Exponentially small expansions associated with a generalised Mathieu series

2016/01/28 by R B Paris, Paris, R B
Mathematics · #30E15 #30E20 #34E05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:30E15 #msc:30E20 #msc:34E05

paper · pdf · doi:10.48550/arxiv.1601.07751

15 pages, 0 figures

arxiv created 2016/01/28 · arxiv updated 2016/01/29

Abstract

We consider the generalised Mathieu series ∑_n=1^∞ (n^γ)/((n^λ+a^λ)^μ) (μ>0) when the parameters λ (>0) and γ are even integers for large complex a in the sector |arg a|<π/λ. The asymptotics in this case consist of a \it finite algebraic expansion together with an infinite sequence of increasingly subdominant exponentially small expansions. When μ is also a positive integer it is possible to give closed-form evaluations of this series. Numerical results are given to illustrate the accuracy of the expansion obtained.

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