Author:

Keywords:

Science & Technology, Physical Sciences, Mathematics, DISCRETE PAINLEVE EQUATIONS, math.CA, 33C45, 33E17, 42C05, 65Q30, 0101 Pure Mathematics, 0102 Applied Mathematics, Numerical & Computational Mathematics, 4901 Applied mathematics, 4904 Pure mathematics

Abstract:

We look at some extensions of the Stieltjes-Wigert weight functions. First we replace the variable x by x^2 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. 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. 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-Painlevé V.