M ar 1 99 4 q - Special Functions , A Tutorial TOM

نویسنده

  • TOM H. KOORNWINDER
چکیده

A tutorial introduction is given to q-special functions and to q-analogues of the classical orthogonal polynomials, up to the level of Askey-Wilson polynomials. 0. Introduction It is the purpose of this paper to give a tutorial introduction to q-hypergeometric functions and to orthogonal polynomials expressible in terms of such functions. An earlier version of this paper was written for an intensive course on special functions aimed at Dutch graduate students, and later it was part of the lecture notes of my course on “Quantum groups and q-special functions” at the European School of Group Theory 1993, Trento, Italy. Eventually, this paper may appear as part of a book containing the notes of the main courses at this School. I now describe the various sections in some more detail. Section 1 gives an introduction to q-hypergeometric functions. The more elementary q-special functions like q-exponential and q-binomial series are treated in a rather self-contained way, but for the higher qhypergeometric functions some identities are given without proof. The reader is referred, for instance, to the encyclopedic treatise by Gasper & Rahman [15]. Hopefully, this section succeeds to give the reader some feeling for the subject and some impression of general techniques and ideas. Section 2 gives an overview of the classical orthogonal polynomials, where “classical” now means “up to the level of Askey-Wilson polynomials” [8]. The section starts with the “very classical” situation of Jacobi, Laguerre and Hermite polynomials and next discusses the Askey tableau of classical orthogonal polynomials (still for q = 1). Then the example of big q-Jacobi polynomials is worked out in detail, as a demonstration how the main formulas in this area can be neatly derived. The section continues with the q-Hahn tableau and then gives a self-contained introduction to the Askey-Wilson polynomials. Both sections conclude with some exercises. Notation The notations N := {1, 2, . . .} and Z+ := {0, 1, 2, . . .} will be used. Acknowledgement I thank René Swarttouw for commenting on a preliminary version of this paper. University of Amsterdam, Faculty of Mathematics and Computer Science, Plantage Muidergracht 24, 1018 TV Amsterdam. The Netherlands, email: [email protected]

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تاریخ انتشار 1994