum _{n=0}^{\infty }}\left(S_n(a_1+[-a_2])-p{S}_{n-1}(a_1+[-a_2])\right)S_{n-1}(e_1+[-e_2])z^n\hfill \\ & =& {\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & & -p{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & =& {\displaystyle \frac{2pz+iqx{z}^2}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{p\left(z-qy{z}^3\right)}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}},\hfill \end{array}$$

After a simple calculation, we have

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{f}_n\left(x,y\right)z^n={\displaystyle \frac{pz+iqx{z}^2+pqy{z}^3}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}}.$$

So, the proof is completed. □

Theorem 11.

For $n\in \mathbb{N},$ the new generating function of the product of $\left(p,q\right)$-modified Pell numbers with bivariate complex Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{l}_n\left(x,y\right)z^n={\displaystyle \frac{2-3ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-ipqxy{z}^3}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}}.$$

(3.40)

Proof. 

We know that (see [26])

$$l_n\left(x,y\right)=2S_n(e_1+[-e_2])-ix{S}_{n-1}(e_1+[-e_2]),$$

We see that

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{l}_n\left(x,y\right)z^n& =& {\displaystyle \sum _{n=0}^{\infty }}\left(\begin{array}{c}\left(S_n(a_1+[-a_2])-p{S}_{n-1}(a_1+[-a_2])\right)\\ \times (2S_n(e_1+[-e_2])-ix{S}_{n-1}(e_1+[-e_2]))\end{array}\right)z^n\hfill \\ & =& 2{\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_n(e_1+[-e_2])z^n\hfill \\ & & -ix{\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \end{array}$$

$$\begin{array}{ccc}& & -2p{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_n(e_1+[-e_2])z^n\hfill \\ & & +ipx{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n.\hfill \end{array}$$

According to relationships (3.35)–(3.38), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{l}_n\left(x,y\right)z^n& =& {\displaystyle \frac{2\left(1-qy{z}^2\right)}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{ix\left(2pz+iqx{z}^2\right)}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{2p\left(ixz+2py{z}^2\right)}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & & +{\displaystyle \frac{ipx\left(z-qy{z}^3\right)}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}}\hfill \\ & =& {\displaystyle \frac{2-3ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-ipqxy{z}^3}{1-2ipxz-(2y\left(2p^2+q\right)-q{x}^2)z^2-2ipqxy{z}^3+q^2{y}^2{z}^4}}.\hfill \end{array}$$

This completes the proof. □

Corollary 6.

By putting $(p=1$ and $q=k)$ and $(p=q=1)$ in Equations (3.39) and (3.40), we give the following new generating functions:

(1) The new generating function of the product of modified k-Pell numbers with bivariate complex Fibonacci polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_{k,n}{f}_n\left(x,y\right)z^n={\displaystyle \frac{z+ikx{z}^2+ky{z}^3}{1-2ixz-(2y\left(2+k\right)-k{x}^2)z^2-2ikxy{z}^3+k^2{y}^2{z}^4}}.$$

(2) The new generating function of the product of modified k-Pell numbers with bivariate complex Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_{k,n}{l}_n\left(x,y\right)z^n={\displaystyle \frac{2-3ixz-(2y\left(2+k\right)-k{x}^2)z^2-ikxy{z}^3}{1-2ixz-(2y\left(2+k\right)-k{x}^2)z^2-2ikxy{z}^3+k^2{y}^2{z}^4}}.$$

(3) The new generating function of the product of modified Pell numbers with bivariate complex Fibonacci polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_n{f}_n\left(x,y\right)z^n={\displaystyle \frac{z+ix{z}^2+y{z}^3}{1-2ixz-(6y-x^2)z^2-2ixy{z}^3+y^2{z}^4}}.$$

(4) The new generating function of the product of modified Pell numbers with bivariate complex Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_n{l}_n\left(x,y\right)z^n={\displaystyle \frac{2-3ixz-\left(6y-x^2\right)z^2-ixy{z}^3}{1-2ixz-(6y-x^2)z^2-2ixy{z}^3+y^2{z}^4}}.$$

