# Construction of general forms of ordinary generating functions for more
families of numbers and multiple variables polynomials

Yilmaz SIMSEK
Department of Mathematics, Faculty of Science
University of Akdeniz TR-07058 Antalya, Turkey
<EMAIL_ADDRESS>
###### Abstract

The aim of this paper is to construct general forms of ordinary generating
functions for special numbers and polynomials involving Fibonacci type numbers
and polynomials, Lucas numbers and polynomials, Chebyshev polynomials, Sextet
polynomials, Humbert-type numbers and polynomials, chain and antichain
polynomials, rank polynomials of the lattices, length of any alphabet of
words, and other graph polynomials. By applying the Euler transform and the
Lambert series to these generating functions, many new identities and
relations are derived. By using differential equations of these generating
functions, some new recurrence relations for these polynomials are found.
General Binet’s type formulas for these polynomials are given. Finally, some
new classes of polynomials and their corresponding certain family of special
numbers are investigated with the help of these generating functions.

Keywords. Generating function, Special functions, Fibonacci polynomials and
numbers, Special polynomials and numbers, Special functions

MSC2020. 05A15, 33-XX, 11B39, 11B83

##  1 Introduction

Special polynomials and numbers with their generating functions play a vital
role not only in many branches of mathematics, but also in almost many applied
sciences, thanks to their important applications. Therefore, the development
of these functions is always destined to be up to date. As a result, many
researchers have been doing very deep research on these functions for more
than a century, constructing mathematical models and findings that will shed
light on solutions to real world problems.

After Fibonacci and Lucas, modified generating functions for the Fibonacci and
the Lucas polynomials have been given by many researchers for almost regularly
every year. Most of this research consists of modifying or unified existing
generating functions by adding either parameters or a few polynomials to the
coefficients of existing generating functions.

For these fundamental reasons, the main motivation of this article is to focus
on the generating functions constructed below and to explore their properties.

We define the following generating functions for new classes of multiple
variables polynomials which denoted respectively by
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$:

$F\left(w,P(\overrightarrow{X_{m}})\right)=\frac{1}{1+\sum\limits_{j=1}^{m}P_{j}(x_{j})w^{j}}=\sum_{n=0}^{\infty}\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)w^{n},$
(1)

and

$\displaystyle
G\left(w,P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)=\frac{\sum\limits_{j=0}^{k}Q_{j}(x_{j})w^{j}}{1+\sum\limits_{j=1}^{m}P_{j}(x_{j})w^{j}}=\sum_{n=0}^{\infty}\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)w^{n},$
(2)

where
$P(\overrightarrow{X_{m}})=\left(P_{1}(x_{1}),P_{2}(x_{2}),\ldots,P_{m}(x_{m})\right)$,
$Q(\overrightarrow{X_{k}})=\left(Q_{1}(x_{1}),Q_{2}(x_{2}),\ldots,Q_{k}(x_{k})\right)$,

$P_{j}(x_{j})=\sum\limits_{v=0}^{d}a_{v}x_{j}^{v},Q_{l}(x_{l})=\sum\limits_{v=0}^{c}b_{v}x_{l}^{v}$

and $m\in\mathbb{N}_{0}$, $c,d,k,m\in\mathbb{N}_{0}$, $0\leq l\leq k$ and
$0\leq j\leq m$.

By using (1) and (2), we investgate many properties of the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$.

In the next sections, we investigate in detail how these two new classes of
polynomials are reduced to some special numbers and polynomials , with some
special cases.

###  1.1 Preliminaries

In order to present our results, we need to give some special classes of
polynomials and numbers with their generating functions.

The Fibonacci polynomials, which are a polynomial sequence, are defined by the
following ordinary generating function:

$\frac{t}{1-xt-t^{2}}=\sum_{n=0}^{\infty}F_{n}\left(x\right)t^{n},$

where $F_{0}\left(x\right)=0$, $F_{1}\left(x\right)=1$ and
$F_{2}\left(x\right)=x$. It is clear that the degree of the polynomials is
$n-1$. The polynomials $F_{n}\left(x\right)$ can also be stateed in terms of
not only well-known Lucas sequences but also the Chebyshev polynomials of the
second kind $U_{n-1}\left(\frac{ix}{2}\right)=i^{n-1}F_{n}\left(x\right)$, $i$
is a imaginary unit $i^{2}=-1$. These polynomials can also be taken into
account as a generalization of the famious Fibonacci numbers $F_{n}$. That is
$F_{n}:=F_{n}\left(1\right)$. The Pell numbers are obtained by evaluating
$F_{n}\left(2\right)$ (cf. [23, p. 411]; see also [22]).

The other important class of the polynomials, which are generated in a similar
way from the Lucas numbers, are known as Lucas polynomials. These polynomials
are defined by

$\frac{2-xt}{1-xt-t^{2}}=\sum_{n=0}^{\infty}L_{n}\left(x\right)t^{n},$

where $L_{0}\left(x\right)=2$, $L_{1}\left(x\right)=x$ and
$L_{2}\left(x\right)=x^{2}+2$ (cf. [23, p. 26]). It is clear that the degree
of the polynomials is $n$. The polynomials $L_{n}\left(x\right)$ can also be
stated in terms of not only well-known Lucas sequences but also the Chebyshev
polynomials of the first kind
$2T_{n}\left(-\frac{ix}{2}\right)=i^{n}L_{n}\left(x\right)$. These polynomials
can also be taken into account as a generalization of the famious Lucas
numbers $L_{n}$. That is $L_{n}:=L_{n}\left(1\right)$ (cf. [23, p. 372]; see
also [22]).

The Humbert polynomials $\left\\{\Pi_{n,m}^{\lambda}\right\\}_{n=0}^{\infty}$
were defined in $1921$ by Humbert [19]. Their generating function is

$\frac{1}{\left(1-mxt+t^{m}\right)^{\lambda}}=\sum_{n=0}^{\infty}\Pi_{n,m}^{\lambda}\left(x\right)t^{n}.$
(3)

The Fibonacci type polynomials in two variables are defined by Ozdemir and
Simsek by the following generating function

$H\left(t;x,y;k,m,n\right)=\sum_{j=0}^{\infty}\mathcal{G}_{j}\left(x,y;k,m,n\right)t^{j}=\frac{1}{1-x^{k}t-y^{m}t^{m+n}},$
(4)

where $k,m,n\in\mathbb{N}_{0}$ (cf. [34]). An explicit formula for the
polynomials $\mathcal{G}_{j}\left(x,y;k,m,n\right)$ is given by

$\mathcal{G}_{j}\left(x,y;k,m,n\right)=\sum_{c=0}^{\left[\frac{j}{m+n}\right]}\binom{j-c\left(m+n-1\right)}{c}y^{mc}x^{jk-
mck-nck},$

where $\left[a\right]$ is the largest integer $\leq a$ (cf. [34]).

Two variable Fibonacci type polynomials of higher order are defined by the
following generating function

$\sum_{j=0}^{\infty}\mathcal{G}_{j}^{\left(h\right)}\left(x,y;k,m,n\right)t^{j}=\frac{1}{\left(1-x^{k}t-y^{m}t^{n+m}\right)^{h}},$
(5)

where $h$ is a positive integer (cf. [35]). Observe that

$\displaystyle\mathcal{G}_{j}^{\left(1\right)}\left(x,y;k,m,n\right)$
$\displaystyle=$ $\displaystyle\mathcal{G}_{j}\left(x,y;k,m,n\right)$
$\displaystyle\mathcal{G}_{j}^{\left(h\right)}\left(ax,-1;1,1,a-1\right)$
$\displaystyle=$ $\displaystyle\Pi_{j,a}^{h}\left(x\right)$

(cf. [35]).

The results of these paper are briefly organized as follows. In Section 2, by
using the Eqs. (1) and (2), we give explicit formulas for the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$.
We also give some relations and identities for these polynomials. In Section
3, we show that many certain families of special numbers and polynomials can
be given in terms of the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$.
We aslo give some special values of these polynomials. Moreover, we give Binet
type formulas for the polynomials. In Sections4, we construct generating
functions for higher order of the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$.
We give some properties of these polynomials with their generating functions
and their special values. Finally, Section 5, we give some partial derivative
equations of the generating functions for the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$.
By using these equation, we derive some recurrence relations for these
polynomials.

##  2 Explicit formulas for polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$

In this section, we give some explicit formulas for the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$
with the aid of the Eqs. (1) and (2).

Under the convergence conditions of the geometric series, we give explicit
formulas for polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$.

Putting $m=2$ in (1), we obtain

$\frac{1}{1+P_{1}(x_{1})w+P_{2}(x_{2})w^{2}}=\sum_{n_{1}=0}^{\infty}\mathbb{Y}_{n_{1}}\left(P(\overrightarrow{X_{2}})\right)w^{n_{1}}.$
(6)

Using (6), we have

$\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{n_{1}}\left(P(\overrightarrow{X_{2}})\right)w^{n_{1}}$
$\displaystyle=$
$\displaystyle\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}(-1)^{n_{1}+n_{2}}\binom{n_{1}+n_{2}}{n_{2}}\left(P_{1}(x_{1})\right)^{n_{1}}$
$\displaystyle\times\left(P_{2}(x_{2})\right)^{n_{2}}w^{n_{1}+2n_{2}}.$

Replacing respectively $n_{1}$ and $n_{2}$ by $n_{1}-2n_{2}\ $ and $n_{2}$ in
the interior of the above sums, we get

$\displaystyle\sum_{n_{1}=0}^{\infty}\mathbb{Y}_{n_{1}}\left(P(\overrightarrow{X_{m}})\right)w^{n_{1}}$
$\displaystyle=$
$\displaystyle\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\left[\frac{n_{1}}{2}\right]}(-1)^{n_{1}-n_{2}}\binom{n_{1}-n_{2}}{n_{2}}\left(P_{1}(x_{1})\right)^{n_{1}-2n_{2}}$
$\displaystyle\times\left(P_{2}(x_{2})\right)^{n_{2}}w^{n_{1}}.$

Comparing the coefficients of $w^{n_{1}}$ on both sides of the above equation
yields the following formula for the polynomials
$\mathbb{Y}_{n_{1}}\left(P_{1}(x),P_{2}(x_{2})\right)$:

