Exponentially small expansions associated with a generalised Mathieu series

R B Paris

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arXiv · 1601.07751

Exponentially small expansions associated with a generalised Mathieu series

R B Paris

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Abstract

We consider the generalised Mathieu series \[\sum_{n=1}^\infty \frac{n^γ}{(n^λ+a^λ)^μ}\qquad (μ>0)\] 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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R B Paris. 2016-01-28. Exponentially small expansions associated with a generalised Mathieu series. https://arxiv.org/abs/1601.07751

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