## Identities for $q$-ultraspherical polynomials and Jacobi functions

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- by H. T. Koelink
- Proc. Amer. Math. Soc.
**123**(1995), 2479-2487 - DOI: https://doi.org/10.1090/S0002-9939-1995-1273504-8
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## 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

- R. Askey and Mourad E. H. Ismail,
*A generalization of ultraspherical polynomials*, Studies in pure mathematics, BirkhΓ€user, Basel, 1983, pp.Β 55β78. MR**820210** - Richard Askey and James Wilson,
*A set of orthogonal polynomials that generalize the Racah coefficients or $6-j$ symbols*, SIAM J. Math. Anal.**10**(1979), no.Β 5, 1008β1016. MR**541097**, DOI 10.1137/0510092 - Richard Askey and James Wilson,
*Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials*, Mem. Amer. Math. Soc.**54**(1985), no.Β 319, iv+55. MR**783216**, DOI 10.1090/memo/0319 - Erich Badertscher and Tom H. Koornwinder,
*Continuous Hahn polynomials of differential operator argument and analysis on Riemannian symmetric spaces of constant curvature*, Canad. J. Math.**44**(1992), no.Β 4, 750β773. MR**1178566**, DOI 10.4153/CJM-1992-044-4 - T. S. Chihara,
*An introduction to orthogonal polynomials*, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR**0481884**
A. ErdΓ©lyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, - George Gasper and Mizan Rahman,
*Basic hypergeometric series*, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR**1052153** - Mourad E. H. Ismail and James A. Wilson,
*Asymptotic and generating relations for the $q$-Jacobi and $_{4}\varphi _{3}$ polynomials*, J. Approx. Theory**36**(1982), no.Β 1, 43β54. MR**673855**, DOI 10.1016/0021-9045(82)90069-7 - H. T. Koelink,
*The addition formula for continuous $q$-Legendre polynomials and associated spherical elements on the $\textrm {SU}(2)$ quantum group related to Askey-Wilson polynomials*, SIAM J. Math. Anal.**25**(1994), no.Β 1, 197β217. MR**1257149**, DOI 10.1137/S0036141090186114
β, - Tom Koornwinder,
*A new proof of a Paley-Wiener type theorem for the Jacobi transform*, Ark. Mat.**13**(1975), 145β159. MR**374832**, DOI 10.1007/BF02386203 - Tom H. Koornwinder,
*Jacobi functions and analysis on noncompact semisimple Lie groups*, Special functions: group theoretical aspects and applications, Math. Appl., Reidel, Dordrecht, 1984, pp.Β 1β85. MR**774055** - M. Alfaro, J. S. Dehesa, F. J. MarcellΓ‘n, J. L. Rubio de Francia, and J. Vinuesa (eds.),
*Orthogonal polynomials and their applications*, Lecture Notes in Mathematics, vol. 1329, Springer-Verlag, Berlin, 1988. MR**973417**, DOI 10.1007/BFb0083349 - Tom H. Koornwinder,
*Jacobi functions as limit cases of $q$-ultraspherical polynomials*, J. Math. Anal. Appl.**148**(1990), no.Β 1, 44β54. MR**1052043**, DOI 10.1016/0022-247X(90)90026-C - Tom H. Koornwinder,
*Askey-Wilson polynomials as zonal spherical functions on the $\textrm {SU}(2)$ quantum group*, SIAM J. Math. Anal.**24**(1993), no.Β 3, 795β813. MR**1215439**, DOI 10.1137/0524049 - A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov,
*Classical orthogonal polynomials of a discrete variable*, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991. Translated from the Russian. MR**1149380**, DOI 10.1007/978-3-642-74748-9 - Masatoshi Noumi and Katsuhisa Mimachi,
*Askey-Wilson polynomials and the quantum group $\textrm {SU}_q(2)$*, Proc. Japan Acad. Ser. A Math. Sci.**66**(1990), no.Β 6, 146β149. MR**1065793**
β,

*Higher transcendental functions*, Vol. 1, McGraw-Hill, New York, 1953.

*Askey-Wilson polynomials and the quantum*$SU(2)$

*group: survey and applications*, Acta Appl. Math. (to appear).

*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.

## Bibliographic Information

- © Copyright 1995 American Mathematical Society
- 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