A new approach to the theory of classical hypergeometric polynomials

Marco, José; Parcet, Javier

doi:10.1090/S0002-9947-04-03620-7

A new approach to the theory of classical hypergeometric polynomials
HTML articles powered by AMS MathViewer

by José Manuel Marco and Javier Parcet PDF

Trans. Amer. Math. Soc. 358 (2006), 183-214 Request permission

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.

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
Richard Askey and James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55. MR 783216, DOI 10.1090/memo/0319
Gaspard Bangerezako, The factorization method for the Askey-Wilson polynomials, J. Comput. Appl. Math. 107 (1999), no. 2, 219–232. MR 1701532, DOI 10.1016/S0377-0427(99)00090-4
Eiichi Bannai and Tatsuro Ito, Algebraic combinatorics. I, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984. Association schemes. MR 882540
A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 18, Springer-Verlag, Berlin, 1989. MR 1002568, DOI 10.1007/978-3-642-74341-2
B. Malcolm Brown and Mourad E. H. Ismail, A right inverse of the Askey-Wilson operator, Proc. Amer. Math. Soc. 123 (1995), no. 7, 2071–2079. MR 1273478, DOI 10.1090/S0002-9939-1995-1273478-X
Ismor Fischer, A Rodrigues-type formula for the $q$-Racah polynomials and some related results, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992) Contemp. Math., vol. 169, Amer. Math. Soc., Providence, RI, 1994, pp. 253–259. MR 1292903, DOI 10.1090/conm/169/01658
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
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
J. H. van Lint and R. M. Wilson, A course in combinatorics, Cambridge University Press, Cambridge, 1992. MR 1207813
Alphonse P. Magnus, Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials, Orthogonal polynomials and their applications (Segovia, 1986) Lecture Notes in Math., vol. 1329, Springer, Berlin, 1988, pp. 261–278. MR 973434, DOI 10.1007/BFb0083366
José Manuel Marco and Javier Parcet, On the natural representation of $S(\Omega )$ into $L^2(\scr P(\Omega ))$: discrete harmonics and Fourier transform, J. Combin. Theory Ser. A 100 (2002), no. 1, 153–175. MR 1932074, DOI 10.1006/jcta.2002.3291
J.M. Marco and J. Parcet, Taylor series for the Askey-Wilson operator and classical summation formulas. Submitted for publication.
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
Mizan Rahman and Sergei K. Suslov, The Pearson equation and the beta integrals, SIAM J. Math. Anal. 25 (1994), no. 2, 646–693. MR 1266583, DOI 10.1137/S003614109222874X
Paul Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl. 330 (2001), no. 1-3, 149–203. MR 1826654, DOI 10.1016/S0024-3795(01)00242-7