Sixthly, the following substitutions,

$$\left\{\begin{array}{c}\hfill a_1-a_2=2p\hfill \\ \hfill a_1{a}_2=q\hfill \end{array}\right.\mathrm{and}\left\{\begin{array}{c}\hfill e_1-e_2=3y\hfill \\ \hfill e_1{e}_2=-2x\hfill \end{array}\right.,$$

in (3.7)–(3.10) would give

$${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}\left(a_1+\left[-a_2\right]\right)S_n\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{3yz-4px{z}^2}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}},$$

(3.41)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(a_1+\left[-a_2\right]\right)S_n\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{1+2qx{z}^2}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}},$$

(3.42)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}\left(a_1+\left[-a_2\right]\right)S_{n-1}\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{z+2qx{z}^3}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}},$$

(3.43)

$${\displaystyle \sum _{n=0}^{\infty }}{S}_n\left(a_1+\left[-a_2\right]\right)S_{n-1}\left(e_1+\left[-e_2\right]\right)z^n={\displaystyle \frac{2pz+3qy{z}^2}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}},$$

(3.44)

respectively; thus, we obtain the following theorems.

Theorem 12.

For $n\in \mathbb{N},$ the new generating function of the product of $\left(p,q\right)$-modified Pell numbers with bivariate Mersenne polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{M}_n\left(x,y\right)z^n={\displaystyle \frac{pz+3qy{z}^2-2pqx{z}^3}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}}.$$

(3.45)

Proof. 

Referring to [20], we have

$$M_n\left(x,y\right)=S_{n-1}(e_1+[-e_2]).$$

We see that

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{M}_n\left(x,y\right)z^n& =& {\displaystyle \sum _{n=0}^{\infty }}\left(S_n(a_1+[-a_2])-p{S}_{n-1}(a_1+[-a_2])\right)S_{n-1}(e_1+[-e_2])z^n\hfill \\ & =& {\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & & -p{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & =& {\displaystyle \frac{2pz+3qy{z}^2}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{p\left(z+2qx{z}^3\right)}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}},\hfill \end{array}$$

After a simple calculation, we have

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{M}_n\left(x,y\right)z^n={\displaystyle \frac{pz+3qy{z}^2-2pqx{z}^3}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}}.$$

So, the proof is completed. □

Theorem 13.

For $n\in \mathbb{N},$ the new generating function of the product of $\left(p,q\right)$-modified Pell numbers with bivariate Mersenne Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{m}_n\left(x,y\right)z^n={\displaystyle \frac{2-9pyz+\left(4x\left(2p^2+q\right)-9q{y}^2\right)z^2+6pqxy{z}^3}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}}.$$

(3.46)

Proof. 

We know that (see [20])

$$m_n\left(x,y\right)=2S_n(e_1+[-e_2])-3y{S}_{n-1}(e_1+[-e_2]),$$

We see that

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{m}_n\left(x,y\right)z^n& =& {\displaystyle \sum _{n=0}^{\infty }}\left(\begin{array}{c}\left(S_n(a_1+[-a_2])-p{S}_{n-1}(a_1+[-a_2])\right)\\ \times (2S_n(e_1+[-e_2])-3y{S}_{n-1}(e_1+[-e_2]))\end{array}\right)z^n\hfill \\ & =& 2{\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_n(e_1+[-e_2])z^n\hfill \\ & & -3y{\displaystyle \sum _{n=0}^{\infty }}{S}_n(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n\hfill \\ & & -2p{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_n(e_1+[-e_2])z^n\hfill \\ & & +3py{\displaystyle \sum _{n=0}^{\infty }}{S}_{n-1}(a_1+[-a_2])S_{n-1}(e_1+[-e_2])z^n.\hfill \end{array}$$

According to relationships (3.41)–(3.44), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}M{P}_{p,q,n}{m}_n\left(x,y\right)z^n& =& {\displaystyle \frac{2\left(1+2qx{z}^2\right)}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{3y\left(2pz+3qy{z}^2\right)}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}}\hfill \\ & & -{\displaystyle \frac{2p\left(3yz-4px{z}^2\right)}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}}\hfill \\ & & +{\displaystyle \frac{3py\left(z+2qx{z}^3\right)}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}}\hfill \\ & =& {\displaystyle \frac{2-9pyz+\left(4x\left(2p^2+q\right)-9q{y}^2\right)z^2+6pqxy{z}^3}{1-6pyz-(9q{y}^2-4x\left(2p^2+q\right))z^2+12pqxy{z}^3+4q^2{x}^2{z}^4}}.\hfill \end{array}$$

This completes the proof. □

Corollary 7.

