Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 026, 14 pages      arXiv:1411.2000
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

On the $q$-Charlier Multiple Orthogonal Polynomials

Jorge Arvesú and Andys M. Ramírez-Aberasturis
Department of Mathematics, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911, Leganés, Spain

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

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.

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

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