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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| Vydáno v: | Journal of mathematical analysis and applications Ročník 421; číslo 2; s. 1101 - 1130 |
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15.01.2015
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Ito, Masahiko Witte, N.S. |
| Author_xml | – sequence: 1 givenname: Masahiko surname: Ito fullname: Ito, Masahiko email: mito@cck.dendai.ac.jp organization: School of Science and Technology for Future Life, Tokyo Denki University, Tokyo 101-8457, Japan – sequence: 2 givenname: N.S. surname: Witte fullname: Witte, N.S. email: nsw@ms.unimelb.edu.au organization: Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia |
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| Cites_doi | 10.4153/CJM-1986-030-6 10.1007/BF01180560 10.1137/0516014 10.1006/jmaa.1994.1339 10.1007/BF02400421 10.1216/RMJ-1989-19-1-223 10.1007/s11139-006-9579-y 10.1088/0305-4470/24/8/009 10.1007/s00220-007-0289-0 10.1137/S0036141092233676 10.1016/S0021-9045(03)00076-5 10.1016/0377-0427(95)00262-6 10.1007/s11139-006-9012-6 10.1088/0305-4470/21/10/015 10.1007/BF00417461 10.1137/S003614109222874X |
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| Keywords | Divided difference operators Basic hypergeometric integrals Non-uniform lattices Semi-classical weights Askey–Wilson polynomials Orthogonal polynomials |
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| Snippet | 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... |
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| SubjectTerms | Askey–Wilson polynomials Basic hypergeometric integrals Divided difference operators Non-uniform lattices Orthogonal polynomials Semi-classical weights |
| Title | On a family of integrals that extend the Askey–Wilson integral |
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