Rapidly Convergent Summation Formulas involving Stirling Series

2016/01/31 by Raphael Schumacher, Schumacher, Raphael
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #Primary 40G99 #Secondary 40A05 #math.NT #msc:40A05 #msc:40G99

paper · pdf · doi:10.48550/arxiv.1602.00336

arxiv created 2016/01/31 · arxiv updated 2016/02/02

Abstract

This paper presents a family of rapidly convergent summation formulas for various finite sums of analytic functions. These summation formulas are obtained by applying a series acceleration transformation involving Stirling numbers of the first kind to the asymptotic, but divergent, expressions for the corresponding sums coming from the Euler-Maclaurin summation formula. While it is well-known that the expressions obtained from the Euler-Maclaurin summation formula diverge, our summation formulas are all very rapidly convergent and thus computationally efficient.

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