Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 023, 9 pages      arXiv:2311.11886      https://doi.org/10.3842/SIGMA.2024.023
Contribution to the Special Issue on Asymptotics and Applications of Special Functions in Memory of Richard Paris

Lerch $\Phi$ Asymptotics

Adri B. Olde Daalhuis
School of Mathematics and Maxwell Institute for Mathematical Sciences, The University of Edinburgh, Edinburgh EH9 3FD, UK

Received November 22, 2023, in final form March 11, 2024; Published online March 21, 2024

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.

Key words: Hurwitz-Lerch zeta function; analytic continuation; asymptotic expansions.

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