Raising and Lowering Operators for Askey-Wilson Polynomials

Abstract

In this paper we describe two pairs of raising/lowering operators for Askey-Wilson polynomials, which result from constructions involving very different techniques. The first technique is quite elementary, and depends only on the ''classical'' properties of these polynomials, viz. the q-difference equation and the three term recurrence. The second technique is less elementary, and involves the one-variable version of the double affine Hecke algebra.

This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

References in corpus (7)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Difference equations and symmetric polynomials defined by their zeros
• Integrality of two variable Kostka functions
• Self-dual Koornwinder-Macdonald polynomials
• The structure relation for Askey-Wilson polynomials
• Creation operators for the Macdonald and Jack polynomials
• Koornwinder polynomials and affine Hecke algebras

Cited by in corpus (6)

• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
• Double Affine Hecke Algebras of Rank 1 and the $Z_3$-Symmetric Askey-Wilson Relations
• The universal DAHA of type $(C_1^{\vee },C_1)$ and Leonard pairs of $q$-Racah type
• Finite size lattice results for the two-boundary Temperley--Lieb loop model
• Double affine Hecke algebra of rank 1 and orthogonal polynomials on the unit circle