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).

Keywords:

• orthogonal polynomials
• difference and q-difference equations
• nonlinear difference equations
• degree raising and lowering operators

AMS Subject Classifications::

• Primary 33D45
• 39A13
• Secondary 30E05

Acknowledgements

The research of Mourad Ismail and Zeinab Mansour is supported by King Saud University, Riyadh through grant DSFP/MATH 01. Part of this article was written while the first author was visiting the Isaac Newton Institute and he wishes to thank the Institute for the hospitality and the excellent scientific environment. The authors also thank the referee for many corrections and detailed comments on the original version of this article.