We study a recurrence relation, originating in combinatorial problems, where the generating function, as a formal power series, satisfies a differential equation that can be solved in a suitable domain; this yields an analytic function in a domain, but the solution is singular at the origin and the generating function has radius of convergence 0. Nevertheless, the solution to the recurrence can be obtained from the analytic solution by finding an asymptotic series expansion. Conversely, the analytic solution can be obtained by summing the generating function by the Borel summation method. This is an explicit example, which we study detail, of a behaviour known to be typical for a large class of holonomic functions. We also express the solution using Bessel functions and Lommel polynomials.

Source : oai:HAL:hal-00990430v1

Volume: Vol. 12 no. 2

Published on: January 1, 2010

Submitted on: March 26, 2015

Keywords: recurrence,divergent generating function,Borel summation,Bessel functions,Lommel polynomials,holonomic function,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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