Remarks on Some Basic Hypergeometric Series

Abstract

Many results in Mathematical Analysis seem to come from some “obvious” computations. For a few years, we have been interested in the analytic theory of linear q-difference equations. One of the problems we are working on is the analytical classification of q-difference equations. Recall that this problem was already considered by G. D. Birkhoff and some of his students ((Birkhoff, 1913), (Birkhoff and Güenther, 1941)). An important goal of these works is to be able to derive transcendental analytical invariants from the divergent power series solutions; that is, to be able to define a good concept of Stokes' multiplier for divergent q-series! Very recently, we noted ((Zhang, 2002), (Ramis et al., 2003)) that this problem can be treated in a satisfactory manner by a new summation theory of divergent power series through the use of Jacobian theta functions and some basic integral calculus. The purpose of the present article is to explain how much “obvious” this mechanism of summation may be if one practises some elementary calculations on q-series. It would be a very interesting question to understand (Di Vizio et al., 2003) whether Ramanujan's mysterious formulas are related to this transcendental invariant analysis…