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We look at some extensions of the Stieltjes-Wigert weight functions. First wereplace the variable x by x^2 in a family of weight functions given by Askey in1989 and we show that the recurrence coefficients of the correspondingorthogonal polynomials can be expressed in terms of a solution of theq-discrete Painlev\'e III equation. Next we consider the q-Laguerre orgeneralized Stieltjes-Wigert weight functions with a quadratic transformationand derive recursive equations for the recurrence coefficients of theorthogonal polynomials. These turn out to be related to the q-discretePainlev\'e V equation. Finally we also consider the little q-Laguerre weightwith a quadratic transformation and show that the recurrence coefficients ofthe orthogonal polynomials are again related to q-discrete Painlev\'e V.

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