###### Lemma 1

Let $n_{1}\in\mathbb{N}_{0}$. Then we have

$\displaystyle\mathbb{Y}_{n_{1}}\left(P_{1}(x),P_{2}(x_{2})\right)$
$\displaystyle=$
$\displaystyle\sum_{n_{2}=0}^{\left[\frac{n_{1}}{2}\right]}(-1)^{n_{1}-n_{2}}\binom{n_{1}-n_{2}}{n_{2}}\left(P_{1}(x_{1})\right)^{n_{1}-2n_{2}}$
$\displaystyle\times\left(P_{2}(x_{2})\right)^{n_{2}}.$

Substituting $m=3$ into (1), we get

$\frac{1}{1+P_{1}(x_{1})w+P_{2}(x_{2})w^{2}+P_{3}(x_{3})w^{3}}=\sum_{n_{1}=0}^{\infty}\mathbb{Y}_{n_{1}}\left(P(\overrightarrow{X_{3}})\right)w^{n_{1}}.$

After some calculations in the above generating function, we obtain

$\displaystyle\sum_{n_{1}=0}^{\infty}\mathbb{Y}_{n_{1}}\left(P(\overrightarrow{X_{3}})\right)w^{n_{1}}$
$\displaystyle=$
$\displaystyle\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}(-1)^{n_{1}+n_{2}+n_{3}}\binom{n_{1}+n_{2}}{n_{2}}\binom{n_{1}+n_{2}+n_{3}}{n_{3}}\prod\limits_{j=1}^{3}\left(P_{j}(x_{j})\right)^{n_{j}}w^{n_{1}+2n_{2}+3n_{3}}.$

Replacing respectively $n_{1}$, $n_{2}$ and $n_{3}$ by
$n_{1}-2(n_{2}-2n_{3})-3n_{3}$, $n_{2}-2n_{3}\ $ and $n_{3}$ in the interior
of the above sums, we get

$\displaystyle\sum_{n_{1}=0}^{\infty}\mathbb{Y}_{n_{1}}\left(P(\overrightarrow{X_{3}})\right)w^{n_{1}}$
$\displaystyle=$
$\displaystyle\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\left[\frac{n_{1}}{2}\right]}\sum_{n_{3}=0}^{\left[\frac{n_{2}}{3}\right]}(-1)^{n_{1}-n_{2}}\binom{n_{1}-n_{2}-n_{3}}{n_{2}-2n_{3}}\binom{n_{1}-n_{2}}{n_{3}}\left(P_{1}(x_{1})\right)^{n_{1}-2n_{2}+n_{3}}$
$\displaystyle\times\left(P_{2}(x_{2})\right)^{n_{2}-2n_{3}}\left(P_{3}(x_{3})\right)^{n_{3}}w^{n_{1}}.$

Comparing the coefficients of $w^{n_{1}}$ on both sides of the above equation
yields the following formula for the polynomials
$\mathbb{Y}_{n_{1}}\left(P_{1}(x),P_{2}(x_{2}),P_{3}(x_{3})\right)$:

###### Lemma 2

Let $n_{1}\in\mathbb{N}_{0}$. Then we have

$\displaystyle\mathbb{Y}_{n_{1}}\left(P(\overrightarrow{X_{3}})\right)$
$\displaystyle=$
$\displaystyle\sum_{n_{2}=0}^{\left[\frac{n_{1}}{2}\right]}\sum_{n_{3}=0}^{\left[\frac{n_{2}}{3}\right]}(-1)^{n_{1}-n_{2}}\binom{n_{1}-n_{2}-n_{3}}{n_{2}-2n_{3}}\binom{n_{1}-n_{2}}{n_{3}}$
$\displaystyle\times\left(P_{1}(x_{1})\right)^{n_{1}-2n_{2}+n_{3}}\left(P_{2}(x_{2})\right)^{n_{2}-2n_{3}}\left(P_{3}(x_{3})\right)^{n_{3}}.$

For $m\geq 4$, by using (1), Lemma 1 and Lemma 2 with mathematical induction,
we obtain

$\sum_{n_{1}=0}^{\infty}\frac{(-1)^{n_{1}}\left(P_{1}(x_{1})w\right)^{n_{1}}}{\left(1+\sum\limits_{j=2}^{m}P_{j}(x_{j})w^{j}\right)^{1+n_{1}}}=\sum_{n_{1}=0}^{\infty}\mathbb{Y}_{n_{1}}\left(P(\overrightarrow{X_{m}})\right)w^{n_{1}}.$

From the above equation, we obtain

$\displaystyle\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}(-1)^{n_{1}+n_{2}}\binom{n_{1}+n_{2}}{n_{2}}\frac{\left(P_{1}(x_{1})\right)^{n_{1}}\left(P_{2}(x_{2})\right)^{n_{2}}w^{n_{1}+2n_{2}}}{\left(1+\sum\limits_{j=3}^{m}P_{j}(x_{j})w^{j}\right)^{1+n_{1}+n_{2}}}$
$\displaystyle=$
$\displaystyle\sum_{n_{1}=0}^{\infty}\mathbb{Y}_{n_{1}}\left(\underset{m\text{
times}}{\underbrace{P_{1}(x_{1}),P_{2}(x_{2}),\ldots,P_{m}(x_{m})}}\right)w^{n_{1}}.$

Continuing these same processes sequentially, we arrive at the following
result:

$\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{n_{1}}\left(P(\overrightarrow{X_{m}})\right)w^{n_{1}}$
$\displaystyle=\underset{m\text{
times}}{\underbrace{\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\cdots\sum_{n_{m}=0}^{\infty}}}\left((-1)^{\sum_{j=0}^{m}n_{j}}\right)\binom{n_{1}+n_{2}}{n_{2}}\binom{n_{1}+n_{2}+n_{3}}{n_{3}}$
$\displaystyle\times\cdots\binom{n_{1}+n_{2}+\cdots+n_{m}}{n_{m}}\prod\limits_{v=1}^{m}\left(P_{v}(x_{v})\right)^{n_{v}}w^{\sum_{j=1}^{m}jn_{j}}.$

Replacing respectively $n_{1}$,$n_{2}$,…,$n_{m-1}$, and $n_{m}$ by
$n_{1}-2(n_{2}-2n_{3})$$-3(n_{3}-3n_{4})-\cdots-\left(m-1\right)(n_{m-1}-\left(m-1\right)n_{m})-mn_{m}$,
$n_{2}-2n_{3}$,…, $n_{m-1}-mn_{m}$, and $n_{m}$ in the interior of the above
sums, and using

$\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}A(n_{1},n_{2})=\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\left[\frac{n_{1}}{a}\right]}A(n_{1},n_{2}-an_{1}),$

where $a\in\mathbb{N}$, we obtain

$\displaystyle\sum_{n_{1}=0}^{\infty}\mathbb{Y}_{n_{1}}\left(P(\overrightarrow{X_{m}})\right)w^{n_{1}}=$
$\displaystyle\underset{m\text{
times}}{\underbrace{\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\left[\frac{n_{1}}{2}\right]}\sum_{n_{3}=0}^{\left[\frac{n_{2}}{3}\right]}\cdots\sum_{n_{m}=0}^{\left[\frac{n_{m-1}}{m}\right]}}}\left((-1)^{\sum_{j=0}^{m}n_{j}}\right)\binom{n_{1}+n_{2}}{n_{2}}\binom{n_{1}+n_{2}+n_{3}}{n_{3}}$
$\displaystyle\times\cdots\binom{n_{1}+n_{2}+\cdots+n_{m}}{n_{m}}\prod\limits_{v=1}^{m}\left(P_{v}(x_{v})\right)^{n_{v}}w^{n_{1}}.$

Comparing the coefficients of $w^{n_{1}}$ on both sides of the above equation,
we arrive at the following theorem:

###### Theorem 3

Let $m\in\mathbb{N}$ with and $n_{1}\in\mathbb{N}_{0}$. Then we have

$\displaystyle\mathbb{Y}_{n_{1}}\left(P(\overrightarrow{X_{m}})\right)=\left(\prod\limits_{j=2}^{m}\sum_{n_{j}=0}^{\left[\frac{n_{j-1}}{j}\right]}\right)(-1)^{\sum_{j=0}^{m}n_{j}}\binom{n_{1}+n_{2}}{n_{2}}$
(9)
$\displaystyle\times\prod\limits_{d=2}^{m}\binom{\sum\limits_{k=1}^{d}n_{k}}{n_{d}}\prod\limits_{v=1}^{m}\left(P_{v}(x_{v})\right)^{n_{v}},$

where

$\left(\prod\limits_{j=2}^{m}\sum_{n_{j}=0}^{\left[\frac{n_{j-1}}{j}\right]}\right)=\sum_{n_{2}=0}^{\left[\frac{n_{1}}{2}\right]}\sum_{n_{3}=0}^{\left[\frac{n_{2}}{2}\right]}\cdots\sum_{n_{m}=0}^{\left[\frac{n_{m-1}}{m}\right]},$

$\prod\limits_{j=2}^{m}\binom{\sum\limits_{k=1}^{d}n_{k}}{n_{d}}=\prod\limits_{k=3}^{m}\binom{n_{1}-\sum\limits_{j=2}^{k-1}j(n_{j}-jn_{j+1})-kn_{k}+\sum\limits_{j=2}^{k-1}(n_{j}-jn_{j+1})+n_{k}}{n_{k}},$

$\prod\limits_{v=1}^{m}\left(P_{v}(x_{v})\right)^{n_{v}}=\left(P_{1}(x_{1})\right)^{n_{1}}\left(P_{2}(x_{2})\right)^{n_{2}}\cdots\left(P_{m}(x_{m})\right)^{n_{m}}$

and

$(-1)^{\sum_{j=0}^{m}n_{j}}=(-1)^{n_{1}-\sum\limits_{j=2}^{m-1}j(n_{j}-jn_{j+1})+\sum\limits_{j=2}^{m-1}(n_{j}-jn_{j+1})+(1-m)n_{m}}.$

Some special values of the equations (1) and (9) are given as follows:

Substituting $m=1$ into (1), we get

$\frac{1}{1+P_{1}(x_{1})w}=\sum_{n=0}^{\infty}\mathbb{Y}_{n}\left(P_{1}(x_{1})\right)w^{n}.$

