On the ω-multiple Meixner polynomials of the first kind

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 ω=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega = 1$\end{document}, 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 ω=1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega =1/2$\end{document} and, for the 1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1/2$\end{document}-multiple Meixner polynomials of the first kind, we state the corresponding result for the main theorems.

The functions The orthogonality conditions give us a linear system of | − → n | = n 1 + n 2 + · · · + n r homogeneous equations for the | − → n | + 1 unknown coefficients of polynomials which always has a nontrivial solution. If the given multi-index − → 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 , . . . , w r defined on [a, b] is called an AT system for the index n ∈ Z r + , n = 0, if w 1 (x), xw 1 (x), . . . , x n 1 -1 w 1 (x), . . . , w r (x), xw r (x), . . . , 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 − → 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 x 1 , xw(x)a x 1 , . . . , x n 1 -1 w(x)a x 1 , . . . , w(x)a x r , xw(x)a x r , . . . , x n r -1 w(x)a x r , with all the a i > 0, i = 1, . . . , r, different and w(x) a continuous function which has no zeros on R + , form a Cheybshev system on R + for every index − → n ∈ 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 ωmultiple 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 is the ω-forward operator and 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 ω = 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.

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 − → n = (n 1 , . . . , n r ), the fixed parameter β > 0 and the parameter − → a = (a 1 , . . . , a r ), (a i =a j whenever i =j), is the unique polynomial of degree | − → n | which satisfies the orthogonality conditions where (a) n,ω = a(a + ω)(a + 2ω) · · · (a + (n -1)ω) is the ω-Pochhammer symbol for a ∈ C and n ∈ N and ω > 0. The weight functions for ω-multiple Meixner polynomials of the first kind are defined as where 0 < a i < 1, for i =, 1, 2, . . . , r, with all the a i different and Γ ω is the ω-gamma function given by 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 ω = 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
Let ω be a positive real number. The raising operator for ω-multiple Meixner polynomials of the first kind is given as Proof By using the product rule for the ω-backward operator Using the ω-summation by parts formula, and the orthogonality conditions, we find the result with P (ω) , which was guaranteed from the uniqueness of the orthogonal polynomials.

Theorem 2.3 Let ω be a positive real number. The Rodrigues formula for ω-multiple
Meixner polynomials of the first kind are introduced by Proof Replacing − → n by − → n -− → e i and β by β + ω in the raising operator formula, we obtain Then for r = 2 the multi-index will be − → n = (n 1 , n 2 ).
and iterating it n 1 times we get w ((ω;β) 1 (x)M (ω;β;a 1 ,a 2 ) and iterating it n 2 times we get By combining these two equations, we obtain the expression for M (ω;β;a 1 ,a 2 ) n 1 ,n 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,

Theorem 2.4
Let ω be a positive real number. The explicit form for ω-multiple Meixner polynomials of the first kind is given by which is exactly the same formula as in [2, equation (4.6), p. 33].
When ω = 1 in Theorem 2.3, we find the Rodrigues formula for multiple Meixner polynomials of the first kind, , which coincides with the formula in [2, equation (4.7), p. 33].
When ω = 1 in Theorem 2.4, we have an explicit form for multiple Meixner polynomials of the first kind, which coincides with the formula in [4, equation (3) The generating function for the multinomial coefficients is given as follows: This series converges absolutely and uniformly for |t 1 | + · · · + |t r | < 1 when x / ∈ N and contains a finite number of terms if x ∈ N.

Theorem 3.2 Let ω be a positive real number. The generating function for the ω-multiple
Meixner polynomials of the first kind is t n 1 1 · · · t n r r n 1 ! · · · n r ! = 1 - Proof Replacing M (ω;β; − → a ) − → n (x) with the explicit form in left hand side of (7) we obtain 1 1 · · · t n r r n 1 ! · · · n r ! .
Changing the order of the summation gives By setting m i = n ik i and putting the factors in m i and k i together we obtain Using the relation between the Pochhammer symbol and the ω-Pochhammer symbol (6), the above equation becomes t n 1 1 · · · t n r r n 1 ! · · · n r ! = 1 - , 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.
Proof Replacing β by βγ + γ in the generating function (7) we obtain From the generating function (7) and Lemma 3.1, the above equation gets the following form: Changing the order of summations and using the relation between the Pochhammer symbol and the ω-Pochhammer symbol (6) we get a ω-1 r k r t n 1 1 · · · t n r r n 1 ! · · · n r ! .
Finally, comparing the coefficients of t n 1 1 ···t nr r n 1 !···n r ! , appearing on both sides of the above equation, we obtain the desired result.

Theorem 3.5 Let ω be a positive real number. The addition formula for ω-multiple Meixner polynomials of the first kind is given by
Proof Changing x with x + y and β with β + γ in the generating function (7), together with changing the order of summations we obtain Finally, comparing the coefficients of Note that in the case when ω = 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. · · · n r k r =0 n 1 k 1 · · · n r k r a k 1 1 · · · a k r r (-γ )

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

1/2-Multiple Meixner polynomials of the first kind
As we mentioned before for the case when ω = 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 ω = 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.
By using these weight functions in (2), the orthogonality conditions for 1/2-multiple Meixner polynomials of the first kind can be written as