On the ω-multiple Meixner polynomials of the first kind

Abstract

In this study, we introduce a new family of discrete multiple orthogonal polynomials, namely ω-multiple Meixner polynomials of the first kind, where ω is a positive real number. Some structural properties of this family, such as the raising operator, Rodrigue’s type formula and an explicit representation are derived. The generating function for ω-multiple Meixner polynomials of the first kind is obtained and by use of this generating function we find several consequences for these polynomials. One of them is a lowering operator which will be helpful for obtaining a difference equation. We give the proof of the lowering operator by use of new technique which is a more elementary proof than the proof of Lee in (J. Approx. Theory 150:132–152, 2008). By combining the lowering operator with the raising operator we obtain the difference equation which has the ω-multiple Meixner polynomials of the first kind as a solution. As a corollary we give a third order difference equation for the ω-multiple Meixner polynomials of the first kind. Also it is shown that, for the special case $\omega = 1$, the obtained results coincide with the existing results for multiple Meixner polynomials of the first kind. In the last section as an illustrative example we consider the special case when $\omega =1/2$ and, for the $1/2$-multiple Meixner polynomials of the first kind, we state the corresponding result for the main theorems.

1 Introduction

Discrete multiple orthogonal polynomials are useful extension of discrete orthogonal polynomials, see [1–13]. The theory of discrete orthogonal polynomials on a linear lattice were extended to such polynomials by Arvesu, Coussement and Van Assche in [2]. Multiple Meixner polynomials are discrete multiple orthogonal polynomials. There are two kinds of multiple Meixner polynomials. In this paper, we concentrate on the multiple Meixner polynomials of the first kind.

Multiple Meixner polynomials of the first kind $M_{\overrightarrow{n}}^{(\beta,\overrightarrow{a})}(x)$, of degree $| \overrightarrow{n}|$, are orthogonal polynomials for the negative binomial distributions when $r>1$. That is,

$$\begin{aligned} \sum_{x=0}^{\infty } M_{\overrightarrow{n}}^{(\beta, \overrightarrow{a})}(x) (-x)_{j} w_{i}^{\beta }(x)=0,\quad j=0,1,\ldots, n_{i}-1, i=1,\ldots,r, \end{aligned}$$

(1)

where $\beta >0$ is the fixed parameter and different values for the $\mathrm{parameter} \overrightarrow{a}=(a_{1},\ldots,a_{r})$, ($a_{i}{\neq }a_{j}$ whenever $i{\neq }j$), where $0< a<1$ with multi-index $\overrightarrow{n}=(n_{1},\ldots,n_{r})$ and $(x)_{k}=x(x+1)(x+2)\cdots (x+k-1)$ is the Pochhammer symbol with $(x)_{0}=1$.

The functions

$$\begin{aligned} w_{i}^{\beta }(x)= \textstyle\begin{cases} \frac{\varGamma (\beta +x)a_{i}^{k}}{\varGamma (\beta )\varGamma (x+1)} & \text{if $x\in \mathbb{R}/ (\{-1,-2,\ldots\}\cup \{-\beta,-\beta -1,\ldots\})$,} \\ 0 & \text{if $x\in \{-1,-2,\ldots\}$,} \end{cases}\displaystyle \end{aligned}$$

are weight functions of the multiple Meixner polynomials of the first kind where $\varGamma (x)$ is the gamma function.

The orthogonality conditions give us a linear system of $|\overrightarrow{n}|=n_{1}+n_{2}+\cdots+n_{r}$ homogeneous equations for the $|\overrightarrow{n}|+1$ unknown coefficients of polynomials which always has a nontrivial solution. If the given multi-index ${\overrightarrow{n}}$ is normal, then the corresponding polynomials will be unique polynomials. For the uniqueness of the polynomials one can use a system called an AT system which was introduced by Nikishin and Sorokin [11].

Definition 1.1

(cf. [11])

A set of continuous real functions $w_{1},w_{2},\ldots,w_{r}$ defined on $[a,b]$ is called an AT system for the index $n\in \mathbb{Z}^{r}_{+}$, $n\neq 0$, if

$$w_{1}(x),xw_{1}(x),\ldots,x^{n_{1}-1}w_{1}(x), \ldots,w_{r}(x),xw_{r}(x),\ldots,x^{n_{r}-1}w_{r}(x) $$

is a Chebyshev system of order $|n|-1$ on $[a,b]$.

In an AT system, all the multi-index ${\overrightarrow{n}}$ are normal. By using the following example it will be easy to show that the weight functions for the multiple Meixner polynomials of the first kind form an AT system.

Example 1.2

(cf. [2])

The functions $w(x)a_{1}^{x},xw(x)a_{1}^{x},\ldots,x^{n_{1}-1}w(x)a_{1}^{x},\ldots, w(x)a_{r}^{x}, xw(x)a_{r}^{x}, \ldots,x^{n_{r}-1}w(x)a_{r}^{x}$, with all the $a_{i}>0,i=1,\ldots,r$, different and $w(x)$ a continuous function which has no zeros on $\mathbb{R}^{+}$, form a Cheybshev system on $\mathbb{R}^{+}$ for every index ${\overrightarrow{n}}\in \mathbb{N}^{r}$.

In [2], Arvesu, Coussement and Van Assche investigated the raising operator and the Rodrigues formula for multiple Meixner polynomials of the first kind. Also, via the Rodrigues formula an explicit formula for these polynomials is obtained by these authors. They investigated these properties for multiple orthogonal polynomials of discrete variables by extending the classical orthogonal polynomials of discrete variables.

Van Assche in [12] obtained a lowering operator for multiple Meixner polynomials of the first kind for the case $r=2$ and then by combining lowering and raising operators he gave the third order difference equation for these polynomials. Later, Lee in [7] obtained a lowering operator for the case r and then by combining lowering and raising operators Lee gave the $(r+1)$th order difference equation for these polynomials.

Ndayiragije and Van Assche in [4] gave generating functions and explicit expressions for the coefficients in the nearest neighbor recurrence relation for multiple Meixner polynomials of the first kind.

In this paper we introduce a new class for discrete multiple polynomials called ω-multiple Meixner polynomials of the first kind. The aim is to obtain some structural properties for these newly introduced polynomials. Firstly we define the orthogonality for the ω-multiple Meixner polynomials of the first kind on the linear lattice by using two important operators, namely the ω-forward and ω-backward difference operators, where

$$\triangle _{\omega }f(x)=f(x+\omega )-f(x) $$

is the ω-forward operator and

$$\nabla _{\omega }f(x)=f(x)-f(x-\omega ) $$

is the ω-backward operator.

In Sect. 2 we obtain some properties of the ω-multiple Meixner polynomials of the first kind, such as the raising operator, Rodrigues’ formula and an explicit form. The generating function for the ω-multiple Meixner polynomials of the first kind is given in Sect. 3 and we obtain some results from the generating function such as connection and addition formulas. Section 4 includes the lowering operator and difference equation for the ω-multiple Meixner polynomials of the first kind. We also show that when $\omega =1$, the results obtained in Sects. 2, 3 and 4, coincide with the existing results for multiple Meixner polynomials of the first kind. One of the main results of this paper is in Sect. 5 where we give some results for the $1 / 2$-multiple Meixner polynomials of the first kind.

