This article is cited in 6 papers

On the $q$-Charlier Multiple Orthogonal Polynomials

Jorge Arvesú, Andys M. Ramírez-Aberasturis

Department of Mathematics, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911, Leganés, Spain

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.

Keywords: multiple orthogonal polynomials; Hermite–Padé approximation; difference equations; classical orthogonal polynomials of a discrete variable; Charlier polynomials; $q$-polynomials.

MSC: 42C05; 33E30; 33C47; 33C65

Received: November 10, 2014; in final form March 23, 2015; Published online March 28, 2015

Language: English

DOI: 10.3842/SIGMA.2015.026

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ArXiv: 1411.2000