q-analogues

Abstract. The study of q-series began in 1748 when Euler considered the generating function for the number of partitions of a positive integer. But the subject itself did not really come into its own until some 100 years later when Heine developed a theory of basic hypergeometric series that contains the theory of the Gauss hypergeometric series as a limiting case. The latter is obtained from the former in the limit as q tends to 1. Although the work stemming from Heine is highly analytic in nature, many of the most beautiful results have combinatorial or number-theoretic significance as well: the q-binomial theorem, Jacobi's triple product identity, and Ramanujan's ₁Y₁ identity come to mind.

In a productive mathematical life spanning the latter part of the nineteenth century and the first half of the twentieth century, F. H. Jackson developed a systematic theory of q-analogues, including the operations of q-differentiation and q-integration, which have recently been made the basis of a undergraduate course in quantum calculus at MIT. An advantage of the q-calculus is that, due to its discrete nature, the concepts of limits and infinitessimals are avoided, yet it reduces to the ordinary infinitessimal calculus of Newton and Leibniz in the limit as q tends to 1.

The purpose of this talk is to provide an accessible introduction to the vast subject of q-analogues. My intention is to give an idea of what the subject is about, why people are interested in it, and how I became involved.