The structure relation for Askey-Wilson polynomials

[摘要] An explicit structure relation for Askey-Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey-Wilson inner product and which sends polynomials of degree n to polynomials of degree n + 1. By specialization of parameters and by taking limits, similar structure relations, as well as lowering and raising relations, can be obtained for other families in the q-Askey scheme and the Askey scheme. This is explicitly discussed for Jacobi polynomials, continuous q-Jacobi polynomials, continuous q-ultraspherical polynomials, and for big q-Jacobi polynomials. An already known structure relation for this last family can be obtained from the new structure relation by using the three-term recurrence relation and the second order q-difference formula. The results are also put in the framework of a more general theory. Their relationship with earlier work by Zhedanov and Bangerezako is discussed. There is also a connection with the string equation in discrete matrix models and with the Sklyanin algebra. (C) 2006 Elsevier B.V. All rights reserved.

[发布日期] 2007-10-15

[效力级别]  Proceedings Paper

[关键词] structure relation;Askey-Wilson polynomials;classical orthogonal polynomials;Askey scheme;lowering and raising relations