Therefore

$\mathbb{Y}_{n}\left(P_{1}(x_{1})\right)=\left(-P_{1}(x_{1})\right)^{n}.$

Substituting $P_{1}(x)=x_{1}$, $P_{2}(x)=x_{2}$, $P_{3}(x)=x_{3}$,$\ldots$,
$P_{m}(x)=x_{m}$ into (1), we obtain

$\frac{1}{1+x_{1}w+x_{2}w^{2}+x_{3}w^{3}+\cdots+x_{m}w^{m}}=\sum_{n_{1}=0}^{\infty}\mathbb{Y}_{n_{1}}\left(x_{1},x_{2},x_{3},\ldots,x_{m}\right)w^{n_{1}}.$

Combining the above equation with (9), we obtain the following result:

###### Corollary 4

Let $m\in\mathbb{N}$ with and $n_{1}\in\mathbb{N}_{0}$. Then we have

$\mathbb{Y}_{n_{1}}\left(x_{1},x_{2},x_{3},\ldots,x_{m}\right)=\left(\prod\limits_{j=2}^{m}\sum_{n_{j}=0}^{\left[\frac{n_{j-1}}{j}\right]}\right)(-1)^{\sum_{j=0}^{m}n_{j}}\prod\limits_{d=2}^{m}\binom{\sum\limits_{k=1}^{d}n_{k}}{n_{d}}\prod\limits_{j=1}^{m}x_{j}^{n_{j}}.$
(10)

Substituting $x_{1}=x$, $x_{2}=x^{2}$, $x_{3}=x^{3}$,$\ldots$, $x_{m}=x^{m}$
into (10), we get the following result:

###### Corollary 5

Let $m\in\mathbb{N}$ with and $n_{1}\in\mathbb{N}_{0}$. Then we have

$\displaystyle\mathbb{Y}_{n_{1}}\left(x,x^{2},x^{3},\ldots,x^{m}\right)$
$\displaystyle=$
$\displaystyle\left(\prod\limits_{j=2}^{m}\sum_{n_{j}=0}^{\left[\frac{n_{j-1}}{j}\right]}\right)(-1)^{\sum_{j=0}^{m}n_{j}}$
$\displaystyle\times\prod\limits_{d=2}^{m}\binom{\sum\limits_{k=1}^{d}n_{k}}{n_{d}}x^{n_{1}+2n_{2}+3n_{3}+\cdots+mn_{m}}.$

Substituting $x_{1}=x$, $x_{2}=x$, $x_{3}=x$,…, $x_{m}=x$ into (10), we get
the following result:

###### Corollary 6

Let $m\in\mathbb{N}$ with and $n_{1}\in\mathbb{N}_{0}$. Then we have

$\displaystyle\mathbb{Y}_{n_{1}}\left(\underset{m{\
times}}{\underbrace{x,x,\ldots,x}}\right)$ $\displaystyle=$
$\displaystyle\left(\prod\limits_{j=2}^{m}\sum_{n_{j}=0}^{\left[\frac{n_{j-1}}{j}\right]}\right)(-1)^{\sum_{j=0}^{m}n_{j}}$
$\displaystyle\times\prod\limits_{d=2}^{m}\binom{\sum\limits_{k=1}^{d}n_{k}}{n_{d}}x^{n_{1}+n_{2}+n_{3}+\cdots+n_{m}}.$

Substituting $x=1$ into (5), we arrive at the following new general family of
special numbers or so-called certain classes of finite sum:

###### Corollary 7

Let $n_{1}\in\mathbb{N}_{0}$. Then we have

$\mathbb{Y}_{n_{1}}\left(\underset{m{\
times}}{\underbrace{1,1,\ldots,1}}\right)=\left(\prod\limits_{j=2}^{m}\sum_{n_{j}=0}^{\left[\frac{n_{j-1}}{j}\right]}\right)(-1)^{\sum_{j=0}^{m}n_{j}}\prod\limits_{d=2}^{m}\binom{\sum\limits_{k=1}^{d}n_{k}}{n_{d}}.$
(13)

Substituting $x=-1$ into (5), we also arrive at the following new general
family of special numbers or so-called certain classes of finite sum:

###### Corollary 8

Let $n_{1}\in\mathbb{N}_{0}$. Then we have

$\displaystyle\mathbb{Y}_{n_{1}}\left(\underset{m{\
times}}{\underbrace{-1,-1,\ldots,-1}}\right)$ $\displaystyle=$
$\displaystyle\left(\prod\limits_{j=2}^{m}\sum_{n_{j}=0}^{\left[\frac{n_{j-1}}{j}\right]}\right)(-1)^{\left(\sum_{j=0}^{m}n_{j}+n_{1}+2n_{2}+3n_{3}+\cdots+mn_{m}\right)}$
(14)
$\displaystyle\times\prod\limits_{d=2}^{m}\binom{\sum\limits_{k=1}^{d}n_{k}}{n_{d}}.$

Observe that the formula given in equation (14) gives us the solution of
Exercise 21 in Charalambides’s book [7, p. 269], given below:

Substituting $P_{j}(x_{j})=-1$, $j\in\left\\{1,2,\ldots,m\right\\}$ into (1),
we get generating function for the Fibonacci numbers of order $m$:

$\frac{1}{1-w-w^{2}-\cdots-w^{m}}=\sum_{n=0}^{\infty}\mathbb{Y}_{n}\left(\underset{m\text{
times}}{\underbrace{-1,-1,\ldots,-1}}\right)w^{n},$

where

$F_{n+1,m}:=\mathbb{Y}_{n}\left(\underset{m\text{
times}}{\underbrace{-1,-1,\ldots,-1}}\right)$

is the number of $n$-permutations of the set $\left\\{0,1\right\\}$ with
repetition and the restriction that no $m$ zeros are consecutive. By using
(14), we have $F_{0,m}=F_{1,m}=1$. Hence, it is easy to show that

$F_{n,m}=2F_{n-1,m}-F_{n-m-1,m},$

where $n=m+1$, $m+2$, $m+3$, $\ldots$,

$F_{n,m}=\sum\limits_{v=0}^{a}F_{n-v,m},$

where $a=\min\left\\{n,m\right\\}$ (cf. [7, p. 269]) and for
$n\in\left\\{1,2,\ldots,m\right\\}$, we also have

$\sum\limits_{v=0}^{n}\binom{n}{v}=2\mathbb{Y}_{n}\left(\underset{m\text{
}times}{\underbrace{-1,-1,\ldots,-1}}\right).$

By combining (2) with (1), we get the following functional equation:

$G\left(w,P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)=F\left(w,P(\overrightarrow{X_{m}})\right)\sum\limits_{j=0}^{k}Q_{j}(x_{j})w^{j}.$

Using the above equation, we get

$\sum_{n=0}^{\infty}\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)w^{n}=\sum\limits_{j=0}^{k}Q_{j}(x_{j})\sum_{n=0}^{\infty}\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)w^{n+j}.$

Therefore

$\sum_{n=0}^{\infty}\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)w^{n}=\sum\limits_{j=0}^{k}Q_{j}(x_{j})\sum_{n=j}^{\infty}\mathbb{Y}_{n-j}\left(P(\overrightarrow{X_{m}})\right)w^{n}.$

Comparing the coefficients of $w^{n}$ on both sides of the above equation, we
arrive at the following theorem:

###### Theorem 9

Let $n\in\mathbb{N}_{0}$. Then we have

$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)=\sum\limits_{j=0}^{k}Q_{j}(x_{j})\mathbb{Y}_{n-j}\left(P(\overrightarrow{X_{m}})\right),$

where $m$ and $k$ any nonegative integers.

##  3 Some special values of the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$

In this section, using some special values of polynomials
$P(\overrightarrow{X_{m}})$ and $Q(\overrightarrow{X_{k}})$, we give special
values of the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$
involving many certain families of the special numbers and polynomials.

Some special certain families of numbers and polynomials can be expressed in
terms of the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$
as follows:

Binomial coefficients: Substituting $m=1$ and $P_{1}(x_{1})=-1-x_{1}$ into
(1), we have the following well-known bivariate generating function for the
binomial coefficients:

$\displaystyle\frac{1}{1+\left(-1-x_{1}\right)w}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{n}\left(-1-x_{1}\right)w^{n}$
$\displaystyle=$ $\displaystyle\sum_{n,m}^{\infty}\binom{n}{m}x_{1}^{m}w^{n}.$

After some calculation above equation, we get

$\displaystyle\frac{1}{1+\left(-1-x_{1}\right)w}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{n}\left(-1-x_{1}\right)w^{n}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\sum_{m=0}^{n}\binom{n}{m}x_{1}^{m}w^{n}.$

Therefore

$\displaystyle\mathbb{Y}_{n}\left(-1-x_{1}\right)=\sum_{m=0}^{n}\binom{n}{m}x_{1}^{m}.$
(15)

Substituting $x_{1}=1$ and $x_{1}=-1$ into (15), respectively, we have the
following well-known results:

$\displaystyle\mathbb{Y}_{n}\left(-2\right)=2^{n}$

and

$\displaystyle\mathbb{Y}_{n}\left(0\right)=0.$

Sextet polynomials of hexagonal systems: Substituting
$P_{1}(x)=-x^{2}-4x-1,P_{2}(x_{2})=x^{2}$ into (6), we get the following
generating functions for the sextet polynomials:

$\frac{1}{1+(-x^{2}-4x-1)w+x^{2}w^{2}}=\sum_{n_{1}=0}^{\infty}\mathbb{Y}_{n_{1}}\left(-x^{2}-4x-1,x^{2}\right)w^{n_{1}}.$
(16)

Thus, combining the above equation with (1), we obtain the following explicit
formula for the sextet polynomials:

$\mathbb{Y}_{n_{1}}\left(-x^{2}-4x-1,x^{2}\right)=\sum_{n_{2}=0}^{\left[\frac{n_{1}}{2}\right]}(-1)^{2n_{1}-3n_{2}}\binom{n_{1}-n_{2}}{n_{2}}\left(x^{2}+4x+1\right)^{n_{1}-2n_{2}}x^{2n_{2}}.$
(17)

Using (17), we have
$\deg\left(\mathbb{Y}_{n_{1}}\left(-x^{2}-4x-1,x^{2}\right)\right)=2n_{1}$.

Li et al. [27] also gave generating function for the sextet polynomials. They
also gave some properties of these polynomials. The sextet polynomials of
hexagonal systems are symmetric, unimodal, log-concave, and asymptotically
normal.