2 Orthogonality for ω-multiple Meixner polynomials of the first kind

In this section, we first give the definition of the orthogonality of ω-multiple Meixner polynomials of the first kind.

Definition 2.1

The monic discrete ω-multiple Meixner polynomial of the first kind, corresponding to the multi-index $\overrightarrow{n}=(n_{1},\ldots,n_{r})$, the fixed parameter $\beta >0$ and the $\mathrm{parameter} \overrightarrow{a}=(a_{1},\ldots,a_{r})$, ($a_{i}{ \neq }a_{j}$ whenever $i{\neq }j$), is the unique polynomial of degree $|\overrightarrow{n}|$ which satisfies the orthogonality conditions

$$\begin{aligned} \sum_{x=0}^{\infty } M_{\overrightarrow{n}}^{(\omega; \beta; \overrightarrow{a})}({ \omega }x) ({-\omega }x)_{j,\omega } w_{i}^{( \omega;\beta )}({\omega }x)=0,\quad j=0,1,\ldots, n_{i}-1, i=1,2, \ldots,r, \end{aligned}$$

(2)

where $(a)_{n,\omega }=a(a+\omega )(a+2\omega )\cdots (a+(n-1)\omega )$ is the ω-Pochhammer symbol for $a\in \mathbb{C}$ and $n\in \mathbb{ N}$ and $\omega >0$.

The weight functions for ω-multiple Meixner polynomials of the first kind are defined as

$$\begin{aligned} w_{i}^{(\omega;\beta )}(x)= \textstyle\begin{cases} \frac{\varGamma _{\omega }(\beta +x)a_{i}^{x}}{\varGamma _{\omega }(\beta )\varGamma _{\omega }(x+\omega )} & \text{if $x\in \mathbb{R}/ (\{-1,-2,\ldots\}\cup \{-\beta,-\beta -1,\ldots\})$,} \\ 0 & \text{if $x\in \{-1,-2,\ldots\}$}, \end{cases}\displaystyle \end{aligned}$$

where $0< a_{i}<1$, for $i=,1,2,\ldots,r$, with all the $a_{i}$ different and $\varGamma _{\omega }$ is the ω-gamma function given by

$$\begin{aligned} \varGamma _{\omega }(x)= \int _{0}^{\infty }{t^{x-1}e^{- \frac{t^{\omega }}{\omega }}} \,dt=\omega ^{\large {\frac{x}{\omega }-1}} \varGamma \biggl(\frac{x}{\omega } \biggr). \end{aligned}$$

By Example 1.2 it is easy to conclude that the weight functions form an AT system which implies the uniqueness of such polynomials. When $\omega =1$, the given orthogonality conditions coincide with the orthogonality conditions in (1).

In the rest of this paper the properties of multiple Meixner polynomials of the first kind are extended to ω-multiple Meixner polynomials of the first kind.

Theorem 2.2

Letωbe a positive real number. The raising operator forω-multiple Meixner polynomials of the first kind is given as

$$\begin{aligned} \nabla _{\omega } \bigl[M_{\overrightarrow{n}}^{(\omega;\beta; \overrightarrow{a})}(x)w_{i}^{(\omega;\beta )}(x) \bigr]= \frac{a_{i}^{\omega }-1}{a_{i}^{\omega }( \beta -\omega )}M_{ \overrightarrow{n}+\overrightarrow{e_{i}}}^{(\omega;\beta -\omega; \overrightarrow{a})}(x)w_{i}^{(\omega;\beta -\omega )}(x). \end{aligned}$$

(3)

Proof

By using the product rule for the ω-backward operator

$\nabla _{\omega }[f(x)g(x)]=f(x)\nabla _{\omega }g(x)+g(x-\omega ) \nabla _{\omega }f(x)$, we obtain

$$\begin{aligned} \nabla _{\omega } \bigl[M_{\overrightarrow{n}}^{(\omega;\beta; \overrightarrow{a})}(x)w_{i}^{(\omega;\beta )}(x) \bigr]= \frac{a_{i}^{\omega }-1}{a_{i}^{\omega }( \beta -\omega )}(x)w_{i}^{( \omega;\beta -\omega )}(x)P^{(\omega )}_{\overrightarrow{n}+ \overrightarrow{e_{i}}}. \end{aligned}$$

Using the ω-summation by parts formula,

$$\begin{aligned} \sum_{k=0}^{\infty }\triangle _{\omega } \bigl[f(\omega k) \bigr]g(\omega k)=- \sum_{k=0}^{\infty } \nabla _{\omega } \bigl[g(\omega k) \bigr]f(\omega k) \quad\text{where } g(-\omega )=0, \end{aligned}$$

and the orthogonality conditions,

$$\begin{aligned} \sum_{k=0}^{\infty } P^{(\omega )}_{\overrightarrow{n}+ \overrightarrow{e_{i}}}({ \omega }x) ({-\omega }x)_{j,\omega } w_{i}^{( \omega;\beta -\omega )}({\omega }x)=0, \end{aligned}$$

we find the result with $P^{(\omega )}_{\overrightarrow{n}+\overrightarrow{e_{i}}}=M_{ \overrightarrow{n}+\overrightarrow{e_{i}}}^{(\omega;\beta -\omega; \overrightarrow{a})}$, which was guaranteed from the uniqueness of the orthogonal polynomials. □

Theorem 2.3

Letωbe a positive real number. The Rodrigues formula forω-multiple Meixner polynomials of the first kind are introduced by

$$\begin{aligned} M_{\overrightarrow{n}}^{(\omega;\beta;\overrightarrow{a})}(x)= {}& ( \beta )_{ \vert \overrightarrow{n} \vert ,\omega } \Biggl[\prod_{k=1}^{r} \biggl({ \frac{a_{k}^{\omega }}{a_{k}^{\omega }-1}} \biggr)^{n_{k}} \frac{\varGamma _{\omega }(\beta )\varGamma _{\omega }(x+\omega )}{\varGamma _{\omega }(\beta +x)} \Biggr] \\ &{}\times \prod_{i=1}^{r}\frac{1}{a^{x}_{i}}\nabla _{\omega }^{n_{i}} \biggl[ \frac{\varGamma _{\omega }(\beta +x+ \vert \overrightarrow{n} \vert \omega ){a^{x}_{i}}}{\varGamma _{\omega }(x+\omega )\varGamma _{\omega }(\beta + \vert \overrightarrow{n} \vert \omega )} \biggr]. \end{aligned}$$

(4)

Proof

Replacing $\overrightarrow{n}$ by $\overrightarrow{n}-\overrightarrow{e_{i}}$ and β by $\beta +\omega$ in the raising operator formula, we obtain

$$\begin{aligned} w_{i}^{(\omega;\beta )}(x)M_{\overrightarrow{n}}^{(\omega;\beta; \overrightarrow{a})}(x)= \frac{{\beta }a_{i}^{\omega }}{a_{i}^{\omega }-1} \nabla _{\omega } \bigl[w_{i}^{(\omega;\beta +\omega )}(x)M_{ \overrightarrow{n}-\overrightarrow{e_{i}}}^{(\omega;\beta +\omega; \overrightarrow{a})}(x) \bigr]. \end{aligned}$$

Then for $r=2$ the multi-index will be $\overrightarrow{n}=(n_{1},n_{2})$.

