2006/01/31 by Tom H. Koornwinder · 40 citations
Mathematics · Physics and Astronomy · #Algebra over a field #Algebraic structures and combinatorial models #Askey–Wilson polynomials #Classical orthogonal polynomials #Difference polynomials #Discrete orthogonal polynomials #Gegenbauer polynomials #Hahn polynomials #Jacobi polynomials #Macdonald polynomials #Mathematical analysis #Mathematics #Nonlinear Waves and Solitons #Orthogonal polynomials #Pure mathematics #Quantum Mechanics and Non-Hermitian Physics #Recurrence relation #Relation (database) #Wilson polynomials #math.CA #msc:33C45 #msc:33D45
paper · pdf · doi:10.1016/j.cam.2006.10.015
published in Journal of Computational and Applied Mathematics 207(2), 214-226 (Elsevier BV) · 18 pages, minor corrections and updated references
openalex publication_date 2006/11/29 · arxiv created 2007/01/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 recurence 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.