Asymptotics of a Mathieu-Gaussian series

Abstract

We consider the asymptotic expansion of the functional series \[S_{μ,γ}(a;λ)=\sum_{n=1}^\infty \frac{n^γe^{-λn^2/a^2}}{(n^2+a^2)^μ}\] for real values of the parameters $\gamma$, $\lambda >0$ and $\mu \ge 0$ as $|a|\to \infty$ in the sector $|\arg a|<\pi /4$. For general values of $\gamma$ the expansion is of algebraic type with terms involving the Riemann zeta function and a terminating confluent hypergeometric function. Of principal interest in this study is the case corresponding to even integer values of $\gamma$, where the algebraic-type expansion consists of a finite number of terms together with a contribution comprising an infinite sequence of increasingly subdominant exponentially small expansions. This situation is analogous to the well-known Poisson-Jacobi formula corresponding to the case $\mu =\gamma =0$. Numerical examples are provided to illustrate the accuracy of these expansions.

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References in corpus (2)

• Asymptotic expansions of several series and their application
• Asymptotic expoansions of mathieu-Bessel series. I

Cited by in corpus (1)

• The asymptotic expansion of a Mathieu-exponential series