For $i=1$ we have

$$\begin{aligned} w_{1}^{(\omega;\beta )}(x)M_{n_{1},n_{2}}^{(\omega;\beta;a_{1},a_{2})}(x)= \frac{{\beta }a_{1}^{\omega }}{a_{1}^{\omega }-1}\nabla _{\omega } \bigl[w_{1}^{( \omega;\beta +\omega )}(x)M_{n_{1}-1,n_{2}}^{(\omega;\beta +\omega;a_{1},a_{2})}(x) \bigr], \end{aligned}$$

and iterating it $n_{1}$ times we get

$$\begin{aligned} w_{1}^{((\omega;\beta )}(x)M_{n_{1},n_{2}}^{(\omega;\beta;a_{1},a_{2})}(x)= {}& (\beta )_{n_{1},\omega } \biggl( \frac{a_{1}^{\omega }}{a_{1}^{\omega }-1} \biggr)^{n_{1}} \\ & {}\times\nabla _{\omega }^{n_{1}} \bigl[w_{1}^{(\omega;\beta +n_{1}\omega )}(x)M_{0,n_{2}}^{( \omega;\beta +n_{1}\omega;a_{1},a_{2})}(x) \bigr]. \end{aligned}$$

For $i=2$ we have

$$\begin{aligned} w_{2}^{(\omega;\beta )}(x)M_{n_{1},n_{2}}^{(\omega;\beta;a_{1},a_{2})}(x)= \frac{{\beta }a_{2}^{\omega }}{a_{2}^{\omega }-1}\nabla _{\omega } \bigl[w_{2}^{( \omega;\beta +\omega )}(x)M_{n_{1},n_{2}-1}^{(\omega;\beta +\omega;a_{1},a_{2})}(x) \bigr], \end{aligned}$$

and iterating it $n_{2}$ times we get

$$\begin{aligned} w_{2}^{(\omega;\beta )}(x)M_{n_{1},n_{2}}^{(\omega;\beta;a_{1},a_{2})}(x)= {} & (\beta )_{n_{2},\omega } \biggl( \frac{a_{2}^{\omega }}{a_{2}^{\omega }-1} \biggr)^{n_{2}} \\ & {}\times\nabla _{\omega }^{n_{2}} \bigl[w_{1}^{(\omega;\beta +n_{2}\omega )}(x)M_{n_{1},0}^{( \omega;\beta +n_{2}\omega;a_{1},a_{2})}(x) \bigr]. \end{aligned}$$

By combining these two equations, we obtain the expression for $M_{n_{1},n_{2}}^{(\omega;\beta;a_{1},a_{2})}(x)$ and if we continue the iteration for r, we derive the Rodrigues formula for ω-multiple Meixner polynomials of the first kind. □

The explicit form can easily be obtained from the Rodrigues formula using the Leibniz rule for the ω-backward operator,

$$\begin{aligned} \nabla _{\omega }^{n}f(x)=\sum_{i=1}^{n}( -1) ^{i} \binom{n}{i} f(x-i \omega ). \end{aligned}$$

Theorem 2.4

Letωbe a positive real number. The explicit form forω-multiple Meixner polynomials of the first kind is given by

$$\begin{aligned} M_{\overrightarrow{n}}^{(\omega;\beta;\overrightarrow{a})}(x)= {} & \sum _{k_{1}=0}^{n_{1}}\sum_{k_{2}=0}^{n_{2}}\cdots \sum_{k_{r}=0}^{n_{r}} \binom{n_{1}}{k_{1}} \binom{n_{2}}{k_{2}}\cdots \binom{n_{r}}{k_{r}}(-x)_{| \overrightarrow{k}|,\omega } \\ & {}\times\prod_{j=1}^{r} \biggl[ \frac{({a_{j}^{\omega }})^{n_{j}-k_{j}}}{(a_{j}^{\omega }-1)^{n_{j}}}( \beta +x)_{|\overrightarrow{n}|-|\overrightarrow{k}|,\omega } \biggr]. \end{aligned}$$

(5)

Corollary 2.5

The special cases of Theorem 2.2, Theorem 2.3and Theorem 2.4when$\omega =1$are easily reduced to results obtained in [2] and [4]. For instance, when$\omega =1$in Theorem 2.2, our result coincides with the raising operator for multiple Meixner polynomials of the first kind,

$$\begin{aligned} \nabla \bigl[M_{\overrightarrow{n}}^{(\beta,\overrightarrow{a})}(x)w_{i}^{\beta }(x) \bigr]=\frac{a_{i}-1}{a_{i}( \beta -1)}M_{\overrightarrow{n}+ \overrightarrow{e_{i}}}^{(\beta -1,\overrightarrow{a})}(x)w_{i}^{ \beta -1}(x), \end{aligned}$$

which is exactly the same formula as in [2, equation (4.6), p. 33].

When$\omega =1$in Theorem 2.3, we find the Rodrigues formula for multiple Meixner polynomials of the first kind,

$$\begin{aligned} M_{\overrightarrow{n}}^{(\beta,\overrightarrow{a})}(x)={}& (\beta )_{ \vert \overrightarrow{n} \vert } \Biggl[\prod_{k=1}^{r} \biggl({ \frac{a_{k}}{a_{k}-1}} \biggr)^{n_{k}} \frac{\varGamma (\beta )\varGamma (x+1)}{\varGamma (\beta +x)} \Biggr] \\ & {}\times \prod_{i=1}^{r} \frac{1}{a^{x}_{i}}\nabla ^{n_{i}} \biggl[ \frac{\varGamma (\beta +x+ \vert \overrightarrow{n} \vert ){a^{x}_{i}}}{\varGamma (x+1)\varGamma (\beta + \vert \overrightarrow{n} \vert )} \biggr], \end{aligned}$$

which coincides with the formula in [2, equation (4.7), p. 33].

When$\omega =1$in Theorem 2.4, we have an explicit form for multiple Meixner polynomials of the first kind,

$$\begin{aligned} M_{\overrightarrow{n}}^{(\beta,\overrightarrow{a})}(x)={}& \sum _{k_{1}=0}^{n_{1}} \sum_{k_{2}=0}^{n_{2}}\cdots \sum_{k_{r}=0}^{n_{r}}\binom{n_{1}}{k_{1}} \binom{n_{2}}{k_{2}}\cdots \binom{n_{r}}{k_{r}}(-x)_{|\overrightarrow{k}|} \\ & {}\times\prod_{j=1}^{r} \biggl[ \frac{(a_{k})^{n_{j}-k_{j}}}{(a_{k}-1)^{n_{j}}}(\beta +x)_{| \overrightarrow{n}|-|\overrightarrow{k}|} \biggr], \end{aligned}$$

which coincides with the formula in [4, equation (3), p. 3]

3 Generating function for ω-multiple Meixner polynomials of the first kind

ω-Multiple Meixner polynomials of the first kind have a multivariate generating function with r variables. The following lemma will be useful for the proof of the theorem for the generating function.