By putting $(p=1$ and $q=k)$ and $(p=q=1)$ in Equations (3.45) and (3.46), we obtain the following new generating functions:

(1) The new generating function of the product of modified k-Pell numbers with bivariate Mersenne polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_{k,n}{M}_n\left(x,y\right)z^n={\displaystyle \frac{z+3ky{z}^2-2kx{z}^3}{1-6yz-(9k{y}^2-4x\left(2+k\right))z^2+12kxy{z}^3+4k^2{x}^2{z}^4}}.$$

(2) The new generating function of the product of modified k-Pell numbers with bivariate Mersenne Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_{k,n}{m}_n\left(x,y\right)z^n={\displaystyle \frac{2-9yz+\left(4x\left(2+k\right)-9k{y}^2\right)z^2+6kxy{z}^3}{1-6yz-(9k{y}^2-4x\left(2+k\right))z^2+12kxy{z}^3+4k^2{x}^2{z}^4}}.$$

(3) The new generating function of the product of modified Pell numbers with bivariate Mersenne polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_n{M}_n\left(x,y\right)z^n={\displaystyle \frac{z+3y{z}^2-2x{z}^3}{1-6yz-(9y^2-12x)z^2+12xy{z}^3+4x^2{z}^4}}.$$

(4) The new generating function of the product of modified Pell numbers with bivariate Mersenne Lucas polynomials is given by

$${\displaystyle \sum _{n=0}^{\infty }}{q}_n{m}_n\left(x,y\right)z^n={\displaystyle \frac{2-9yz+\left(12x-9y^2\right)z^2+6xy{z}^3}{1-6yz-(9y^2-12x)z^2+12xy{z}^3+4x^2{z}^4}}.$$

• Putting $x=y=1$ in Theorems 12 and 13 and Corollary 7, we obtain the following table (

  Table 5. New generating functions of the products of some numbers and polynomials.

  Coefficient of ${\mathit{z}}^{\mathit{n}}$	Generating Function
  $M{P}_{p,q,n}{M}_n$	${\displaystyle \frac{pz+3q{z}^2-2pq{z}^3}{1-6pz-(5q-8p^2)z^2+12pq{z}^3+4q^2{z}^4}}$
  $M{P}_{p,q,n}{m}_n$	${\displaystyle \frac{2-9pz+\left(8p^2-5q\right)z^2+6pq{z}^3}{1-6pz-(5q-8p^2)z^2+12pq{z}^3+4q^2{z}^4}}$
  $q_{k,n}{M}_n$	${\displaystyle \frac{z+3k{z}^2-2k{z}^3}{1-6z-(5k-8)z^2+12k{z}^3+4k^2{z}^4}}$
  $q_{k,n}{m}_n$	${\displaystyle \frac{2-9z+\left(8-5k\right)z^2+6k{z}^3}{1-6z-(5k-8)z^2+12k{z}^3+4k^2{z}^4}}$
  $q_n{M}_n$	${\displaystyle \frac{z+3z^2-2z^3}{1-6z+3z^2+12z^3+4z^4}}$
  $q_n{m}_n$	${\displaystyle \frac{2-9z+3z^2+6z^3}{1-6z+3z^2+12z^3+4z^4}}$

  ):

4. Conclusions

In this paper, using Theorem 1, we have derived new generating functions of the products of $\left(p,q\right)$-modified Pell numbers with various bivariate polynomials. The derived theorems and corollaries are based on symmetric functions and products of these numbers and polynomials.