Lecture notes for an introductory minicourse on q-series

Abstract

These lecture notes were written for a mini-course that was designed to introduce students and researchers to {\it $q$-series,} which are also called {\it basic hypergeometric series} because of the parameter $q$ that is used as a base in series that are ``{\it over, above or beyond}'' the {\it geometric series}. We start by considering $q$-extensions (also called $q$-analogues) of the binomial theorem, the exponential and gamma functions, and of the beta function and beta integral, and then progress on to the derivations of rather general summation, transformation, and expansion formulas, integral representations, and applications. Our main emphasis is on methods that can be used to {\bf derive} formulas, rather than to just {\it verify} previously derived formulas.