The relation between Pochhammer symbol and ω-Pochhammer symbol is given as follows:

$$\begin{aligned} (a)_{n,\omega }=\omega ^{n} \biggl(\frac{a}{\omega } \biggr)_{n}. \end{aligned}$$

(6)

Lemma 3.1

(cf. [4, Lemma 1, p. 4])

The generating function for the multinomial coefficients is given as follows:

$$\begin{aligned} \sum_{m_{1}=0}^{\infty }\cdots \sum _{m_{r}=0}^{\infty } \frac{(-x)_{|\overrightarrow{m}|}}{m_{1}!\cdots m_{r}!}t_{1}^{m_{1}}\cdots t_{r}^{m_{r}}=(1-t_{1}\cdots -t_{r})^{x}. \end{aligned}$$

This series converges absolutely and uniformly for$|t_{1}|+\cdots+|t_{r}|<1$when$x\notin \mathbb{N}$and contains a finite number of terms if$x\in \mathbb{N}$.

Theorem 3.2

Letωbe a positive real number. The generating function for theω-multiple Meixner polynomials of the first kind is

$$\begin{aligned} \sum_{n_{1}=0}^{\infty }\sum _{n_{2}=0}^{\infty }\cdots \sum_{n_{r}=0}^{ \infty }M_{\overrightarrow{n}}^{(\omega;\beta;\overrightarrow{a})}(x) \frac{t_{1}^{n_{1}}\cdots t_{r}^{n_{r}}}{n_{1}!\cdots n_{r}!}={}& \biggl(1- \frac{\omega t_{1}}{a_{1}^{\omega }-1}-\cdots- \frac{\omega t_{r}}{a_{r}^{\omega }-1} \biggr)^{\frac{x}{\omega }} \\ & {}\times \biggl(1-\frac{\omega t_{1} a_{1}^{\omega }}{a_{1}^{\omega }-1}-\cdots- \frac{\omega t_{r} a_{r}^{\omega }}{a_{r}^{\omega }-1} \biggr)^{\frac{-(\beta +x)}{\omega }}. \end{aligned}$$

(7)

Proof

Replacing $M_{\overrightarrow{n}}^{(\omega;\beta;\overrightarrow{a})}(x)$ with the explicit form in left hand side of (7) we obtain

$$\begin{aligned} &\sum_{n_{1}=0}^{\infty }\cdots \sum _{n_{r}=0}^{\infty }\sum_{k_{1}=0}^{n_{1}}\cdots \sum_{k_{r}=0}^{n_{r}} \binom{n_{1}}{k_{1}} \binom{n_{2}}{k_{2}}\cdots \binom{n_{r}}{k_{r}}(-x)_{|\overrightarrow{k}|,\omega } \\ & \quad{}\times\prod_{j=1}^{r} \biggl[ \frac{({a_{k}^{\omega }})^{n_{j}-k_{j}}}{(a_{k}^{\omega }-1)^{n_{j}}}( \beta +x)_{|\overrightarrow{n}|-|\overrightarrow{k}|,\omega } \biggr] \frac{t_{1}^{n_{1}}\cdots t_{r}^{n_{r}}}{n_{1}!\cdots n_{r}!}. \end{aligned}$$

Changing the order of the summation gives

$$\begin{aligned} &\sum_{k_{1}=0}^{\infty }\cdots \sum _{k_{r}=0}^{\infty }\sum_{n_{1}=k_{1}}^{ \infty }\cdots \sum_{n_{r}=k_{r}}^{\infty } \frac{(-x)_{|\overrightarrow{k}|,\omega }}{k_{1}!(n_{1}-k_{1})!\cdots k_{r}!(n_{r}-k_{r})!} \frac{a_{1}^{\omega (n_{1}-k_{1})}\cdots a_{r}^{\omega (n_{r}-k_{r})}}{(a_{1}^{\omega }-1)^{n_{1}}\cdots (a_{r}^{\omega }-1)^{n_{r}}} \\ & \quad{}\times x t_{1}^{n_{1}}\cdots t_{r}^{n_{r}}( \beta +x)_{|\overrightarrow{n}|-| \overrightarrow{k}|,\omega }. \end{aligned}$$

By setting $m_{i}=n_{i}-k_{i}$ and putting the factors in $m_{i}$ and $k_{i}$ together we obtain

$$\begin{aligned} &\sum_{k_{1}=0}^{\infty }\cdots \sum _{k_{r}=0}^{\infty } \frac{(-x)_{|\overrightarrow{k}|,\omega }}{k_{1}!\cdots k_{r}!} \biggl( \frac{t_{1}}{a_{1}^{\omega }-1} \biggr)^{k_{1}}\cdots \biggl( \frac{t_{r}}{a_{r}^{\omega }-1} \biggr)^{k_{r}} \\ &\quad {}\times\sum_{m_{1}=0}^{\infty }\cdots \sum _{m_{r}=0}^{\infty } \frac{(\beta +x)_{|\overrightarrow{m}|,\omega }}{m_{1}!\cdots m_{r}!} \biggl( \frac{t_{1}a_{1}^{\omega }}{a_{1}^{\omega }-1} \biggr)^{m_{1}}\cdots \biggl(\frac{t_{r}a_{r}^{\omega }}{a_{r}^{\omega }-1} \biggr)^{m_{r}}. \end{aligned}$$

Using the relation between the Pochhammer symbol and the ω-Pochhammer symbol (6), the above equation becomes

$$\begin{aligned} &\sum_{k_{1}=0}^{\infty }\cdots \sum _{k_{r}=0}^{\infty } \frac{ (\frac{-x}{\omega } )_{|\overrightarrow{k}|}}{k_{1}!\cdots k_{r}!} \biggl( \frac{\omega t_{1}}{a_{1}^{\omega }-1} \biggr)^{k_{1}}\cdots \biggl( \frac{\omega t_{r}}{a_{r}^{\omega }-1} \biggr)^{k_{r}} \\ & \quad{}\times\sum_{m_{1}=0}^{\infty }\cdots \sum _{m_{r}=0}^{\infty } \frac{ (\frac{\beta +x}{\omega } )_{|\overrightarrow{m}|}}{m_{1}!\cdots m_{r}!} \biggl( \frac{\omega t_{1}a_{1}^{\omega }}{a_{1}^{\omega }-1} \biggr)^{m_{1}}\cdots \biggl(\frac{\omega t_{r}a_{r}^{\omega }}{a_{r}^{\omega }-1} \biggr)^{m_{r}}. \end{aligned}$$

Now using Lemma 3.1, we obtain the generating function for ω-multiple Meixner polynomials of the first kind, which is the desired result. □

Corollary 3.3

When$\omega =1$, the special case of Theorem 3.2reduces to the result of [4]. For instance, when$\omega =1$in Theorem 3.2, we have a generating function for multiple Meixner polynomials of the first kind,

$$\begin{aligned} \sum_{n_{1}=0}^{\infty }\sum _{n_{2}=0}^{\infty }\cdots \sum_{n_{r}=0}^{ \infty }M_{\overrightarrow{n}}^{(\beta,\overrightarrow{a})}(x) \frac{t_{1}^{n_{1}}\cdots t_{r}^{n_{r}}}{n_{1}!\cdots n_{r}!} ={}& \biggl(1- \frac{t_{1}}{a_{1}-1}-\cdots-\frac{t_{r}}{a_{r}-1} \biggr)^{x} \\ & {}\times \biggl(1-\frac{t_{1} a_{1}}{a_{1}-1}-\cdots- \frac{t_{r} a_{r}}{a_{r}-1} \biggr)^{-(\beta +x)}, \end{aligned}$$

which coincides with the formula in [4, equation (7), p. 4].

