Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 002, 11 pages      math.QA/0701134      http://dx.doi.org/10.3842/SIGMA.2007.002
Contribution to the Vadim Kuznetsov Memorial Issue

Raising and Lowering Operators for Askey-Wilson Polynomials

Siddhartha Sahi
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA

Received September 20, 2006, in final form December 27, 2006; Published online January 04, 2007

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.

Key words: orthogonal polynomials; Askey-Wilson polynomials; q-difference equation; three term recurrence; raising operators; lowering operators; root systems; double affine Hecke algebra.

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