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 [7], of Gasper and Rahman [18] or of Bailey [12] 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 [13] 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 [17]. These notes are summarized from a forthcoming Cambridge University Press monograph.