The asymptotic expansion of a generalisation of the Euler-Jacobi series

Abstract

We consider the asymptotic expansion of the sum \[S_p(a;w)=\sum_{n=1}^\infty n^{-w}\e^{-an^p}\] as $a\to 0$ in $|\arg a|<\pi /2$ for arbitrary finite $p>$ and $w>0$. Our attention is concentrated mainly on the case when $p$ and $w$ are both even integers, where the expansion consists of a {\it finite} algebraic expansion together with a sequence of increasingly subdominant exponential expansions. This exponentially small component produces a transformation for $S_p(a;w)$ analogous to the well-known Poisson-Jacobi transformation for the sum with $p=2$ and $w=0$. Numerical results are given to illustrate the accuracy of the expansion obtained.

15 pages, 0 figures, 3 tables

References in corpus (1)

• A Poisson-Jacobi-type transformation for the sum $\sum_{n=1}^\infty n^{-2m} \exp (-an^2}$ for positive integer $m$

Cited by in corpus (1)

• Asymptotics of a sum of modified Bessel functions with non-linear argument