On the q-Charlier Multiple Orthogonal Polynomials

Abstract

Abstract. We introduce a new family of special functions, namely q-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to q-analogues of Poisson distributions. We focus our attention on their structural properties. Raising and lowering operators as well as Rodrigues-type formulas are obtained. An explicit representation in terms of a q-analogue of the second of Appell's hypergeometric functions is given. A high-order linear q-difference equation with polynomial coefficients is deduced. Moreover, we show how to obtain the nearest neighbor recurrence relation from some difference operators involved in the Rodrigues-type formula.

Multiple Meixner Polynomials on a Non-Uniform Lattice

“…Finally, we combine the lowering and the raising operators to derive the q-difference equation. A similar procedure is given in [31,32,36,[39][40][41]. Finally, the recurrence relations will be derived from some specific difference operators used in Theorems 2 and 4.…”

“…We will find a lowering operator for the q-Meixner multiple orthogonal polynomials of the first kind. We will follow a similar strategy used in [32].…”

“…We will use Lemma 4 involving this function f n (s; β) as well as difference operator (32). Consider equation…”

“…Let us recursively use Lemma 4 involving the product of r difference operators acting on the function f n (s; β), which in this case is the operator M α q, n (see expression (32)). Thus,…”

“…They are related to the simultaneous rational approximation of a system of r analytic functions [18,19] and play an important role both in pure and applied mathematics (see for instance [20][21][22] as well as [23][24][25][26][27]). In this context, some families of continuous and discrete multiple orthogonal polynomials have been studied [3,[28][29][30] as well as some multiple q-orthogonal polynomials [31][32][33]. The goal of the present paper is to study some multiple Meixner polynomials on a non-uniform lattice x(s) = q s − 1/q − 1, s = 0, 1, .…”