The sextet polynomials are related to the Clar covering polynomial of
hexagonal systems and chromatic polynomials, which have many applications in
graphs theory and and in the theory of hexagonal systems. The sextet
polynomials are also very important to analyze hexagonal system or benzenoid
system with the aid of finite connected plane graph without cut vertices with
every interior face is bounded by a regular hexagon of side length $1$.
Moreover, topological properties of hexagonal systems have many importan
applications in various quantum mechanical models of the electronic structure
of benzenoid hydrocarbons and also in resonance theory, Huckel molecular
orbital theory, Clar’s aromatic sextet theory and the theory of conjugated
circuits (cf. [6, 18, 27]).

Rank polynomials: Substituting $Q_{0}(x)=1$, $Q_{1}(x)=-x^{2}$ and
$P_{1}(x)=-1-x-x^{2}$, $P_{2}(x)=x^{2}$, $P_{3}(x_{3})=x^{3}$ into (2), we get
the following rank polynomials of the lattices $\mathcal{J}(\mathcal{G}_{m})$,
which is the set of all ideals of the partially ordered set $\mathcal{G}_{m}$
which is known as garland of order $m$:

$g_{m}(x)=\mathbb{S}_{m}\left(-1-x-x^{2},x^{2},x^{3};1,-x^{2}\right),$

where

$g_{m}(x)=\sum\limits_{k=0}^{m}g_{mk}x^{k}.$

The rank polynomials of the lattices have various interesting applications in
the theory of Ockham algebras, and also in combinatorics involving fences (or
zigzag posets), crowns, garlands (or double fences) and many of their
generalizations of posets.

The garland of order $m$ is the partially ordered set $\mathcal{G}_{m}$
briefly defined as follows: $\mathcal{G}_{0}$ is the empty poset,
$\mathcal{G}_{1}$ is the chain of length $1$ and, for any other $m\geq 2$,
$\mathcal{G}_{m}$ is the poset on $2m$ elements $x_{1}$, $x_{2}$,…,$x_{m}$ and
$y_{1}$, $y_{2}$,…,$y_{m}$ with cover relations $x_{1}<y_{1}$,
$x_{2}<y_{2}$,…,$x_{j}<y_{j-1}$, $x_{j}<y_{j}$ and $x_{j}<y_{j+1}$, for $j=2$,
$3$,…,$m-1$ and $x_{m}<y_{m-1}$ and $x_{m}<y_{m}$. The ideals and the
antichains of a garland can be represented as words of a regular language.
Here the maximal elements of an ideal of a poset is an antichain, and this
establishes a bijective correspondence between all ideals of poset (cf. [32]).
An explicit formula for the number $g_{m}$ of all antichains or ideals of
$\mathcal{G}_{m}$ is given as follows:

$g_{m}=\mathbb{S}_{m}\left(-2,-1;1,1\right)=\frac{\left(1+\sqrt{2}\right)^{m+1}+\left(1-\sqrt{2}\right)^{m+1}}{2}$

(cf. [32]).

Note that the formula of the number $g_{m}$ of all antichains is also related
to Exercise 11 in Charalambides’s book [7, p. 267]. Due to the Exercise 11, we
also observe that $g_{m}$ is the number of $m$-permutations of the set
$\left\\{0,1,2\right\\}$ with repetition and the restriction now two zeros and
no two ones are consecutive.

Figurate numbers: We give relations between special values of the polynomials
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$
and the figurate numbers.

The $j$-gonal numbers, which are members of the space figurate numbers:

$\frac{x+(j-3)x^{2}}{1-3x+3x^{2}-x^{3}}=\sum\limits_{n=0}^{\infty}\mathbb{S}_{n}\left(-3,3,-1;0,1,j-3\right)w^{n}$

is the generating functions for the sequence of hexagonal prism numbers. That
is,

$S_{j}(n):=\mathbb{S}_{n}\left(-3,3,-1;0,1,j-3\right).$

For $j=3$, $S_{3}(n)$ reduces to the triangular numbers, which was studied by
Theon of Smyrna in the II-th century AC.

Hexagonal prism numbers, which are members of the space figurate numbers:

$\frac{x+10x^{2}+7x^{3}}{1-4x+6x^{2}-4x^{3}+x^{4}}=\sum\limits_{n=0}^{\infty}\mathbb{S}_{n}\left(-4,6,-4,1;0,1,10,7\right)w^{n}$

is the generating functions for the sequence of hexagonal prism numbers (cf.
[9, p. 145]). That is,

$PCS_{6}(n):=\mathbb{S}_{n}\left(-4,6,-4,1;0,1,10,7\right).$

The centered $j$-pyramidal numbers

$CS_{j}^{3}(n):=\mathbb{S}_{n}\left(-4,6,-4,1;0,1,j-2,1\right).$

The centered dodecahedron numbers

$\overline{D}(n):=\mathbb{S}_{n}\left(-4,6,-4,1;0,1,17,17,1\right).$

The centered icosahedron number

$\overline{I}(n):=\mathbb{S}_{n}\left(-4,6,-4,1;0,1,9,9,1\right).$

(Sloane’s A005902).

The centered octahedron number

$\overline{O}(n):=\mathbb{S}_{n}\left(-4,6,-4,1;0,1,3,3,1\right),$

for detail, see [9, p. 145].

Antichain polynomials: The antichain polynomials, which are also related to
not only the Chebyshev polynomials and the Sturm sequence but also the Euler
transform of a formal series, are given by

$G\left(w,-1-x,-x;1,x\right)=\sum\limits_{n=0}^{\infty}\mathbb{S}_{n}\left(-1-x,-x;1,x\right)w^{n}.$

From the above equation, we get

$a_{n}(x)=\mathbb{S}_{n}\left(-1-x,-x;1,x\right),$

where

$a_{n}(x)=\sum\limits_{k=0}^{n}a_{nk}x^{k}.$

These polynomials are log-concave and unimodal, for detail see [32].

$\sum\limits_{n=0}^{\infty}a_{2n,n}x^{n}$ is the generatin functions,
associated with diagonal of the double series, for the following matrix, which
rows are given by the sequence A035607,

$A=\left[a_{n,m}\right]=\left[\begin{array}[]{ccccccc}1&&&&&&\\\ 1&4&2&&&&\\\
1&6&8&2&&&\\\ 1&8&18&12&2&&\\\ 1&10&32&38&16&2&\\\ 1&12&50&88&66&20&2\\\
\ldots&&&&&&\end{array}\right]$

is given by

$\displaystyle G\left(w,-1-4x-x^{2},x^{2};1,x^{2}\right)$ $\displaystyle=$
$\displaystyle\sum\limits_{n=0}^{\infty}\mathbb{S}_{n}\left(-1-4x-x^{2},x^{2};1,x^{2}\right)w^{n}$
$\displaystyle=$
$\displaystyle\frac{1+x^{2}w}{1+\left(-1-4x-x^{2}\right)w+x^{2}w^{2}},$

for detail see [32].

Euler transform: The Euler transform of a formal series
$F(w)=\sum\limits_{j\geq 0}u_{j}w^{j}$ is given by

$\displaystyle T^{\theta}\left(F(w)\right)$ $\displaystyle=$
$\displaystyle\frac{1}{1-\theta w}F\left(\frac{w}{1-\theta w}\right)$
$\displaystyle=$
$\displaystyle\sum\limits_{j=0}^{\infty}\sum\limits_{v=0}^{j}\binom{j}{v}\theta^{j-v}u_{v}w^{j}$

Thus by applying the Euler transform $T^{\theta}$ to the formal series
$G\left(w,-1-x,-x;1,x\right)$ with respect to $w$, we obtain

$\displaystyle T^{x}\left\\{G\left(w,-1-x,-x;1,x\right)\right\\}$
$\displaystyle=$ $\displaystyle F\left(w,-1-3x,2x^{2}\right)$ $\displaystyle=$
$\displaystyle F_{U}\left(\sqrt{2}xw,\frac{1+3x}{2\sqrt{2}x}\right),$

where $F_{U}\left(w,x\right)$ denotes the following generating functions for
the Chebyshev polynomials of the second kind

$\displaystyle F_{U}\left(w,x\right)$ $\displaystyle=$ $\displaystyle
G\left(w,-2x,1;1\right)$ $\displaystyle=$ $\displaystyle
F\left(w,-2x,1\right)$ $\displaystyle=$
$\displaystyle\sum\limits_{n=0}^{\infty}U_{n}(x)w^{n},$

where

$U_{n}(x):=\mathbb{S}_{n}\left(-2x,1;1\right)=\mathbb{Y}_{n}\left(-2x,1\right).$

Since the Euler transform is an invertible operator. By applying the Euler
invertible operator $T^{-\theta}=\left(T^{\theta}\right)^{-1}$ to Eq. (3), we
obtain

$\displaystyle\mathbb{S}_{n}\left(-1-x,-x;1,x\right)$ $\displaystyle=$
$\displaystyle T^{-x}\left\\{F\left(w,-1-3x,2x^{2}\right)\right\\}$
$\displaystyle=$ $\displaystyle
T^{-x}\left\\{F_{U}\left(\sqrt{2}xw,\frac{1+3x}{2\sqrt{2}x}\right)\right\\}.$

For detail applications the Euler transform of a formal series to the
antichain polynomials and the Chebyshev polynomials see [32].

Convolved Fermat quotients: Goubi [14] gave combinatorial formulation of the
convolved Fermat quotients, which are related to the polynomials
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{2m}});Q(\overrightarrow{X_{k}})\right)$.

