Structure relations for q-polynomials and some applications

Abstract

We derive structure relations for polynomials orthogonal on a half-line or on the real line. Among other things, we derive their degree raising and lowering first-order q-difference operators. We study the properties of the basis of solutions of the corresponding second-order q-difference equation. This generalizes the results of Ismail and Simeonov [M.E.H. Ismail and P. Simeonov, q-difference operators for orthogonal polynomials, J. Comput. Appl. Math. 233 (3) (2009), 749–761]. We apply these structure relations and similar known ones in differential equations to derive the nonlinear difference equations satisfied by the sequence {β n }, where β n are the coefficients of the three-term recurrence relation satisfied by orthogonal polynomials. The polynomials under consideration are orthogonal with respect to q-analogues of exponential weights (Freud weights).