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Finite, closed-form expressions for the partition function and for Euler, Bernoulli, and Stirling numbers

By Jerome Malenfant

Abstract

We find general solutions to the generating-function equation sum c_q^{(X)} z^q = F(z)^X, where X is a complex number and F(z) is a convergent power series with |F(0)| >0. We then use these results to derive finite expressions containing only integers or simple fractions for partition functions and for Euler, Bernoulli, and Stirling numbers.Comment: 19 page

Topics: Mathematics - Number Theory, 05A17 (Primary) 11Y35, 11B68 (Secondary)
Year: 2011
OAI identifier: oai:arXiv.org:1103.1585
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