Variations of Stieltjes–Wigert and q-Laguerre polynomials and their recurrence coefficients

We look at some extensions of the Stieltjes–Wigert weight functions. First we replace the variable x by x2 in a family of weight functions given by Askey in 1989 and we show that the recurrence coefficients of the corresponding orthogonal polynomials can be expressed in terms of a solution of the q-...

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Bibliographic Details
Published in:Journal of approximation theory 2015-05, Vol.193, p.56-73
Main Authors: Boelen, Lies, Van Assche, Walter
Format: Article
Language:English
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Summary:We look at some extensions of the Stieltjes–Wigert weight functions. First we replace the variable x by x2 in a family of weight functions given by Askey in 1989 and we show that the recurrence coefficients of the corresponding orthogonal polynomials can be expressed in terms of a solution of the q-discrete Painlevé III equation q-PIII. Next we consider the q-Laguerre or generalized Stieltjes–Wigert weight functions with a quadratic transformation and derive recursive equations for the recurrence coefficients of the orthogonal polynomials. These turn out to be related to the q-discrete Painlevé V equation q-PV. Finally we also consider the little q-Laguerre weight with a quadratic transformation and show that the recurrence coefficients of the orthogonal polynomials are again related to q-PV.
ISSN:0021-9045
1096-0430
DOI:10.1016/j.jat.2014.06.012