On the?-multiple Meixner polynomials of the first kind

Abstract

In this study, we introduce a new family of discrete multiple orthogonal polynomials, namely omega-multiple Meixner polynomials of the first kind, where omega 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 omega-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 omega-multiple Meixner polynomials of the first kind as a solution. As a corollary we give a third order difference equation for the omega-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.