Bell numbers, their relatives, and algebraic differential equations

Authors:
Martin Klazar
Affiliations:
Department of Applied Mathematics, KAM, Charles University, Malostranské and Institute for Theoretical Computer Science, ITI, Charles University, Malostranské námestí 25, 118 0 ...
Venue:
Journal of Combinatorial Theory Series A
Year:
2003

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Abstract

We prove that the ordinary generating function of Bell numbers satisfies no algebraic differential equation over C(x) (in fact, over a larger field). We investigate related numbers counting various set partitions (the Uppuluri-Carpenter numbers, the numbers of partitions with j mod i blocks, the Bessel numbers, the numbers of connected partitions, and the numbers of crossing partitions) and prove for their ogf's analogous results. Recurrences, functional equations, and continued fraction expansions are derived.