Fibonacci polynomials: The Fibonacci polynomials can also be expressed in
terms of the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$
as

$\displaystyle\mathbb{S}_{n}\left(-x,-1;0,1\right)$ $\displaystyle=$
$\displaystyle\mathbb{Y}_{n-1}\left(-x,-1\right)=\sum_{n_{2}=0}^{\left[\frac{n-1}{2}\right]}\binom{n-n_{2}-1}{n_{2}}x^{n-2n_{2}-1}$
$\displaystyle=$ $\displaystyle F_{n}(x).$

Let $m\in\mathbb{N}$. We define the following certain series of reciprocals of
the Fibonacci numbers:

$\mathcal{R}_{m}(w)=\sum\limits_{j=1}^{\infty}\frac{w^{j}}{F_{mj}}.$ (19)

Let $\lambda+\mu=1$, $\lambda-\mu=\sqrt{5}$ and $\lambda\mu=-1$. Using (19),
we have

$\displaystyle\mathcal{R}_{m}(\lambda^{2}w)-\mathcal{R}_{m}(\mu^{2}w)$
$\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{\infty}\frac{\lambda^{2j}-\mu^{2j}}{F_{mj}}w^{j}$
$\displaystyle=$
$\displaystyle\left(\lambda-\mu\right)\sum\limits_{j=1}^{\infty}\frac{F_{2j}}{F_{mj}}w^{j}.$

Substituting $m=1$ into (19), we get

$\mathcal{R}_{1}(w)=\sum\limits_{j=1}^{\infty}\frac{w^{j}}{F_{j}},$

(cf. [15]).

Substituting $m=1$ into (3), we also get

$\mathcal{R}_{1}(\lambda^{2}w)-\mathcal{R}_{1}(\mu^{2}w)=\sum\limits_{j=1}^{\infty}\frac{F_{2j}}{F_{j}}w^{j}=G\left(w,-1,-1;1,2\right).$

Combining the above equation with (2), we obtain

$\sum\limits_{j=1}^{\infty}\frac{F_{2j}}{F_{j}}w^{j}=\sum_{j=1}^{\infty}\mathbb{S}_{j}\left(-1,-1;1,2\right)w^{j}.$

Comparing the coefficients of $w^{j}$ on both sides of the above equality, we
arrive at the following theorem:

###### Theorem 10

Let $j\in\mathbb{N}$. Then we have

$\mathbb{S}_{j}\left(-1,-1;1,2\right)=\frac{F_{2j}}{F_{j}}.$

Substituting $m=2$ and $w=1$ into (19), the following certain series of
reciprocals of the Fibonacci numbers can be expressed in terms of the Lambert
series, which is defined by

$L(w)=\sum\limits_{j=1}^{\infty}\frac{w^{j}}{1-w^{j}}$
$\sum\limits_{j=1}^{\infty}\frac{1}{F_{2j}}=L\left(\frac{3-\sqrt{5}}{2}\right)-L\left(\frac{7-3\sqrt{5}}{2}\right),$

(cf. [15]).

Lucas polynomials: The Lucas polynomials can also be expressed in terms of the
polynomials $\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$
as follows:

$\displaystyle\mathbb{S}_{n}\left(-x,-1;2,-x\right)$ $\displaystyle=$
$\displaystyle
2\mathbb{Y}_{n}\left(-x,-1\right)-x\mathbb{Y}_{n-1}\left(-x,-1\right)$
$\displaystyle=$ $\displaystyle
2\sum_{n_{1}=0}^{\left[\frac{n_{1}}{2}\right]}\binom{n_{1}-n_{2}}{n_{2}}x^{n_{1}-2n_{2}}-\sum_{n_{1}=0}^{\left[\frac{n_{1}-1}{2}\right]}\binom{n_{1}-n_{2}-1}{n_{2}}x^{n_{1}-2n_{2}-2}$
$\displaystyle=$ $\displaystyle L_{n}(x).$

Using (19), we have

$\displaystyle\mathcal{R}_{m}(\lambda^{2}w)+\mathcal{R}_{m}(\mu^{2}w)$
$\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{\infty}\frac{\lambda^{2j}+\mu^{2j}}{F_{mj}}w^{j}$
$\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{\infty}\frac{L_{2j}}{F_{mj}}w^{j}.$

Putting $m=4$ in the above equation after that combining the well-known
Binet’s formulas for the Fibonacci numbers and the Lucas numbers, we get

$\mathcal{R}_{4}(\lambda^{2}w)+\mathcal{R}_{4}(\mu^{2}w)=\frac{1}{\lambda-\mu}\sum\limits_{j=1}^{\infty}\frac{1}{F_{2j}}w^{j}.$
$\sum\limits_{j=1}^{\infty}\frac{L_{2j}}{F_{4j}}w^{j}=\frac{1}{\lambda-\mu}\sum\limits_{j=1}^{\infty}\frac{1}{F_{2j}}w^{j}.$

When $w=1$, the above equation reduces to the following result:

###### Corollary 11

$\sum\limits_{j=1}^{\infty}\frac{L_{2j}}{F_{4j}}=\frac{L\left(\frac{3-\sqrt{5}}{2}\right)-L\left(\frac{7-3\sqrt{5}}{2}\right)}{\sqrt{5}}.$

Fibonacci type polynomials and Lucas type polynomials:

Fibonacci type polynomials and Lucas type polynomial can also be expressed in
terms of the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$
as follows:

The Tribonacci-Lucas polynomials $t_{l,n}(x)$:

$t_{l,n}(x)=\mathbb{S}_{n}\left(-x^{2},-x,-1;3,-2x^{2},x\right),$

the Tribonacci polynomials $T_{t,n}(x)$:

$T_{t,n}(x)=\mathbb{S}_{n}\left(-x^{2},-x,-1;3,0,1\right),$

(cf. [10], [36]).

The Tribonacci quaternion polynomials $P_{t,n}(x)$:

$P_{t,n}(x)=\mathbb{S}_{n}\left(-x^{2},-x,-1;i+x^{2}j+(x+x^{4})k,i+xj+(1+x^{3})k,j+x^{2}k\right),$

where a quaternion $V$ with real components $x_{0}$, $x_{1}$, $x_{2}$, $x_{3}$
and basis $1$, $i$, $j$, $k$ is a number of the form
$V=1x_{0}+x_{1}i+x_{2}j+x_{3}k$, where $i^{2}=j^{2}=k^{2}=1$, $ij=k=-ji$,
$jk=i=-kj$, $ki=j=-ik$, and the conjuget $\overline{V}$ of $V$ is
$\overline{V}=1x_{0}+x_{1}i+x_{2}j+x_{3}k$ and the norm is
$V\overline{V}=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$ (cf. [17]).

The Chebyshev polynomials of the first kind $T_{n}(x)$:

$T_{n}(x):=\mathbb{S}_{n}\left(-2x,1;1,-x\right),$

the Chebyshev polynomials of the second kind $U_{n}(x)$:

$U_{n}(x):=\mathbb{S}_{n}\left(-2x,1;1\right)=\mathbb{Y}_{n}\left(-2x,1\right),$

the Chebyshev polynomials of the third kind $T_{3,n}(x)$:

$T_{3,n}(x):=\mathbb{S}_{n}\left(-2x,1;1,-1\right),$

the Chebyshev polynomials of the fourtht kind $T_{4,n}(x)$:

$T_{4,n}(x):=\mathbb{S}_{n}\left(-2x,1;1,1\right),$

the monic $2$-orthogonal Chebyshev polynomial $\widehat{T}_{n}(x)$:

$\widehat{T}_{n}(x):=\mathbb{S}_{n}\left(x,\alpha,\gamma\right),$

where where $\alpha$, $\gamma$ are constants (cf. [Abromovic]-[43]).

The Generalized Padovan sequences ($m$\- Padovan numbers):

$\mathcal{P}_{n}^{(m)}:=\mathbb{S}_{n}\left(0,\underset{m\text{ times
}}{\underbrace{-1,-1,\ldots,-1}};0,1,1\right)$

(cf. [4]).

Generating functions of the set $F$ of words over $\left\\{a,b\right\\}$: The
following generating functions of the set $\mathcal{F}$ of words over
$\left\\{a,b\right\\}$ without factor $w=a^{n}$ is given by

$\displaystyle G\left(z,-2,\underset{m\text{
times}}{\underbrace{0,0,\ldots,0}},1;x,\underset{m-1\text{
times}}{\underbrace{0,0,\ldots,0}},-1\right)$ $\displaystyle=$
$\displaystyle\sum\limits_{m=0}^{\infty}\mathbb{S}_{m}\left(-2,\underset{m\text{
times }}{\underbrace{0,0,\ldots,0}},1;x,\underset{m-1\text{
times}}{\underbrace{0,0,\ldots,0}},-1\right)z^{m}$ $\displaystyle=$
$\displaystyle\frac{1-z^{m}}{1-2z+z^{m+1}},$

for detail, see [28, p. 98].

###  3.1 Binet type formulas the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$

In this section, we give Binet type formulas for the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$
with the aid of generating functions.

Substituting $m=2$ into (1), we now give Binet type formula for the
polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{2}})\right)=\mathbb{Y}_{n}\left(P_{1}(x_{1}),P_{2}(x_{2})\right)$
as follows.

$\frac{1}{1+P_{1}(x_{1})w+P_{2}(x_{2})w^{2}}=\sum_{n=0}^{\infty}\mathbb{Y}_{n}\left(P_{1}(x_{1}),P_{2}(x_{2})\right)w^{n}.$

The partial fraction decomposition of the left side of above equation is given
by

$\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{n}\left(P_{1}(x_{1}),P_{2}(x_{2})\right)w^{n}$
$\displaystyle=$
$\displaystyle\frac{1}{\left(1-\frac{w}{a_{1}(x_{1},x_{2})}\right)\left(1-\frac{w}{a_{2}(x_{1},x_{2})}\right)}$
$\displaystyle=$
$\displaystyle-\frac{a_{2}(x_{1},x_{2})}{\left(a_{1}(x_{1},x_{2})-a_{2}(x_{1},x_{2})\right)\left(1-\frac{w}{a_{1}(x_{1},x_{2})}\right)}$
$\displaystyle+\frac{a_{1}(x_{1},x_{2})}{\left(a_{1}(x_{1},x_{2})-a_{2}(x_{1},x_{2})\right)\left(1-\frac{w}{a_{2}(x_{1},x_{2})}\right)},$

where

$a_{1}(x_{1},x_{2})=\frac{-P_{1}(x_{1})+\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}{2P_{2}(x_{2})}$

and

$a_{2}(x_{1},x_{2})=\frac{-P_{1}(x_{1})-\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}{2P_{2}(x_{2})}.$

Therefore

$\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{n}\left(P_{1}(x_{1}),P_{2}(x_{2})\right)w^{n}$
$\displaystyle=$
$\displaystyle-\frac{a_{2}(x_{1},x_{2})}{a_{1}(x_{1},x_{2})-a_{2}(x_{1},x_{2})}\sum_{n=0}^{\infty}\left(\frac{w}{a_{1}(x_{1},x_{2})}\right)^{n}$
$\displaystyle+\frac{a_{1}(x_{1},x_{2})}{a_{1}(x_{1},x_{2})-a_{2}(x_{1},x_{2})}\sum_{n=0}^{\infty}\left(\frac{w}{a_{2}(x_{1},x_{2})}\right)^{n}.$

