Remarks on Some Basic Hypergeometric Series

  • Changgui Zhang
Part of the Developments in Mathematics book series (DEVM, volume 13)


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…


Power Series Theta Function Laurent Series Removable Singularity Convergent Power Series 
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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Changgui Zhang
    • 1
  1. 1.Laboratoire AGAT (UMR — CNRS 8524) UFR Math.Université des Sciences et Technologies de LilleVilleneuve d'Ascq cedexFrance

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