Lectures on q-Orthogonal Polynomials
These notes form a brief introduction to the theory of Askey-Wilson polynomials and related topics. We have used a novel approach to the development of those parts needed from the theory of basic hypergeometric functions. One important difference between our approach to basic hypergeometric functions and other approaches, for example those of Andrews, Askey, and Roy , of Gasper and Rahman  or of Bailey  is our frequent use of the Askey-Wilson divided difference operators, the q-difference operator, and the identity theorem for analytic functions.
We apply the bootstrap method from  and an attachment procedure to derive orthogonality relations for the Askey-Wilson polynomials in two steps using the Al-Salam-Chihara polynomials as an intermediate step. The continuous q-ultraspherical polynomialsand continuous q-Hermite polynomials are studied from a different approach. As an application of connection coefficient formulas we give the recent proof and generalizations of the Rogers-Ramanujan identities from .
These notes are summarized from a forthcoming Cambridge University Press monograph.
KeywordsOrthogonal Polynomial Orthogonality Relation Basic Hypergeometric Series Ultraspherical Polynomial Wilson Polynomial
Unable to display preview. Download preview PDF.
- 4.G. E. Andrews, q-series: Their development and application in analysis, number theory, combinatorics, physics, and computer algebra, CBMS Regional Conference Series, number 66, American Mathematical Society, Providence, R. I. 1986.Google Scholar
- 6.G. E. Andrews and R. A. Askey, Classical orthogonal polynomials, in: “Polynomes Orthogonaux et Applications”,C. Breziniski, et al., eds., Lecture Notes in Mathematics, Vol. 1171, Springer-Verlag, Berlin, 1984, pp. 36–63.Google Scholar
- 9.R. A. Askey and M. E. H. Ismail, A generalization of ultraspherical polynomials, in: “Studies in Pure Mathematics”, P. Erdös, ed., Birkhauser, Basel, 1983, pp. 55–78.Google Scholar
- 10.R. A. Askey and M. E. H. Ismail, Recurrence relations, continued fractions and orthogonal polynomials, Memoirs Amer. Math. Soc. Number 300 (1984).Google Scholar
- 11.R. A. Askey and J. A. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs Amer. Math. Soc. Number 319 (1985).Google Scholar
- 15.A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.Google Scholar