The generating function will be used to establish the connection formula and the addition formula for ω-multiple Meixner polynomials of the first kind.

Theorem 3.4

Letωbe a positive real number. ω-Multiple Meixner polynomials of the first kind$M_{\overrightarrow{n}}^{(\omega;\beta; \overrightarrow{a})}(x)$and$M_{\overrightarrow{n}-\overrightarrow{k}}^{(\omega;\beta +\gamma; \overrightarrow{a})}(x)$satisfy the following connection formula:

$$\begin{aligned} M_{\overrightarrow{n}}^{(\omega;\beta;\overrightarrow{a})}(x)= {}& \sum _{k_{1}=0}^{n_{1}}\cdots \sum_{k_{r}=0}^{n_{r}} \binom{n_{1}}{k_{1}}\cdots \binom{n_{r}}{k_{r}}(-\gamma )_{| \overrightarrow{k}|,\omega } \\ & {}\times \biggl(\frac{a_{1}^{\omega }}{a_{1}^{\omega -1}} \biggr)^{k_{1}}\cdots \biggl( \frac{a_{r}^{\omega }}{a_{r}^{\omega -1}} \biggr)^{k_{r}}M_{ \overrightarrow{n}-\overrightarrow{k}}^{(\omega;\beta +\gamma; \overrightarrow{a})}(x). \end{aligned}$$

(8)

Proof

Replacing β by $\beta -\gamma +\gamma$ in the generating function (7) we obtain

$$\begin{aligned} \sum_{n_{1}=0}^{\infty }\cdots \sum _{n_{r}=0}^{\infty } M_{ \overrightarrow{n}}^{(\omega;\beta;\overrightarrow{a})}(x) \frac{t_{1}^{n_{1}}\cdots t_{r}^{n_{r}}}{n_{1}!\cdots n_{r}!}= {}& \biggl(1- \frac{\omega t_{1}}{a_{1}^{\omega }-1}-\cdots- \frac{\omega t_{r}}{a_{r}^{\omega }-1} \biggr)^{\frac{x}{\omega }} \\ & {}\times \biggl(1-\frac{\omega t_{1} a_{1}^{\omega }}{a_{1}^{\omega }-1}-\cdots- \frac{\omega t_{r} a_{r}^{\omega }}{a_{r}^{\omega }-1} \biggr)^{\frac{-(\beta +\gamma +x)}{\omega }} \\ & {}\times \biggl(1-\frac{\omega t_{1} a_{1}^{\omega }}{a_{1}^{\omega }-1}-\cdots- \frac{\omega t_{r} a_{r}^{\omega }}{a_{r}^{\omega }-1} \biggr)^{\frac{\gamma }{\omega }}. \end{aligned}$$

From the generating function (7) and Lemma 3.1, the above equation gets the following form:

$$\begin{aligned} \sum_{n_{1}=0}^{\infty }\cdots \sum _{n_{r}=0}^{\infty }M_{ \overrightarrow{n}}^{(\omega;\beta;\overrightarrow{a})}(x) \frac{t_{1}^{n_{1}}\cdots t_{r}^{n_{r}}}{n_{1}!\cdots n_{r}!} ={}& \sum_{n_{1}=0}^{ \infty }\cdots \sum_{n_{r}=0}^{\infty }M_{\overrightarrow{n}}^{(\beta + \gamma,\overrightarrow{a})}(x) \frac{t_{1}^{n_{1}}\cdots t_{r}^{n_{r}}}{n_{1}!\cdots n_{r}!} \\ & {}\times\sum_{k_{1}=0}^{\infty }\cdots \sum _{k_{r}=0}^{\infty } \frac{ (\frac{-\gamma }{\omega } )_{|\overrightarrow{k}|}}{k_{1}!\cdots k_{r}!} \biggl( \frac{\omega a_{1}^{\omega }}{a_{1}^{\omega }-1} \biggr)^{k_{1}}\cdots \biggl(\frac{\omega a_{r}^{\omega }}{a_{r}^{\omega }-1} \biggr)^{k_{r}} \\ & {}\times t_{1}^{k_{1}}\cdots t_{r}^{k_{r}}. \end{aligned}$$

Changing the order of summations and using the relation between the Pochhammer symbol and the ω-Pochhammer symbol (6) we get

$$\begin{aligned} \sum_{n_{1}=0}^{\infty }\cdots \sum _{n_{r}=0}^{\infty }M_{ \overrightarrow{n}}^{(\omega;\beta;\overrightarrow{a})}(x) \frac{t_{1}^{n_{1}}\cdots t_{r}^{n_{r}}}{n_{1}!\cdots n_{r}!}={}&\sum_{n_{1}=0}^{ \infty }\cdots \sum _{n_{r}=0}^{\infty }\sum_{k_{1}=0}^{n_{1}}\cdots \sum_{k_{r}=0}^{n_{r}} \binom{n_{1}}{k_{1}}\cdots \binom{n_{r}}{k_{r}} \\ & {}\times M_{\overrightarrow{n}-\overrightarrow{k}}^{(\omega;\beta +\gamma; \overrightarrow{a})}(x) (-\gamma )_{|\overrightarrow{k}|,\omega } \biggl(\frac{a_{1}^{\omega }}{a_{1}^{\omega -1}} \biggr)^{k_{1}}\cdots \\ & {}\times \biggl(\frac{a_{r}^{\omega }}{a_{r}^{\omega -1}} \biggr)^{k_{r}} \frac{t_{1}^{n_{1}}\cdots t_{r}^{n_{r}}}{n_{1}!\cdots n_{r}!}. \end{aligned}$$

Finally, comparing the coefficients of $\frac{t_{1}^{n_{1}}\cdots t_{r}^{n_{r}}}{n_{1}!\cdots n_{r}!}$, appearing on both sides of the above equation, we obtain the desired result. □

Theorem 3.5

Letωbe a positive real number. The addition formula forω-multiple Meixner polynomials of the first kind is given by

$$\begin{aligned} M_{\overrightarrow{n}}^{(\omega;\beta +\gamma;\overrightarrow{a})}(x+y)= \sum_{k_{1}=0}^{n_{1}}\cdots \sum_{k_{r}=0}^{n_{r}}\binom{n_{1}}{k_{1}}\cdots \binom{n_{r}}{k_{r}}M_{\overrightarrow{n}-\overrightarrow{k}}^{( \omega;\beta;\overrightarrow{a})}(x)M_{\overrightarrow{k}}^{( \omega;\gamma;\overrightarrow{a})}(y). \end{aligned}$$

(9)