Comparing the coefficients of $w^{n}$ on both sides of the above equation, we
get

$\displaystyle\mathbb{Y}_{n}\left(P_{1}(x_{1}),P_{2}(x_{2})\right)=\frac{\left(a_{1}(x_{1},x_{2})\left(\frac{1}{a_{2}(x_{1},x_{2})}\right)^{n}-a_{2}(x_{1},x_{2})\left(\frac{1}{a_{1}(x_{1},x_{2})}\right)^{n}\right)}{a_{1}(x_{1},x_{2})-a_{2}(x_{1},x_{2})}.$

After some elemntary calculations in the above equation, the following binet-
type formula is obtained:

###### Theorem 12

Let $n\in\mathbb{N}_{0}$. Then we have

$\displaystyle\mathbb{Y}_{n}\left(P_{1}(x_{1}),P_{2}(x_{2})\right)$
$\displaystyle=$
$\displaystyle\frac{\left(-P_{1}(x_{1})+\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}\right)\left(\frac{2P_{2}(x_{2})}{-P_{1}(x_{1})-\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}\right)^{n}}{2P_{2}(x_{2})\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}$
$\displaystyle+\frac{\left(P_{1}(x_{1})+\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}\right)\left(\frac{2P_{2}(x_{2})}{-P_{1}(x_{1})+\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}\right)^{n}}{2P_{2}(x_{2})\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}.$

Substituting $m=2$ and $k=1$ into (2), we now give Binet type formula for the
polynomials
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{2}});Q(\overrightarrow{X_{1}})\right)=\mathbb{S}_{n}\left(P_{1}(x_{1}),P_{2}(x_{2});Q_{0}(x_{0}),Q_{1}(x_{1})\right)$
as follows:

$\frac{Q_{0}(x_{0})+Q_{1}(x_{1})w}{1+P_{1}(x_{1})w+P_{2}(x_{2})w^{2}}=\sum_{n=0}^{\infty}\mathbb{S}_{n}\left(P(\overrightarrow{X_{2}});Q(\overrightarrow{X_{1}})\right)w^{n}.$

The partial fraction decomposition of the left side of above equation is given
by

$\displaystyle\sum_{n=0}^{\infty}\mathbb{S}_{n}\left(P(\overrightarrow{X_{2}});Q(\overrightarrow{X_{1}})\right)w^{n}$
$\displaystyle=$
$\displaystyle\frac{Q_{0}(x_{0})+Q_{1}(x_{1})w}{\left(1-\frac{w}{a_{1}(x_{1},x_{2})}\right)\left(1-\frac{w}{a_{2}(x_{1},x_{2})}\right)}$
$\displaystyle=$
$\displaystyle-\frac{a_{1}(x_{1},x_{2})a_{2}(x_{1},x_{2})\left(\frac{Q_{0}(x_{0})}{a_{1}(x_{1},x_{2})}+Q_{1}(x_{1})\right)}{\left(a_{1}(x_{1},x_{2})-a_{2}(x_{1},x_{2})\right)\left(1-\frac{w}{a_{1}(x_{1},x_{2})}\right)}$
$\displaystyle+\frac{a_{2}(x_{1},x_{2})a_{1}(x_{1},x_{2})\left(\frac{Q_{0}(x_{0})}{a_{2}(x_{1},x_{2})}+Q_{1}(x_{1})\right)}{\left(a_{1}(x_{1},x_{2})-a_{2}(x_{1},x_{2})\right)\left(1-\frac{w}{a_{2}(x_{1},x_{2})}\right)},$

where

$a_{1}(x_{1},x_{2})=\frac{-P_{1}(x_{1})+\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}{2P_{2}(x_{2})}$

and

$a_{2}(x_{1},x_{2})=\frac{-P_{1}(x_{1})-\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}{2P_{2}(x_{2})}.$

Therefore

$\displaystyle\sum_{n=0}^{\infty}\mathbb{S}_{n}\left(P_{1}(x_{1}),P_{2}(x_{2});Q_{0}(x_{0}),Q_{1}(x_{1})\right)w^{n}$
$\displaystyle=$
$\displaystyle-\frac{a_{1}(x_{1},x_{2})a_{2}(x_{1},x_{2})\left(\frac{Q_{0}(x_{0})}{a_{1}(x_{1},x_{2})}+Q_{1}(x_{1})\right)}{a_{1}(x_{1},x_{2})-a_{2}(x_{1},x_{2})}\sum_{n=0}^{\infty}\left(\frac{1}{a_{1}(x_{1},x_{2})}\right)^{n}w^{n}$
$\displaystyle+\frac{a_{2}(x_{1},x_{2})a_{1}(x_{1},x_{2})\left(\frac{Q_{0}(x_{0})}{a_{2}(x_{1},x_{2})}+Q_{1}(x_{1})\right)}{\left(a_{1}(x_{1},x_{2})-a_{2}(x_{1},x_{2})\right)}\sum_{n=0}^{\infty}\left(\frac{1}{a_{2}(x_{1},x_{2})}\right)^{n}w^{n}.$

Comparing the coefficients of $w^{n}$ on both sides of the above equation, we
get

$\displaystyle\mathbb{S}_{n}\left(P_{1}(x_{1}),P_{2}(x_{2});Q_{0}(x_{0}),Q_{1}(x_{1})\right)$
$\displaystyle=$
$\displaystyle\frac{\left(\frac{2P_{2}(x_{2})Q_{0}(x_{0})}{-P_{1}(x_{1})-\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}+Q_{1}(x_{1})\right)}{P_{2}(x_{2})\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}\left(\frac{2P_{2}(x_{2})}{-P_{1}(x_{1})-\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}\right)^{n}$
$\displaystyle-\frac{\left(\frac{2P_{2}(x_{2})Q_{0}(x_{0})}{-P_{1}(x_{1})+\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}+Q_{1}(x_{1})\right)}{P_{2}(x_{2})\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}\left(\frac{2P_{2}(x_{2})}{-P_{1}(x_{1})+\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}\right)^{n}.$

After some elementary calculations in the above equation, the following binet-
type formula is obtained:

###### Theorem 13

Let $n\in\mathbb{N}_{0}$. Then we have

$\displaystyle\left(\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}\right)\mathbb{S}_{n}\left(P_{1}(x_{1}),P_{2}(x_{2});Q_{0}(x_{0}),Q_{1}(x_{1})\right)$
$\displaystyle=$ $\displaystyle
Q_{0}(x_{0})\left(\left(\frac{2P_{2}(x_{2})}{-P_{1}(x_{1})-\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}\right)^{n+1}-\left(\frac{2P_{2}(x_{2})}{-P_{1}(x_{1})+\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}\right)^{n+1}\right)$
$\displaystyle+Q_{1}(x_{1})\left(\left(\frac{2P_{2}(x_{2})}{-P_{1}(x_{1})-\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}\right)^{n}-\left(\frac{2P_{2}(x_{2})}{-P_{1}(x_{1})+\sqrt{P_{1}^{2}(x_{1})-4P_{2}(x_{2})}}\right)^{n}\right).$

Substituting $P_{1}(x_{1})=-1$, $P_{2}(x_{2})=-1$, $Q_{0}(x_{0})=0$, and
$Q_{1}(x_{1})=1$, we get the following corollary:

###### Corollary 14

Let $n\in\mathbb{N}_{0}$. Then we have

$\mathbb{S}_{n}\left(-1,-1;0,1\right)=\frac{(-2)^{n}}{\sqrt{5}}\left(\frac{1}{\left(1+\sqrt{5}\right)^{n}}-\frac{1}{\left(1-\sqrt{5}\right)^{n}}\right)=F_{n}.$
(23)

Substituting $P_{1}(x_{1})=-1$, $P_{2}(x_{2})=-1$, $Q_{0}(x_{0})=2$, and
$Q_{1}(x_{1})=-1$, we get the following corollary:

###### Corollary 15

Let $n\in\mathbb{N}_{0}$. Then we have

$\displaystyle\mathbb{S}_{n}\left(-1,-1;2,-1\right)$ $\displaystyle=$
$\displaystyle\frac{2(-2)^{n+1}}{\sqrt{5}}\left(\frac{1}{\left(1-\sqrt{5}\right)^{n+1}}-\frac{1}{\left(1+\sqrt{5}\right)^{n+1}}\right)$
$\displaystyle-\frac{(-2)^{n}}{\sqrt{5}}\left(\frac{1}{\left(1-\sqrt{5}\right)^{n}}-\frac{1}{\left(1+\sqrt{5}\right)^{n}}\right)$
$\displaystyle=$ $\displaystyle L_{n}.$

Substituting $P_{1}(x_{1})=-2$, $P_{2}(x_{2})=-1$, $Q_{0}(x_{0})=1$, and
$Q_{1}(x_{1})=1$, we get the following corollary:

###### Corollary 16

Let $n\in\mathbb{N}_{0}$. Then we have

$\displaystyle\left(2\sqrt{2}\right)\mathbb{S}_{n}\left(-2,-1;1,1\right)$
$\displaystyle=$
$\displaystyle\left(\frac{1}{-1+\sqrt{2}}\right)^{n+1}-\left(\frac{-1}{1+\sqrt{2}}\right)^{n+1}+\left(\frac{1}{-1+\sqrt{2}}\right)^{n}-\left(\frac{-1}{1+\sqrt{2}}\right)^{n}.$

Observe that it is easy to see that the formula given in equation (16) gives
us the solution of Exercise 11 in Charalambides’s book [7, p. 267], given
below:

$\mathbb{S}_{n}\left(-2,-1;1,1\right)=y_{n},$

where $y_{n}$ is the number of $n$-permutations of the set
$\left\\{0,1,2\right\\}$ with repetition and the restriction now two zeros and
no two ones are consecutive.

The Pell numbers

$\mathbb{S}_{n}\left(-2,-1;0,1\right)=P_{n}.$

The Pell-Lucas numbers

$\mathbb{S}_{n}\left(-1,-1;2,-2\right)=Pl_{n},$

(cf. [1]).

