Abstract
An infinite-dimensional representation of the double affine Hecke algebra of rank 1 and type \((C_1^{\vee },C_1)\) in which all generators are tridiagonal is presented. This representation naturally leads to two systems of polynomials that are orthogonal on the unit circle. These polynomials can be considered as circle analogs of the Askey–Wilson polynomials. The corresponding polynomials orthogonal on an interval are constructed and discussed.
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Acknowledgements
The authors wish to thank Paul Terwilliger for his comments on the manuscript. The research of ST is supported by JSPS KAKENHI (Grant No. 16K13761) and that of LV by a discovery grant of the Natural Sciences and Engineering Research Council (NSERC) of Canada. The work of AZ is supported by the National Science Foundation of China (Grant No.11771015).
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Communicated by Sergey Denisov.
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Tsujimoto, S., Vinet, L. & Zhedanov, A. Double Affine Hecke Algebra of Rank 1 and Orthogonal Polynomials on the Unit Circle. Constr Approx 50, 209–241 (2019). https://doi.org/10.1007/s00365-019-09468-z
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DOI: https://doi.org/10.1007/s00365-019-09468-z
Keywords
- Double affine Hecke algebra
- Orthogonal polynomials on the unit circle
- Delsarte-Genin map
- Askey–Wilson polynomials and algebra