Non-Symmetric Askey--Wilson Shift Operators

2024/12/04 by van Horssen, Max, Schlösser, Philip
#20C08 #33D45 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2412.03169

Abstract

We classify the shift operators for the symmetric Askey-Wilson polynomials and construct shift operators for the non-symmetric Askey-Wilson polynomials using two decompositions of non-symmetric Askey-Wilson polynomials in terms of symmetric ones. These shift operators are difference-reflection operators, and we discuss the conditions under which they restrict to shift operators for the symmetric Askey-Wilson polynomials. We use them to compute the norms of the non-symmetric Askey-Wilson polynomials and compute their specialisations for q→1. These turn out to be shift operators for the non-symmetric Heckman-Opdam polynomials of type BC₁ that have recently been found.

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