Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

INVERSE FACTORIAL-SERIES SOLUTIONS OF DIFFERENCE EQUATIONS

A. B. Olde Daalhuisa1

a1 School of Mathematics, King’s Buildings, University of Edinburgh, Edinburgh EH9 3JZ, UK (adri@maths.ed.ac.uk)

Abstract

We obtain inverse factorial-series solutions of second-order linear difference equations with a singularity of rank one at infinity. It is shown that the Borel plane of these series is relatively simple, and that in certain cases the asymptotic expansions incorporate simple resurgence properties. Two examples are included. The second example is the large $a$ asymptotics of the hypergeometric function ${}_2F_1(a,b;c;x)$.

AMS 2000 Mathematics subject classification: Primary 34E05; 39A11. Secondary 33C05

(Received July 08 2003)

Keywords

  • asymptotic expansions;
  • Borel transform;
  • difference equations;
  • exponentially improved asymptotics;
  • factorial series;
  • hypergeometric functions