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

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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. 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-discrete Painlevé V.