The Fibonacci polynomials

$\mathbb{S}_{n}\left(-x,-1;0,1\right)=F_{n}(x)=\frac{1}{\sqrt{x^{2}+4}}\left(\left(\frac{-2}{x-\sqrt{x^{2}+4}}\right)^{n}-\left(\frac{-2}{x+\sqrt{x^{2}+4}}\right)^{n}\right).$

The Lucas polynomials

$\mathbb{S}_{n}\left(-x,-1;2,-x\right)=L_{n}(x)$

That is

$\displaystyle\left(\sqrt{x^{2}+4}\right)\mathbb{S}_{n}\left(-x,-1;2,-x\right)$
$\displaystyle=$ $\displaystyle
2\left(\left(\frac{-2}{x-\sqrt{x^{2}+4}}\right)^{n+1}-\left(\frac{-2}{x+\sqrt{x^{2}+4}}\right)^{n+1}\right)$
$\displaystyle-$ $\displaystyle
x\left(\left(\frac{-2}{x-\sqrt{x^{2}+4}}\right)^{n}-\left(\frac{-2}{x+\sqrt{x^{2}+4}}\right)^{n}\right).$

##  4 Higher order of the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$

In this section, we define higher order of the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$.
We give some properties of these polynomials with their special values.

The polynomials $\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ of
order $\beta$, which is denoted by
$\mathbb{Y}_{n}^{(\beta)}\left(P(\overrightarrow{X_{m}})\right)$, are defined
by the following generating function

$F\left(w,P(\overrightarrow{X_{m}});\beta\right)=\frac{1}{\left(1+\sum\limits_{j=1}^{m}P_{j}(x_{j})w^{j}\right)^{\beta}}=\sum_{n=0}^{\infty}\mathbb{Y}_{n}^{(\beta)}\left(P(\overrightarrow{X_{m}})\right)w^{n}.$
(26)

Here we note that

$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right):=\mathbb{Y}_{n}^{(1)}\left(P(\overrightarrow{X_{m}})\right).$

Some special values of the function
$F\left(w,P(\overrightarrow{X_{m}});\beta\right)$ are given as follows:

$F\left(w,-1;\beta\right)=\frac{1}{\left(1-w\right)^{\beta}}=\sum_{n=0}^{\infty}\mathbb{Y}_{n}^{(\beta)}\left(-1\right)w^{n}.$

For $\left|w\right|<1$, using binomial theorem, we get

$\mathbb{Y}_{n}^{(\beta)}\left(-1\right)=(-1)^{n}\binom{-\beta}{n}=\binom{\beta+n-1}{n}=\frac{\beta\left(\beta+1\right)\cdots\left(\beta+n-1\right)}{n!}=\frac{\left(\beta\right)_{n}}{n!},$

where $\left(\beta\right)_{n}$ is denoted the rising factorial (the Pochhammer
function). Few values of $\mathbb{Y}_{n}^{(\beta)}\left(-1\right)$ are given
by

$\displaystyle\mathbb{Y}_{n}^{(1)}\left(-1\right)$ $\displaystyle=$
$\displaystyle\frac{\left(1\right)_{n}}{n!}=1,$
$\displaystyle\mathbb{Y}_{n}^{(2)}\left(-1\right)$ $\displaystyle=$
$\displaystyle\frac{\left(2\right)_{n}}{n!}=n+1,$
$\displaystyle\mathbb{Y}_{n}^{(3)}\left(-1\right)$ $\displaystyle=$
$\displaystyle\frac{\left(3\right)_{n}}{n!}=\frac{\left(n+1\right)\left(n+2\right)}{2},$
$\displaystyle\mathbb{Y}_{n}^{(4)}\left(-1\right)$ $\displaystyle=$
$\displaystyle\frac{\left(4\right)_{n}}{n!}=\frac{\left(n+1\right)\left(n+2\right)(n+3)}{6}$
$\displaystyle\vdots$ $\displaystyle\mathbb{Y}_{n}^{(k)}\left(-1\right)$
$\displaystyle=$
$\displaystyle\frac{\left(k+1\right)_{n}}{n!}=\binom{k+n}{n},$

and so on.

Substituting $P_{1}(x)=-2x$, and $P_{2}(x)=1$, into (26), we get the most
important family of orthogonal polynomials which are so-called the
ultraspherical polynomials or the Gegenbauer polynomials, for
$0\leq\left|x\right|<1$, $\beta>0$,

$C_{n}^{(\beta)}(x):=\mathbb{Y}_{n}^{(\beta)}\left(-2x,1\right),$

which are particular solutions of the Gegenbauer differential equation, and
these polynomials are also represented by the following Gaussian
hypergeometric series:

$\mathbb{Y}_{n}^{(\beta)}\left(-2x,1\right)=\frac{\left(2\beta\right)_{n}}{n!}\text{
}_{2}F_{1}\left(-n,2\beta+n;\frac{1}{2}+\beta;\frac{1-x}{2}\right),$

and also special cases of the Jacobi polynomials are given by

$P_{n}^{\left(-\frac{1}{2}+\beta,-\frac{1}{2}+\beta\right)}(x):=\frac{\left(\frac{1}{2}+\beta\right)_{n}}{\left(2\beta\right)_{n}}\mathbb{Y}_{n}^{(\beta)}\left(-2x,1\right).$

For detail about these polynomials see also [Abromovic], [40], and [43].

With the aid of the Rodrigues formula for the Gegenbauer polynomials, we have

$\displaystyle\mathbb{Y}_{n}^{(\beta)}\left(-2x,1\right)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{n}\sqrt{\pi}\Gamma(n+2\beta)(1-x^{2})^{-\beta+\frac{1}{2}}\binom{n-\frac{1}{2}}{n}}{2^{n}\Gamma(2\beta)\Gamma\left(n+\beta+\frac{1}{2}\right)}\frac{d^{n}}{dx^{n}}\left\\{(1-x^{2})^{n+\beta-\frac{1}{2}}\right\\}$
$\displaystyle=$
$\displaystyle\frac{(-1)^{n}\sqrt{\pi}(n+1)C_{n}\Gamma(n+2\beta)(1-x^{2})^{-\beta+\frac{1}{2}}}{2^{3n}\Gamma(2\beta)\Gamma\left(n+\beta+\frac{1}{2}\right)}\frac{d^{n}}{dx^{n}}\left\\{(1-x^{2})^{n+\beta-\frac{1}{2}}\right\\},$

where $C_{n}$ denotes the Catalan numbers,
$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ and

$\Gamma\left(n+\frac{1}{2}\right)=\binom{n-\frac{1}{2}}{n}n!\sqrt{\pi}=\frac{(2n)!}{4^{n}n!}\sqrt{\pi}.$

When $\beta=\frac{1}{2}$ and $\beta=1$, the Gegenbauer polynomials reduce to
the Legendre polynomials and the Chebyshev polynomials of the second kind,
respectively.

The Rogers polynomials or the Rogers-Askey-Ismail polynomials, which are so
called continuous $q$-ultraspherical polynomials, are a family of orthogonal
polynomials introduced by Rogers (1892, 1893, 1894) in the course of his work
on the Rogers–Ramanujan identities, are defined in terms of the $q$-Pochhammer
symbol and the basic hypergeometric series

$C_{n}^{(\beta\left|q\right.)}(x):=\mathbb{Y}_{n}^{(\beta\left|q\right.)}\left(-2x,1\right)=\frac{\left(\beta;q\right)}{\left(q;q\right)}e^{in\theta}\text{
}_{2}\phi_{1}\left(q^{-n},\beta;\beta^{-1}q^{1-n};q,q\beta^{-1}e^{-2i\theta}\right),$

where $x=\cos\left(\theta\right)$,

${}_{d}\phi_{c}\left(a_{1},\ldots,a_{d};b_{1},\ldots,b_{c};q,z\right)=\sum\frac{(a_{1},\ldots,a_{d};q)_{n}}{(b_{1},\ldots,b_{c};q)_{n}}\left((-1)^{n}q^{\binom{n}{2}}\right)^{1+c-d}z^{n}$

and

$(a_{1},\ldots,a_{d};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\cdots(a_{d};q)_{n},(a_{1};q)_{n}$

denotes the $q$-Pochhammer symbol ($q$-shifted factorial) or $q$-analog of the
Pochhammer symbol $(x)_{n}$,

$(a_{j};q)_{n}=\left(1-a_{j}\right)\left(1-a_{j}q\right)\cdots\left(1-a_{j}q^{n-1}\right)$

and $(a_{j};q)_{0}=1$ and also

$\lim_{q\rightarrow 1}\frac{(q^{x};q)_{n}}{(1-q)^{n}}=(x)_{n}$

(cf. [2]).

Substituting $\beta=\frac{1}{2}$, $P_{1}(x)=-2x$ and $P_{2}(x)=1$ into (26),
we get other most important family of orthogonal polynomials which are so-
called the Legendre polynomials

$P_{n}(x):=\mathbb{Y}_{n}^{(\frac{1}{2})}\left(-2x,1\right).$

Substituting $\beta=-\frac{1}{2}$, $P_{1}(x)=-3x,P_{2}(x)=0$, and $P_{3}(1)=1$
into (26), we get the Pincherle polynomials introduced by Humbert in 1921,

$P_{n}(x):=\mathbb{Y}_{n}^{(-\frac{1}{2})}\left(-3x,0,1\right)$

and

$\mathbb{Y}_{2n}^{(-\frac{1}{2})}\left(-3x,0,1\right)=\frac{d^{n}}{dx^{n}}\left\\{x^{n}(x^{2}-1)^{n}\right\\}$

(cf. [19]).

Substituting $P_{1}(x)=-mx,P_{2}(x)=P_{m}(x)=0,\ldots,P_{m}(1)=1$ into (26),
we get the Humbert polynomials, which are generalization of the Pincherle
polynomials introduced by Humbert in 1921,

$\Pi_{n,m}^{\beta}(x):=\mathbb{Y}_{n}^{(\beta)}\left(-mx,\underset{m-1\text{
times}}{\underbrace{0,0,\ldots,0}},1\right).$

For other important properties and applications of these polynomials, the
references given here, among other references, may be reviewed (cf.
[Abromovic], [2], [40], [43]).