Proof

Changing x with $x+y$ and β with ${\beta +\gamma }$ in the generating function (7), together with changing the order of summations we obtain

$$\begin{aligned} \sum_{n_{1}=0}^{\infty }\cdots \sum _{n_{r}=0}^{\infty }M_{ \overrightarrow{n}}^{(\beta +\gamma,\overrightarrow{a})}(x+y) \frac{t_{1}^{n_{1}}\cdots t_{r}^{n_{r}}}{n_{1}!\cdots n_{r}!}={}& \sum_{n_{1}=0}^{ \infty }\cdots \sum_{n_{r}=0}^{\infty }\sum _{k_{1}=0}^{n_{1}}\cdots \sum_{k_{r}=0}^{n_{r}} \frac{t_{1}^{k_{1}}\cdots t_{r}^{k_{r}}}{k_{1}!\cdots k_{r}!}M_{ \overrightarrow{k}}^{(\gamma,\overrightarrow{a})}(y) \\ &{}\times \frac{t_{1}^{n_{1}-k_{1}}\cdots t_{r}^{n_{r}-k_{r}}}{(n_{1}-k_{1})!\cdots (n_{r}-k_{r})!}M_{ \overrightarrow{n}-\overrightarrow{k}}^{(\beta,\overrightarrow{a})}(x). \end{aligned}$$

Finally, comparing the coefficients of $\frac{t_{1}^{n_{1}}\cdots t_{r}^{n_{r}}}{n_{1}!\cdots n_{r}!}$, appearing on both sides of the above equation we get the result. □

Note that in the case when $\omega =1$, the connection and addition formula for multiple Meixner polynomials are also new. So, here we have new relations for multiple Meixner polynomials of the first kind, which are given below.

Corollary 3.6

Multiple Meixner polynomials of the first kind satisfy the following connection and addition formulas, respectively:

$$\begin{aligned} &M_{\overrightarrow{n}}^{(\beta,\overrightarrow{a})}(x)=\sum_{k_{1}=0}^{n_{1}}\cdots \sum_{k_{r}=0}^{n_{r}}\binom{n_{1}}{k_{1}}\cdots \binom{n_{r}}{k_{r}}a_{1}^{k_{1}} \cdots a_{r}^{k_{r}}(- \gamma )_{|\overrightarrow{k}|}M_{ \overrightarrow{n}-\overrightarrow{k}}^{(\beta +\gamma, \overrightarrow{a})}(x), \\ &M_{\overrightarrow{n}}^{(\beta +\gamma,\overrightarrow{a})}(x+y)= \sum_{k_{1}=0}^{n_{1}}\cdots \sum_{k_{r}=0}^{n_{r}}\binom{n_{1}}{k_{1}}\cdots \binom{n_{r}}{k_{r}}M_{\overrightarrow{n}-\overrightarrow{k}}^{( \beta,\overrightarrow{a})}(x)M_{\overrightarrow{k}}^{(\gamma, \overrightarrow{a})}(y). \end{aligned}$$

4 Difference equation for ω-multiple Meixner polynomials of the first kind

In this section, the aim is to introduce the difference equation for the ω-multiple Meixner polynomials of the first kind by combining the lowering and raising operators.

Theorem 4.1

Letωbe a positive real number. The raising operator forω-multiple Meixner polynomials of the first kind is given as

$$\begin{aligned} L_{a_{i}}^{(\beta )} \bigl[M_{\overrightarrow{n}}^{(\omega;\beta; \overrightarrow{a})} \bigr]=-M_{\overrightarrow{n}+e_{i}}^{(\omega;\beta - \omega;\overrightarrow{a})} \end{aligned}$$

(10)

where$L_{a_{i}}^{(\beta )}[\cdot]$is defined by

$$\begin{aligned} L_{a_{i}}^{(\beta )}[y]=\frac{x}{1-a_{i}^{\omega }}\nabla _{\omega }y- \biggl[\frac{a_{i}^{\omega }(\beta -\omega )}{1-a_{i}^{\omega }}-x \biggr]y. \end{aligned}$$

Proof

The proof follows directly from the Rodrigues formula (4) for the ω-multiple Meixner polynomials of the first kind. □

Theorem 4.2

Letωbe a positive real number. The lowering operator forω-multiple Meixner polynomials of the first kind is

$$\begin{aligned} \Delta _{\omega }M_{\overrightarrow{n}}^{(\omega;\beta; \overrightarrow{a})}=\sum _{i=1}^{r}\omega n_{i}M_{\overrightarrow{n}-e_{i}}^{( \omega;\beta +\omega;\overrightarrow{a})}. \end{aligned}$$

(11)

In particular, for$r=2$

$$\begin{aligned} \Delta _{\omega }M_{n_{1},n_{2}}^{(\omega;\beta; a_{1},a_{2})}= \omega n_{1}M_{n_{1}-1,n_{2}}^{(\omega;\beta +\omega; a_{1},a_{2})}+ \omega n_{2} M_{n_{1},n_{2}-1}^{(\omega;\beta +\omega; a_{1},a_{2})}. \end{aligned}$$

Proof

Changing β for $\beta -\omega$ and applying the operator $\Delta _{\omega }$ to both sides of the generating function (7) for the case $r=2$ we obtain

$$\begin{aligned} \sum_{n_{1}=0}^{\infty }\sum _{n_{2}=0}^{\infty }\Delta _{\omega }M_{n_{1},n_{2}}^{( \omega;\beta -\omega;a_{1},a_{2})} \frac{t_{1}^{n_{1}}t_{2}^{n_{2}}}{n_{1}!n_{2}!}= {}& (\omega t_{1}+ \omega t_{2}) \biggl(1-\frac{\omega t_{1}}{a_{1}^{\omega }-1}- \frac{\omega t_{2}}{a_{2}^{\omega }-1} \biggr)^{\frac{x}{\omega }} \\ & {}\times \biggl(1-\frac{\omega t_{1} a_{1}^{\omega }}{a_{1}^{\omega }-1}- \frac{\omega t_{2} a_{2}^{\omega }}{a_{2}^{\omega }-1} \biggr)^{\frac{-(\beta +x)}{\omega }}. \end{aligned}$$

By use of the generating function (7) the above equation gets the following form:

$$\begin{aligned} \sum_{n_{1}=0}^{\infty }\sum _{n_{2}=0}^{\infty }\Delta _{\omega }M_{n_{1},n_{2}}^{( \omega;\beta -\omega;a_{1},a_{2})} \frac{t_{1}^{n_{1}}t_{2}^{n_{2}}}{n_{1}!n_{2}!} ={}& \omega \sum_{n_{1}=0}^{ \infty } \sum_{n_{2}=0}^{\infty }M_{n_{1},n_{2}}^{(\beta,a_{1},a_{2})} \frac{t_{1}^{n_{1}+1}t_{2}^{n_{2}}}{n_{1}!n_{2}!} \\ & {}+ \omega \sum_{n_{1}=0}^{\infty }\sum _{n_{2}=0}^{\infty }M_{n_{1},n_{2}}^{( \beta,a_{1},a_{2})} \frac{t_{1}^{n_{1}}t_{2}^{n_{2}+1}}{n_{1}!n_{2}!}. \end{aligned}$$

