Difference equations and quantized discriminants for q-orthogonal polynomials

Abstract

We show that orthogonal polynomials on generalized q-linear grid have raising and lowering operators and satisfy a second-order q-difference equation. It is shown that for a general class of weight functions on general q-linear grids, the functions of the second kind and the orthogonal polynomials are linear independent solutions of the same second-order q-difference equation. We introduce q-analogues of the discriminant and evaluate the quantized discriminant for general q-orthogonal polynomials in terms of the recursion coefficients. The q-discriminants of the discrete q-Hermite and little q-Jacobi polynomials, as well as a generalized discriminant of the continuous q-Jacobi polynomials, are given explicitly. A q-analogue of the Freud weights is introduced and we derive the second-order q-difference equation they satisfy.