A new approach to the theory of classical hypergeometric polynomials

Authors:
José Manuel Marco and Javier Parcet

Journal:
Trans. Amer. Math. Soc. **358** (2006), 183-214

MSC (2000):
Primary 33D15, 33D45

DOI:
https://doi.org/10.1090/S0002-9947-04-03620-7

Published electronically:
December 28, 2004

MathSciNet review:
2171229

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we present a unified approach to the spectral analysis of a hypergeometric type operator whose eigenfunctions include the classical orthogonal polynomials. We write the eigenfunctions of this operator by means of a new Taylor formula for operators of Askey-Wilson type. This gives rise to some expressions for the eigenfunctions, which are unknown in such a general setting. Our methods also give a general Rodrigues formula from which several well-known formulas of Rodrigues-type can be obtained directly. Moreover, other new Rodrigues-type formulas come out when seeking for regular solutions of the associated functional equations. The main difference here is that, in contrast with the formulas appearing in the literature, we get non-ramified solutions which are useful for applications in combinatorics. Another fact, that becomes clear in this paper, is the role played by the theory of elliptic functions in the connection between ramified and non-ramified solutions.

**1.**G.E. Andrews, R. Askey and R. Roy,*Special Functions*, Encyclopedia of Mathematics and its Applications**71**, Cambridge Univ. Press, 1999. MR**2000g:33001****2.**R. Askey and J.A. Wilson,*Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials*, Mem. Amer. Math. Soc.**54**(1985), no. 319. MR**87a:05023****3.**G. Bangerezako,*The factorization method for the Askey-Wilson polynomials*, J. Computational and Applied Math.**107**(1999), 219-232. MR**2000h:33015****4.**E. Bannai and T. Ito,*Algebraic Combinatorics I: Association Schemes*, Benjamin/Cummings Publishing Co., Inc., Menlo Park, 1984.MR**87m:05001****5.**A.E. Brouwer, A.M. Cohen and A. Neumaier,*Distance regular graphs*, Springer-Verlag, 1989. MR**90e:05001****6.**B.M. Brown and M.E.H. Ismail,*A right inverse for the Askey-Wilson operator*, Proc. Amer. Math. Soc.**123**(1995), 2071-2079. MR**95i:33019****7.**I. Fischer,*A Rodrigues-Type formula for the -Racah polynomials and some related results*, Contemporary Mathematics**169**(1994), 253-259. MR**95j:33049****8.**G. Gasper and M. Rahman,*Basic hypergeometric series. With a foreword by Richard Askey*, Encyclopedia of Mathematics and its Applications**35**, Cambridge Univ. Press, 1990.MR**91d:33034****9.**D.A. Leonard,*Orthogonal polynomials, duality and association schemes*, SIAM J. Math. Anal.**13**(1982), 656-663.MR**83m:42014****10.**J.H. van Lint and R.M. Wilson,*A course in combinatorics*, Cambridge Univ. Press, 1992. MR**94g:05003****11.**A.P. Magnus,*Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials*, Lecture Notes in Math., vol. 1329, Springer, Berlin, 1986, pp. 261-278. MR**90d:33008****12.**J.M. Marco and J. Parcet,*On the natural representation of into : discrete harmonics and Fourier transform*, J. Combin. Theory Ser. A**100**(2002), 153-175. MR**2003m:05209****13.**J.M. Marco and J. Parcet,*Taylor series for the Askey-Wilson operator and classical summation formulas*. Submitted for publication.**14.**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.MR**92m:33019****15.**M. Rahman and S.K. Suslov,*The Pearson equation and the beta integrals*, SIAM J. Math. Anal.**25**(1994), 646-693.MR**95f:33001****16.**P. Terwilliger,*Two linear transformations each tridiagonal with respect to an eigenbasis for the other*, Linear Alg. Appl.**330**(2001), 149-203. MR**2002h:15021**

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Additional Information

**José Manuel Marco**

Affiliation:
Department of Mathematics, Universidad Autónoma de Madrid, Madrid 28049, Spain

**Javier Parcet**

Affiliation:
Department of Mathematics, Universidad Autónoma de Madrid, Madrid 28049, Spain

Email:
javier.parcet@uam.es

DOI:
https://doi.org/10.1090/S0002-9947-04-03620-7

Keywords:
P-sequence,
hypergeometric operator,
Taylor and Rodrigues formula

Received by editor(s):
July 17, 2003

Received by editor(s) in revised form:
January 8, 2004

Published electronically:
December 28, 2004

Article copyright:
© Copyright 2004
American Mathematical Society