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arXiv · 1503.07329

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

R. B. Paris
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Abstract

We consider the asymptotic expansion of the sum \[S_p(a;w)=\sum_{n=1}^\infty n^{-w}\e^{-an^p}\] as $a\rightarrow 0$ in $|\arg\,a|<π/2$ 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.

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R. B. Paris. 2015-03-25. The asymptotic expansion of a generalisation of the Euler-Jacobi series. https://arxiv.org/abs/1503.07329

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