Exponentially small expansions associated with a generalised Mathieu series

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 $\lambda$ ($>0$) and $\gamma$ are even integers for large complex $a$ in the sector $|\arg a|<\pi /\lambda$. The asymptotics in this case consist of a {\it finite} algebraic expansion together with an infinite sequence of increasingly subdominant exponentially small expansions. When $\mu$ 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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References in corpus (1)

• On some series formed by values of the Riemann Zeta function