On a family of integrals that extend the Askey–Wilson integral

We study a family of integrals parameterised by N=2,3,… generalising the Askey–Wilson integral N=2 which has arisen in the theory of q-analogs of monodromy preserving deformations of linear differential systems and in theory of the Baxter Q operator for the XXZ open quantum spin chain. These integra...

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Bibliographic Details
Published in:	Journal of mathematical analysis and applications Vol. 421; no. 2; pp. 1101 - 1130
Main Authors:	Ito, Masahiko, Witte, N.S.
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 15.01.2015
Subjects:	Askey–Wilson polynomials Basic hypergeometric integrals Divided difference operators Non-uniform lattices Orthogonal polynomials Semi-classical weights Divided difference operators Basic hypergeometric integrals Non-uniform lattices Semi-classical weights Askey–Wilson polynomials Orthogonal polynomials
ISSN:	0022-247X, 1096-0813
Online Access:	Get full text
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Summary:	We study a family of integrals parameterised by N=2,3,… generalising the Askey–Wilson integral N=2 which has arisen in the theory of q-analogs of monodromy preserving deformations of linear differential systems and in theory of the Baxter Q operator for the XXZ open quantum spin chain. These integrals are particular examples of moments defined by weights generalising the Askey–Wilson weight and we show the integrals are characterised by various (N−1)-th order linear q-difference equations which we construct. In addition we demonstrate that these integrals can be evaluated as a finite sum of (N−1)BC1-type Jackson integrals or φ2N+12N+2 basic hypergeometric functions.
ISSN:	0022-247X 1096-0813
DOI:	10.1016/j.jmaa.2014.07.056