The non-symmetric Wilson polynomials are the Bannai–Ito polynomials
HTML articles powered by AMS MathViewer
- by Vincent X. Genest, Luc Vinet and Alexei Zhedanov PDF
- Proc. Amer. Math. Soc. 144 (2016), 5217-5226 Request permission
Abstract:
The one-variable non-symmetric Wilson polynomials are shown to coincide with the Bannai–Ito polynomials. The isomorphism between the corresponding degenerate double affine Hecke algebra of type $(C_1^{\vee }, C_1)$ and the Bannai–Ito algebra is established. The Bannai–Ito polynomials are seen to satisfy an orthogonality relation with respect to a positive-definite and continuous measure on the real line. A non-compact form of the Bannai–Ito algebra is introduced and a four-parameter family of its infinite-dimensional and self-adjoint representations is exhibited.References
- George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
- Eiichi Bannai, Orthogonal polynomials in coding theory and algebraic combinatorics, Orthogonal polynomials (Columbus, OH, 1989) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 294, Kluwer Acad. Publ., Dordrecht, 1990, pp. 25–53. MR 1100287, DOI 10.1007/978-94-009-0501-6_{2}
- Eiichi Bannai and Tatsuro Ito, Algebraic combinatorics. I, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984. Association schemes. MR 882540
- Vincent X. Genest, Luc Vinet, and Alexei Zhedanov, A Laplace-Dunkl equation on $S^2$ and the Bannai-Ito algebra, Comm. Math. Phys. 336 (2015), no. 1, 243–259. MR 3322373, DOI 10.1007/s00220-014-2241-4
- Pavel Etingof, Alexei Oblomkov, and Eric Rains, Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces, Adv. Math. 212 (2007), no. 2, 749–796. MR 2329319, DOI 10.1016/j.aim.2006.11.008
- Vincent X. Genest, Luc Vinet, and Alexei Zhedanov, Bispectrality of the complementary Bannai-Ito polynomials, SIGMA Symmetry Integrability Geom. Methods Appl. 9 (2013), Paper 018, 20. MR 3033560, DOI 10.3842/SIGMA.2013.018
- Vincent X. Genest, Luc Vinet, and Alexei Zhedanov, The Bannai-Ito polynomials as Racah coefficients of the $sl_{-1}(2)$ algebra, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1545–1560. MR 3168462, DOI 10.1090/S0002-9939-2014-11970-8
- Vincent X. Genest, Luc Vinet, and Alexei Zhedanov, A Laplace-Dunkl equation on $S^2$ and the Bannai-Ito algebra, Comm. Math. Phys. 336 (2015), no. 1, 243–259. MR 3322373, DOI 10.1007/s00220-014-2241-4
- Wolter Groenevelt, Wilson function transforms related to Racah coefficients, Acta Appl. Math. 91 (2006), no. 2, 133–191. MR 2249545, DOI 10.1007/s10440-006-9024-7
- Wolter Groenevelt, Fourier transforms related to a root system of rank 1, Transform. Groups 12 (2007), no. 1, 77–116. MR 2308030, DOI 10.1007/s00031-005-1124-5
- Wolter Groenevelt, Multivariable Wilson polynomials and degenerate Hecke algebras, Selecta Math. (N.S.) 15 (2009), no. 3, 377–418. MR 2551187, DOI 10.1007/s00029-009-0005-3
- Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2009. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey; Reprint of the 2005 original. MR 2542683
- M. E. H. Ismail, E. Rains, and D. Stanton, Orthogonality of very well-poised series, 2015.
- Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096, DOI 10.1007/978-3-642-05014-5
- Tom H. Koornwinder, Askey-Wilson polynomials for root systems of type $BC$, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 189–204. MR 1199128, DOI 10.1090/conm/138/1199128
- Tom H. Koornwinder, The relationship between Zhedanov’s algebra $\textrm {AW}(3)$ and the double affine Hecke algebra in the rank one case, SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007), Paper 063, 15. MR 2299864, DOI 10.3842/SIGMA.2007.063
- Tom H. Koornwinder, Zhedanov’s algebra $\rm AW(3)$ and the double affine Hecke algebra in the rank one case. II. The spherical subalgebra, SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008), Paper 052, 17. MR 2425640, DOI 10.3842/SIGMA.2008.052
- Douglas A. Leonard, Orthogonal polynomials, duality and association schemes, SIAM J. Math. Anal. 13 (1982), no. 4, 656–663. MR 661597, DOI 10.1137/0513044
- I. G. Macdonald, Symmetric functions and orthogonal polynomials, University Lecture Series, vol. 12, American Mathematical Society, Providence, RI, 1998. Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, NJ. MR 1488699, DOI 10.1090/ulect/012
- S. Shahi, Non-symmetric Koornwinder polynomials and duality, Annals of Mathematics, 150:267–282, 1999.
- Jasper V. Stokman, Koornwinder polynomials and affine Hecke algebras, Internat. Math. Res. Notices 19 (2000), 1005–1042. MR 1792347, DOI 10.1155/S1073792800000520
- Paul Terwilliger, The universal Askey-Wilson algebra and DAHA of type $(C^\vee _1,C_1)$, SIGMA Symmetry Integrability Geom. Methods Appl. 9 (2013), Paper 047, 40. MR 3116183, DOI 10.3842/SIGMA.2013.047
- Satoshi Tsujimoto, Luc Vinet, and Alexei Zhedanov, Dunkl shift operators and Bannai-Ito polynomials, Adv. Math. 229 (2012), no. 4, 2123–2158. MR 2880217, DOI 10.1016/j.aim.2011.12.020
- Satoshi Tsujimoto, Luc Vinet, and Alexei Zhedanov, Dual $-1$ Hahn polynomials: “classical” polynomials beyond the Leonard duality, Proc. Amer. Math. Soc. 141 (2013), no. 3, 959–970. MR 3003688, DOI 10.1090/S0002-9939-2012-11469-8
- J. F. van Diejen, Commuting difference operators with polynomial eigenfunctions, Compositio Math. 95 (1995), no. 2, 183–233. MR 1313873
- J. F. van Diejen, Multivariable continuous Hahn and Wilson polynomials related to integrable difference systems, J. Phys. A 28 (1995), no. 13, L369–L374. MR 1352361
- J. F. van Diejen and E. Emsiz, Discrete harmonic analysis on a Weyl alcove, J. Funct. Anal. 265 (2013), no. 9, 1981–2038. MR 3084495, DOI 10.1016/j.jfa.2013.06.023
- Jan Felipe van Diejen and Luc Vinet (eds.), Calogero-Moser-Sutherland models, CRM Series in Mathematical Physics, Springer-Verlag, New York, 2000. MR 1843558, DOI 10.1007/978-1-4612-1206-5
- Luc Vinet and Alexei Zhedanov, A limit $q=-1$ for the big $q$-Jacobi polynomials, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5491–5507. MR 2931336, DOI 10.1090/S0002-9947-2012-05539-5
- A. S. Zhedanov, “Hidden symmetry” of Askey-Wilson polynomials, Teoret. Mat. Fiz. 89 (1991), no. 2, 190–204 (Russian, with English summary); English transl., Theoret. and Math. Phys. 89 (1991), no. 2, 1146–1157 (1992). MR 1151381, DOI 10.1007/BF01015906
Additional Information
- Vincent X. Genest
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 970414
- Email: vxgenest@mit.edu
- Luc Vinet
- Affiliation: Centre de recherches mathématiques, Université de Montréal, Montréal, QC, Canada H3C 3J7
- MR Author ID: 178665
- ORCID: 0000-0001-6211-7907
- Email: vinetl@crm.umontreal.ca
- Alexei Zhedanov
- Affiliation: Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine
- MR Author ID: 234560
- Email: zhedanov@yahoo.com
- Received by editor(s): July 10, 2015
- Received by editor(s) in revised form: February 8, 2016
- Published electronically: May 24, 2016
- Additional Notes: This work was presented on January 9, 2016 by the first author at the Joint Mathematics Meetings in the AMS Special Session on Special Functions and $q$-Series
- Communicated by: Mourad Ismail
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 5217-5226
- MSC (2010): Primary 33C80, 20C08
- DOI: https://doi.org/10.1090/proc/13141
- MathSciNet review: 3556266