Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Identities for $ q$-ultraspherical polynomials and Jacobi functions


Author: H. T. Koelink
Journal: Proc. Amer. Math. Soc. 123 (1995), 2479-2487
MSC: Primary 33C45; Secondary 33D55
DOI: https://doi.org/10.1090/S0002-9939-1995-1273504-8
MathSciNet review: 1273504
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A q-analogue of a result by Badertscher and Koornwinder [Canad. J. Math. 44 (1992), 750-773] relating the action of a Hahn polynomial of differential operator argument on ultraspherical polynomials to an ultraspherical polynomial of shifted order and degree is derived. The q-analogue involves q-Hahn polynomials, continuous q-ultraspherical polynomials, and a shift operator. Another limit as q tends to 1 yields an identity for Jacobi functions. Combination with another result of Badertscher and Koornwinder gives a curious formula for Jacobi functions.


References [Enhancements On Off] (What's this?)

  • [1] R. Askey and M. E. H. Ismail, A generalization of ultraspherical polynomials, Studies in Pure Mathematics (P. Erdös, ed.), Birkhäuser, Basel, 1983, pp. 55-78. MR 820210 (87a:33015)
  • [2] R. Askey and J. Wilson, A set of orthogonal polynomials that generalize the Racah coefficients or the 6-j symbols, SIAM J. Math. Anal. 10 (1979), 1008-1016. MR 541097 (80k:33012)
  • [3] -, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., vol. 54, no. 319, Amer. Math. Soc., Providence, RI, 1985. MR 783216 (87a:05023)
  • [4] E. Badertscher and T. H. Koornwinder, Continuous Hahn polynomials of differential operator argument and analysis on Riemannian symmetric spaces of constant curvature, Canad. J. Math. 44 (1992), 750-773. MR 1178566 (94k:43011)
  • [5] T. S. Chihara, An introduction to orthogonal polynomials, Math. Appl., vol. 13, Gordon and Breach, New York, 1978. MR 0481884 (58:1979)
  • [6] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, Vol. 1, McGraw-Hill, New York, 1953.
  • [7] G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia Math. Appl., vol. 35, Cambridge Univ. Press, London and New York, 1990. MR 1052153 (91d:33034)
  • [8] M. E. H. Ismail and J. Wilson, Asymptotic and generating relations for the q-Jacobi and $ _4{\varphi _3}$ polynomials, J. Approx. Theory 36 (1982), 43-54. MR 673855 (84e:33012)
  • [9] H. T. Koelink, The addition formula for continuous q-Legendre polynomials and associated spherical elements on the $ SU(2)$ quantum group related to Askey-Wilson polynomials, SIAM J. Math. Anal. 25 (1994), 197-217. MR 1257149 (95f:33023)
  • [10] -, Askey-Wilson polynomials and the quantum $ SU(2)$ group: survey and applications, Acta Appl. Math. (to appear).
  • [11] T. H. Koornwinder, A new proof of a Paley-Wiener type theorem for the Jacobi transform, Ark. Mat. 13 (1975), 145-159. MR 0374832 (51:11028)
  • [12] -, Jacobi functions and analysis on noncompact semisimple Lie groups, Special Functions: Group Theoretical Aspects and Applications (R. A. Askey, T. H. Koornwinder, and W. Schempp, eds.), Reidel, Dordrecht, 1984, pp. 1-85. MR 774055 (86m:33018)
  • [13] -, Group theoretic interpretations of Askey's scheme of hypergeometric orthogonal polynomials, Orthogonal Polynomials and Their Applications (M. Alfaro, J. S. Dehesa, F. J. Marcellan, J. L. Rubio de Francia, and J. Vinuesa, eds.), Lecture Notes in Math., vol. 1329, Springer, New York, 1988, pp. 46-72. MR 973417 (89f:00027)
  • [14] -, Jacobi functions as limit cases of q-ultraspherical polynomials, J. Math. Anal. Appl. 148 (1990), 44-54. MR 1052043 (91c:33033)
  • [15] -, Askey-Wilson polynomials as zonal spherical functions on the $ SU(2)$ quantum group, SIAM J. Math. Anal. 24 (1993), 795-813. MR 1215439 (94k:33042)
  • [16] A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical orthogonal polynomials of a discrete variable, Springer Ser. Comput. Phys., Springer, New York, 1991. MR 1149380 (92m:33019)
  • [17] M. Noumi and K. Mimachi, Askey-Wilson polynomials and the quantum group $ S{U_q}(2)$, Proc. Japan Acad. Ser. A 66 (1990), 146-149. MR 1065793 (91k:33015)
  • [18] -, Askey-Wilson polynomials as spherical functions on $ S{U_q}(2)$, Quantum Groups (P. P. Kulish, ed.), Lecture Notes in Math., vol. 1510, Springer, New York, 1992, pp. 98-103.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 33C45, 33D55

Retrieve articles in all journals with MSC: 33C45, 33D55


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1273504-8
Keywords: Continuous q-ultraspherical polynomial, q-Hahn polynomial, Jacobi function, ultraspherical polynomial, continuous symmetric Hahn polynomial
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society