The Fibonacci type polynomials of higher order in two variables
$\mathcal{G}_{v}^{(h)}(x,y,k,m,n)$:

$\mathcal{G}_{v}^{(h)}(x_{1},x_{m+n},k,m,n):=\mathbb{Y}_{v}^{(h)}\left(-x_{1}^{k},\underset{m+n-1\text{
times}}{\underbrace{0,0,\ldots,0}},-x_{m+n}^{m}\right)$

(cf. [34]).

By using (26), we get

$\mathbb{Y}_{n}^{(\beta+\gamma)}\left(P(\overrightarrow{X_{m}})\right)=\sum\limits_{j=0}^{n}\mathbb{Y}_{n}^{(\beta)}\left(P(\overrightarrow{X_{m}})\right)\mathbb{Y}_{n-j}^{(\gamma)}\left(P(\overrightarrow{X_{m}})\right).$

The polynomials
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$
of order $\alpha,\beta$, which is denoted by
$\mathbb{S}_{n}^{(\alpha,\beta)}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$
, are defined by the following generating function

$\displaystyle
H\left(w,P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}});\alpha,\beta\right)$
$\displaystyle=$
$\displaystyle\frac{\left(\sum\limits_{j=0}^{k}Q_{j}(x_{j})w^{j}\right)^{\alpha}}{\left(1+\sum\limits_{j=1}^{m}P_{j}(x_{j})w^{j}\right)^{\beta}}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\mathbb{S}_{n}^{(\alpha,\beta)}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)w^{n}.$

It is clear that

$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right):=\mathbb{S}_{n}^{(1,1)}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right).$

Combining (4) with (26), we get the following functional equation:

$H\left(w,P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}});1,\beta\right)=\left(\sum\limits_{j=0}^{k}Q_{j}(x_{j})w^{j}\right)F\left(w,P(\overrightarrow{X_{m}});\beta\right).$

Combining (4) with (2), we get

$\sum_{n=0}^{\infty}\mathbb{S}_{n}^{(1,\beta)}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)w^{n}=\left(\sum\limits_{j=0}^{k}Q_{j}(x_{j})w^{j}\right)\sum_{n=0}^{\infty}\mathbb{Y}_{n}^{(\beta)}\left(P(\overrightarrow{X_{m}})\right)w^{n}.$

Therefore

$\sum_{n=0}^{\infty}\mathbb{S}_{n}^{(1,\beta)}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)w^{n}=\sum\limits_{j=0}^{k}\sum_{n=j}^{\infty}Q_{j}(x_{j})\mathbb{Y}_{n}^{(\beta)}\left(P(\overrightarrow{X_{m}})\right)w^{n}.$

Comparing the coefficients of $w^{n}$ on both sides of the above equality, we
arrive at the following theorem:

###### Theorem 17

Let $n,k\in\mathbb{N}$ with $k\leq n$. Then we have

$\mathbb{S}_{n}^{(1,\beta)}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)=\sum\limits_{j=0}^{k}Q_{j}(x_{j})\mathbb{Y}_{n-j}^{(\beta)}\left(P(\overrightarrow{X_{m}})\right).$
(28)

The generalized Catalan polynomials $\mathcal{P}_{v,m}^{h,Q_{1}}(x_{1})$:

$\mathcal{P}_{v,m}^{h,Q_{1}}(x_{1}):=\mathbb{S}_{v}^{(1,h)}\left(-m,\underset{m-1\text{
times}}{\underbrace{0,0,\ldots,0}},-x_{1};1,Q_{1}(x_{1})\right)$

(cf. [13]).

Generating functions for two $2$-variable Simsek polynomials: Replacing
$\left(\sum\limits_{j=0}^{k}Q_{j}(x_{j})w^{j}\right)^{\alpha}$ and
$\left(1+\sum\limits_{j=1}^{m}P_{j}(x_{j})w^{j}\right)^{\beta}$ by
$\left(1+\lambda w\right)^{\alpha_{1}}\left(1+\delta
w^{2}\right)^{\alpha_{2}}$ and $\lambda-1+\lambda^{2}w$ in the equation (4),
repectively, Khan et al. ([20], [21]) defined the following generating
function:

$\frac{\left(1+\lambda w\right)^{\alpha_{1}}\left(1+\delta
w^{2}\right)^{\alpha_{2}}}{\lambda-1+\lambda^{2}w}=\sum_{n=0}^{\infty}\mathbb{S}_{n}^{(\alpha_{1}+\alpha_{2},1)}\left(\lambda-1,\lambda^{2};1,1,\delta\right)w^{n},$

where

$Y_{n}(\alpha_{1},\alpha_{2};\lambda,\delta):=\mathbb{S}_{n}^{(\alpha_{1}+\alpha_{2},1)}\left(\lambda-1,\lambda^{2};1,1,\delta\right)$

denotes the two $2$-variable Simsek polynomials and
$Y_{n}(0,0;\lambda,\delta)$ denotes the Simsek numbers.
$Y_{n}(\alpha;\lambda):=Y_{n}(\alpha_{1},0;\lambda,\delta)$ known as the
Simsek polynomials (cf. for detail, see [24], [25], [26], [38], [39], [42]).

##  5 Recurrence relation

In this section we give recurrence relations of the polynomials
$\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)$ and
$\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$.

We show that the differential of the functions
$F\left(w,P(\overrightarrow{X_{m}})\right)$ and
$G\left(w,P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$ are
given as follows:

$\frac{\partial}{\partial
w}\left\\{F\left(w,P(\overrightarrow{X_{m}})\right)\right\\}=-\sum\limits_{j=1}^{m}jP_{j}(x_{j})w^{j-1}F\left(w,P(\overrightarrow{X_{m}});2\right),$
(29) $\displaystyle\frac{\partial}{\partial
w}\left\\{G\left(w,P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)\right\\}$
$\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{k}lQ_{l}(x_{l})w^{l-1}F\left(w,P(\overrightarrow{X_{m}})\right)$
$\displaystyle-\sum\limits_{l=0}^{k}Q_{l}(x_{l})w^{l}\frac{\partial}{\partial
w}\left\\{F\left(w,P(\overrightarrow{X_{m}})\right)\right\\},$

and

$\displaystyle
F\left(w,P(\overrightarrow{X_{m}});2\right)\frac{\partial}{\partial
w}\left\\{G\left(w,P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)\right\\}$
$\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{k}lQ_{l}(x_{l})w^{l-1}+\sum\limits_{l=1}^{k}\sum\limits_{j=1}^{m}lQ_{l}(x_{l})P_{j}(x_{j})w^{j+l-1}-\sum\limits_{l=0}^{k}\sum\limits_{j=1}^{m}jQ_{l}(x_{l})P_{j}(x_{j})w^{l+j-1}.$

The above differential equation can also be given by

$\displaystyle\frac{\partial}{\partial
w}\left\\{G\left(w,P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)\right\\}$
$\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{k}lQ_{l}(x_{l})w^{l-1}F\left(w,P(\overrightarrow{X_{m}})\right)-$
$\displaystyle-\sum\limits_{l=0}^{k}\sum\limits_{j=1}^{m}jQ_{l}(x_{l})P_{j}(x_{j})w^{l+j-1}F\left(w,P(\overrightarrow{X_{m}});2\right).$

or

$\displaystyle\frac{\partial}{\partial
w}\left\\{G\left(w,P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)\right\\}$
$\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{k}lQ_{l}(x_{l})w^{l-1}F\left(w,P(\overrightarrow{X_{m}})\right)$
$\displaystyle-\sum\limits_{j=1}^{m}jP_{j}(x_{j})w^{j-1}G\left(w,P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)F\left(w,P(\overrightarrow{X_{m}})\right).$

Combining (29) with (1) and (26), we get

$\sum_{n=1}^{\infty}n\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)w^{n-1}=-\sum\limits_{j=1}^{m}jP_{j}(x_{j})\sum_{n=0}^{\infty}\mathbb{Y}_{n}^{(2)}\left(P(\overrightarrow{X_{m}})\right)w^{n+j-1}.$

Therefore

$\sum_{n=0}^{\infty}(n+1)\mathbb{Y}_{n+1}\left(P(\overrightarrow{X_{m}})\right)w^{n}=-\sum\limits_{j=1}^{m}jP_{j}(x_{j})\sum_{n=j-1}^{\infty}\mathbb{Y}_{n-j+1}^{(2)}\left(P(\overrightarrow{X_{m}})\right)w^{n}.$

After the necessary operations in the previous equation, the $w^{n}$
coefficients on both sides of this equation are compared and the following
result is easily obtained:

###### Theorem 18

Let $n\in\mathbb{N}_{0}$. Then we have

$\mathbb{Y}_{n+1}\left(P(\overrightarrow{X_{m}})\right)=-\frac{1}{n+1}\sum\limits_{j=1}^{m}jP_{j}(x_{j})\mathbb{Y}_{n-j+1}^{(2)}\left(P(\overrightarrow{X_{m}})\right).$

Combining (5)with (1) and (2), we get

$\displaystyle\sum_{n=1}^{\infty}n\mathbb{S}_{n}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)w^{n-1}$
$\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{k}lQ_{l}(x_{l})\sum_{n=0}^{\infty}\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)w^{n+l-1}$
$\displaystyle-\sum\limits_{l=0}^{k}Q_{l}(x_{l})\sum_{n=1}^{\infty}n\mathbb{Y}_{n}\left(P(\overrightarrow{X_{m}})\right)w^{n+l-1}.$

After the necessary operations in the previous equation, the $w^{n}$
coefficients on both sides of this equation are compared and the following
result is easily obtained:

###### Theorem 19

Let $n\in\mathbb{N}$. Then we have

$\displaystyle\mathbb{S}_{n+1}\left(P(\overrightarrow{X_{m}});Q(\overrightarrow{X_{k}})\right)$
$\displaystyle=$
$\displaystyle\frac{1}{n+1}\sum\limits_{l=1}^{k}lQ_{l}(x_{l})\mathbb{Y}_{n+1-l}\left(P(\overrightarrow{X_{m}})\right)$
$\displaystyle-\sum\limits_{l=0}^{k}\left(\frac{n+1-l}{n+1}\right)Q_{l}(x_{l})\mathbb{Y}_{n-l+1}\left(P(\overrightarrow{X_{m}})\right).$

Note that if the above operations are applied to the derivative equations from
Eq. (5) to Eq. (5), other different recurrence relations are obtained. We omit
these relations and solution extraction here.

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