Skip to Main content Skip to Navigation
Journal articles

A divergent generating function that can be summed and analysed analytically

Abstract : 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.
Document type :
Journal articles
Complete list of metadatas

Cited literature [18 references]  Display  Hide  Download

https://hal.inria.fr/hal-00990430
Contributor : Service Ist Inria Sophia Antipolis-Méditerranée / I3s <>
Submitted on : Tuesday, May 13, 2014 - 3:36:37 PM
Last modification on : Wednesday, November 29, 2017 - 10:26:17 AM
Long-term archiving on: : Monday, April 10, 2017 - 10:07:16 PM

File

1305-4850-1-PB.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-00990430, version 1

Collections

Citation

Svante Janson. A divergent generating function that can be summed and analysed analytically. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2010, 12 (2), pp.1-22. ⟨hal-00990430⟩

Share

Metrics

Record views

120

Files downloads

564