Changing β for $\beta +\omega$ and then replacing $n_{1}$ with $n_{1}-1$ and $n_{2}$ with $n_{2}-1$ in the right side of equation we obtain

$$\begin{aligned} &\sum_{n_{1}=0}^{\infty }\sum _{n_{2}=0}^{\infty } \Delta _{\omega }M_{n_{1},n_{2}}^{( \omega;\beta;a_{1},a_{2})} \frac{t_{1}^{n_{1}}t_{2}^{n_{2}}}{n_{1}!n_{2}!}\\ &\quad = \sum_{n_{1}=0}^{ \infty }\sum _{n_{2}=0}^{\infty } \bigl[\omega n_{1}M_{n_{1}-1,n_{2}}^{( \omega;\beta +\omega;a_{1},a_{2})}+\omega n_{2} M_{n_{1},n_{2}-1}^{(\omega;\beta +\omega;a_{1},a_{2})} \bigr] \frac{t_{1}^{n_{1}}t_{2}^{n_{2}}}{n_{1}!n_{2}!}. \end{aligned}$$

Finally, comparing the coefficients of $\frac{t_{1}^{n_{1}}t_{2}^{n_{2}}}{n_{1}!n_{2}!}$, appearing on both sides of the above equation the desired result is obtained for the case $r=2$. □

Remark 4.1

In the case $\omega =1$, the lowering operator for the ω-multiple Meixner polynomials of the first kind coincide with the lowering operator for the multiple Meixner polynomials of the first kind which is given in [7]. For the proof of the ω type we consider a different approach, where the generating function plays an important role and the proof becomes simpler than the corresponding proof when $\omega =1$ [7, Theorem 2.4, p. 138].

Theorem 4.3

Letωbe a positive real number. The difference equation for theω-multiple Meixner polynomials${{ M_{\overrightarrow{n}}^{(\omega;\beta;\overrightarrow{a})}(x)}_{| \overrightarrow{n}|=0}^{\infty }}$is given by

$$\begin{aligned} &L_{a_{1}}^{(\beta +2\omega -r\omega )} L_{a_{2}}^{(\beta +3\omega -r \omega )} \cdots L_{a_{r}}^{(\beta +\omega )}\Delta _{\omega } M_{ \overrightarrow{n}}^{(\omega;\beta;\overrightarrow{a})}(x) \\ &\quad{}+ \sum_{i=1}^{r}\omega n_{i}L_{a_{1}}^{(\beta +2\omega -r\omega )} \cdots L_{a_{i-1}}^{(\beta +i\omega -r\omega )}L_{a_{i+1}}^{(\beta +(i+1) \omega -r\omega )} \cdots \\ &\quad {}\times L_{a_{r}}^{(\beta )}L_{a_{i}}^{(\beta +\omega )}M_{ \overrightarrow{n}}^{(\omega;\beta;\overrightarrow{a})}(x)=0. \end{aligned}$$

(12)

Proof

Since $L_{a_{k}}^{(n)}L_{a_{m}}^{(n+\omega )}=L_{a_{m}}^{(n)}L_{a_{k}}^{(n+ \omega )}$ for $n, a_{k}, a_{m} \in \mathbb{R}$, we obtain

$$\begin{aligned} L_{a_{1}}^{(\beta +2\omega -r\omega )}L_{a_{2}}^{(\beta +3\omega -r \omega )} \cdots L_{a_{r}}^{(\beta +\omega )}={}& L_{a_{1}}^{(\beta +2 \omega -r\omega )}\cdots L_{a_{i-1}}^{(\beta +i\omega -r\omega )} \\ & {}\times L_{a_{i+1}}^{(\beta +(i+1)\omega -r\omega )}\cdots L_{a_{r}}^{( \beta )}L_{a_{i}}^{(\beta +\omega )}, \end{aligned}$$

for $i=1,\ldots, r$.

Now applying $L_{a_{1}}^{(\beta +2\omega -r\omega )}L_{a_{2}}^{(\beta +3\omega -r \omega )}\cdots L_{a_{r}}^{(\beta +\omega )}$ to the lowering operator (11) and using the raising operator (10), we get the result. □

Theorem 4.4

Letωbe a positive real number. The third order difference equation for theω-multiple Meixner polynomials${{ M_{\overrightarrow{n}}^{(\omega;\beta;\overrightarrow{a})}(x)}_{n_{1}+n_{2}=0}^{ \infty }}$of the first kind is given as

$$\begin{aligned} &x(x-\omega )\nabla _{\omega }^{2}\Delta _{\omega }y +x \bigl[\beta \bigl(a_{1}^{\omega }+a_{2}^{\omega } \bigr)+(x-\omega ) \bigl(a_{1}^{\omega }+a_{2}^{\omega }-2 \bigr) \bigr] \nabla _{\omega }\Delta _{\omega }y \\ &\quad{} + \bigl[a_{1}^{\omega }\beta -x \bigl(1-a_{1}^{\omega } \bigr) (a_{2}\omega \beta -x \omega \bigl(1-a_{1}^{\omega } \bigr) \bigl(1-a_{2}^{\omega } \bigr)-a_{1}^{\omega }a_{2}^{\omega } \beta \omega \bigr]\Delta _{\omega }y \\ & \quad{}+\bigl[n_{1} \bigl(1-a_{1}^{\omega } \bigr)+n_{2} \bigl(1-a_{2}^{\omega } \bigr) \bigr] \omega x \nabla _{\omega }y \\ &\quad{} +\omega (\beta -\omega ) \bigl[ \bigl[n_{1}a_{2}^{\omega }+n_{2}a_{1}^{\omega }-a_{1}^{\omega }a_{2}^{\omega }(n_{1}+n_{2}) \bigr] \\ &\quad{}- \bigl(1-a_{1}^{\omega } \bigr) \bigl(1-a_{2}^{\omega } \bigr) (n_{1}+n_{2})x \bigr]y=0. \end{aligned}$$

(13)

Proof

Considering the case for $r=2$ in Theorem 4.3, we have

$$\begin{aligned} L_{a_{1}}^{(\beta )}L_{a_{2}}^{(\beta +\omega )}\Delta _{\omega }y+ \omega n_{1}L_{a_{2}}^{(\beta )}y+ \omega n_{2}L_{a_{1}}^{(\beta )}y=0, \end{aligned}$$

where $y= M_{n_{1},n_{2}}^{(\omega;\beta; a_{1},a_{2})}(x)$, which gives the proof. □

Corollary 4.5

The special cases of Theorem 4.1, Theorem 4.2and Theorem 4.4, Theorem 4.4when$\omega =1$can easily be reduced to the results obtained in [7]. For instance,

when$\omega =1$in Theorem 4.1, we have the raising operator$L_{a_{i}}^{(\beta )}[\cdot]$for any polynomial,

$$\begin{aligned} L_{a_{i}}^{(\beta )}[y]=\frac{x}{1-a_{i}}\nabla y- \biggl[ \frac{a_{i}(\beta -1)}{1-a_{i}}-x \biggr]y, \end{aligned}$$

which is exactly the same formula as in [7, equation (2.4), p. 138].

When$\omega =1$in Theorem 4.2the lowering operator for multiple Meixner polynomials of the first kind has the following form:

$$\begin{aligned} \Delta M_{\overrightarrow{n}}^{(\beta,\overrightarrow{a})}=\sum_{i=1}^{r}n_{i}M_{ \overrightarrow{n}-e_{i}}^{(\beta +1,\overrightarrow{a})}, \end{aligned}$$

which coincides with the formula in [7, equation (2.5), p. 138].

