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.
Highlights
Discrete multiple orthogonal polynomials are useful extension of discrete orthogonal polynomials, see [1,2,3,4,5,6,7,8,9,10,11,12,13]
The theory of discrete orthogonal polynomials on a linear lattice were extended to such polynomials by Arvesu, Coussement and Van Assche in [2]
K=0 P−→+−→ei(–ωx)j,ωw(iω;β–ω)(ωx) = 0, we find the result with P−→+−→ei = M−→, which was guaranteed from the uniqueness of the orthogonal polynomials
Summary
Discrete multiple orthogonal polynomials are useful extension of discrete orthogonal polynomials, see [1,2,3,4,5,6,7,8,9,10,11,12,13]. In [2], Arvesu, Coussement and Van Assche investigated the raising operator and the Rodrigues formula for multiple Meixner polynomials of the first kind. Van Assche in [12] obtained a lowering operator for multiple Meixner polynomials of the first kind for the case r = 2 and by combining lowering and raising operators he gave the third 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. 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.
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