An asymptotic expansion for the q-binomial series using singularity analysis for generating functions

Abstract

For each k, 1≤k≤n, let denote the Gaussian (or q-binomial) Coefficients, where . Then the q-binomial series, or Rogers-Szegö polynomials, are defined as for fixed 0<q<1. The q-binomial series is one of a great variety of series that generalize corresponding hypergeometric series by replacing a number a by (1−q a )/(1−q), or basic number, recovering the hypergeometric series when q→∞. There exists a vast literature on hypergeometric and basic hyper geometric series as well as their applications in many disciplines. Our purpose in this paper is twofold. We first present a review of the concept of a basic analogue of a hypergeometric function and some applications of the q-binomial series and the Gauss coefficients. Then, we outline an important asymptotic analysis method called singularity analysis that can be applied to generating functions of combinatorial enumeration sequences in order to obtain asymptotic expressions of their coefficients (i.e. the sequence). Finally, we use this method in order to derive an asymptotic expression for the q-binomial series. Therefore, we present another example of the power of the singularity analysis method, giving accurate estimates for a frequently occurring hypergeometric expression.