When$\omega =1$in Theorem 4.4, the$(n+1)$th order difference equation for multiple Meixner polynomials of the first kind becomes

$$\begin{aligned} &L_{a_{1}}^{(\beta +2-r} L_{a_{2}}^{(\beta +3-r)} \cdots L_{a_{r}}^{( \beta +1)}\Delta M_{\overrightarrow{n}}^{(\beta,\overrightarrow{a})}(x) \\ & \quad{}+\sum_{i=1}^{r} n_{i}L_{a_{1}}^{(\beta +2 -r)} \cdots L_{a_{i-1}}^{( \beta +i-r)}L_{a_{i+1}}^{(\beta +(i+1)-r)}\cdots L_{a_{r}}^{(\beta )}L_{a_{i}}^{( \beta +1)}M_{\overrightarrow{n}}^{(\beta,\overrightarrow{a})}(x)=0, \end{aligned}$$

which reduces to the formula in [7, Theorem 2.5, p. 139].

When$\omega =1$in Theorem 4.4, the third order difference equation for multiple Meixner polynomials of the first kind is obtained as

$$\begin{aligned} & x(x-1)\nabla ^{2}\Delta y+ x \bigl[\beta (a_{1}+a_{2})+(x-1) (a_{1}+a_{2}-2) \bigr] \nabla \Delta y \\ &\quad{}+ \bigl[ \bigl(a_{1} \beta -x(1-a_{1}) \bigr) (a_{2}\beta -x(1-a_{1}) (1-a_{2})-a_{1} a_{2} \beta \bigr]\Delta y \\ & \quad{}+\bigl[n_{1}(1-a_{1})+n_{2}(1-a_{2}) \bigr] x\nabla y \\ & \quad{}+(\beta -1) \bigl[ \bigl[n_{1}a_{2}+n_{2}a_{1}-a_{1} a_{2}(n_{1}+n_{2}) \bigr]-(1-a_{1}) (1-a_{2}) (n_{1}+n_{2})x \bigr]y=0, \end{aligned}$$

which coincides with the formula in [7, Corollary 2.6, p. 139].

5 $1/2$-Multiple Meixner polynomials of the first kind

As we mentioned before for the case when $\omega =1$, ω-multiple Meixner polynomials of the first kind reduce to the known multiple Meixner polynomials of the first kind. For the other values of ω we have new classes for multiple Meixner polynomials of the first kind where ω is positive real number. In this section we exhibit the case $\omega =1/2$ and state some relations for $1/2$-multiple Meixner polynomials of the first kind such as weight functions, orthogonality conditions, the explicit form, the generating function and a third order difference equation.

$1/2$-Multiple Meixner polynomials of the first kind have the following weight functions:

$$\begin{aligned} w_{i}^{(1/2;\beta )}(x)= \frac{\varGamma _{1/2}(\beta +x)a_{i}^{x}}{\varGamma _{1/2}(\beta )\varGamma _{1/2}(x+1/2)}= \frac{\varGamma (2(\beta +x))a_{i}^{x}}{\varGamma (2\beta )\varGamma (2(x+1/2))}\quad i=1,2,\ldots,r. \end{aligned}$$

By using these weight functions in (2), the orthogonality conditions for $1/2$-multiple Meixner polynomials of the first kind can be written as

$$\begin{aligned} \sum_{x=0}^{\infty } M_{\overrightarrow{n}}^{(1/2;\beta; \overrightarrow{a})} \biggl(\frac{x}{2} \biggr) (-x)_{j} \frac{\varGamma (2\beta +x)a_{i}^{x/2}}{\varGamma (2\beta )\varGamma (x+1)} =0,\quad j=0,1,\ldots, n_{i}-1. \end{aligned}$$

The explicit form for $1/2$-multiple Meixner polynomials can easily be obtained from (5) as follows:

$$\begin{aligned} M_{\overrightarrow{n}}^{(1/2;\beta;\overrightarrow{a})}(x)= {}& \sum _{k_{1}=0}^{n_{1}}\cdots \sum_{k_{r}=0}^{n_{r}} \binom{n_{1}}{k_{1}}\cdots \binom{n_{r}}{k_{r}} \\ & {}\times\prod_{j=1}^{r} \biggl[ \frac{({a_{j}^{1/2}})^{n_{j}-k_{j}}}{(2a_{j}^{1/2}-2)^{n_{j}}}(-2x)_{| \overrightarrow{k}|}(2\beta +2x)_{|\overrightarrow{n}|-| \overrightarrow{k}|} \biggr]. \end{aligned}$$

$1/2$-Multiple Meixner polynomials of the first kind have the following generating function:

$$\begin{aligned} \sum_{n_{1}=0}^{\infty }\cdots \sum _{n_{r}=0}^{\infty }M_{ \overrightarrow{n}}^{(1/2;\beta;\overrightarrow{a})}(x) \frac{t_{1}^{n_{1}}\cdots t_{r}^{n_{r}}}{n_{1}!\cdots n_{r}!}= {}& \biggl(1- \frac{ t_{1}}{2(a_{1}^{1/2}-1)}- \cdots - \frac{ t_{r}}{2(a_{r}^{1/2}-1)} \biggr)^{2x} \\ & {}\times \biggl(1-\frac{ t_{1} a_{1}^{1/2}}{2(a_{1}^{1/2}-1)}-\cdots - \frac{ t_{r} a_{r}^{1/2}}{2(a_{r}^{1/2}-1)} \biggr)^{-2(\beta +x)}. \end{aligned}$$

$1/2$-Multiple Meixner polynomials of the first kind satisfy the following third order difference equation:

$$\begin{aligned} & x(x-1/2)\nabla _{1/2}^{2}\Delta _{1/2} y +x \bigl[\beta \bigl(a_{1}^{1/2}+a_{2}^{1/2} \bigr)+(x-1/2) \bigl(a_{1}^{1/2}+a_{2}^{1/2}-2 \bigr) \bigr] \nabla _{1/2}\Delta _{1/2} y \\ &\quad{} +\bigl[a_{1}^{1/2} \beta -x \bigl(1-a_{1}^{1/2} \bigr) ((a_{2}\beta )/2- \bigl[x \bigl(1-a_{1}^{1/2} \bigr) \bigl(1-a_{2}^{1/2} \bigr) \bigr]/2- \bigl[a_{1}^{1/2} a_{1}^{1/2} \beta \bigr]/2 \bigr]\Delta _{1/2} y \\ &\quad{} +\bigl[n_{1} \bigl(1-a_{1}^{1/2} \bigr)+ n_{2} \bigl(1-a_{2}^{1/2} \bigr) \bigr]x/2\nabla _{1/2} y \\ &\quad{}+(\beta -1/2)/2 \bigl[ \bigl[n_{1}a_{2}^{1/2}+n_{2}a_{1}^{1/2}-a_{1}^{1/2} a_{2}^{1/2}(n_{1}+n_{2}) \bigr] \\ &\quad{}- \bigl(1-a_{1}^{1/2} \bigr) \bigl(1-a_{2}^{1/2} \bigr) (n_{1}+n_{2})x \bigr]y=0, \end{aligned}$$

where $\Delta _{1/2} f(x)=f(x+1/2)-f(x)$ and $\nabla _{1/2}f(x)=f(x)-f(x-1/2)$.