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A divergent generating function that can be summed and analysed analytically
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
2010 (English)In: Discrete mathematics and theoretical computer science (Online), ISSN 1462-7264, E-ISSN 1365-8050, Vol. 12, no 2, 1-22 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2010. Vol. 12, no 2, 1-22 p.
Keyword [en]
recurrence, divergent generating function, Borel summation, Bessel functions, Lommel polynomials, holonomic function
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-137618ISI: 000276310200001OAI: oai:DiVA.org:uu-137618DiVA: diva2:378581
Available from: 2010-12-15 Created: 2010-12-15 Last updated: 2017-12-11Bibliographically approved

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