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

We look at some extensions of the Stieltjes–Wigert weight functions. First we replace the variable xx by x2x2 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 qq-discrete Painlevé III equation qq-PIII. Next we consider the qq-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 qq-discrete Painlevé V equation qq-PV. Finally we also consider the little qq-Laguerre weight with a quadratic transformation and show that the recurrence coefficients of the orthogonal polynomials are again related to qq-PV.