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.

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关于第一类ω-多重Meixner多项式

在这项研究中，我们介绍了一个新的离散多重正交多项式族，即第一类ω-多重Meixner多项式，其中ω是一个正实数。推导了该族的一些结构特性，例如加注运算符，Rodrigue的类型公式和显式表示。获得了第一类ω-多重Meixner多项式的生成函数，通过使用该生成函数，我们发现了这些多项式的几种结果。其中之一是降低算子，这将有助于获得差分方程。我们使用新技术给出了降级算子的证明，它比Lee中的证明更基础（J.近似论150：132-152，2008）。通过组合下降算子和上升算子，我们得到了以第一类ω-倍Meixner多项式为解的差分方程。作为推论，我们给出了第一类ω-多重Meixner多项式的三阶差分方程。还表明，对于特殊情况$ \ omega = 1 $，获得的结果与多个第一类Meixner多项式的现有结果一致。在最后一部分中，作为说明性示例，我们考虑了$ \ omega = 1/2 $的特殊情况，对于第一种类型的$ 1/2 $乘以Meixner多项式，我们陈述了主要定理的相应结果。还表明，对于特殊情况$ \ omega = 1 $，获得的结果与多个第一类Meixner多项式的现有结果一致。在最后一部分中，作为说明性示例，我们考虑了$ \ omega = 1/2 $的特殊情况，对于第一种类型的$ 1/2 $乘以Meixner多项式，我们陈述了主要定理的相应结果。还表明，对于特殊情况$ \ omega = 1 $，获得的结果与多个第一类Meixner多项式的现有结果一致。在最后一部分中，作为说明性示例，我们考虑了$ \ omega = 1/2 $的特殊情况，对于第一种类型的$ 1/2 $乘以Meixner多项式，我们陈述了主要定理的相应结果。