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arXiv · 2311.11886

Lerch $\Phi$ asymptotics

Adri B. Olde Daalhuis
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Abstract

We use a Mellin-Barnes integral representation for the Lerch transcendent $\Phi(z,s,a)$ to obtain large $z$ asymptotic approximations. The simplest divergent asymptotic approximation terminates in the case that $s$ is an integer. For non-integer $s$ the asymptotic approximations consists of the sum of two series. The first one is in powers of $(\ln z)^{-1}$ and the second one is in powers of $z^{-1}$. Although the second series converges, it is completely hidden in the divergent tail of the first series. We use resummation and optimal truncation to make the second series visible.

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BibTeXRIS

Adri B. Olde Daalhuis. 2023-11-20. Lerch $\Phi$ asymptotics. https://doi.org/10.3842/sigma.2024.023

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