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Lerch $Φ$ asymptotics

Adri B. Olde Daalhuis

Abstract

We use a Mellin-Barnes integral representation for the Lerch transcendent $Φ(z,s,a)$ 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.

Lerch $Φ$ asymptotics

Abstract

We use a Mellin-Barnes integral representation for the Lerch transcendent to obtain large asymptotic approximations. The simplest divergent asymptotic approximation terminates in the case that is an integer. For non-integer the asymptotic approximations consists of the sum of two series. The first one is in powers of and the second one is in powers of . 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.
Paper Structure (3 sections, 1 theorem, 32 equations, 1 table)

This paper contains 3 sections, 1 theorem, 32 equations, 1 table.

Table of Contents

  1. Introduction and summary
  2. Large $\boldsymbol{z}$ asymptotics
  3. A factorial series expansion

Key Result

Theorem 1

We take $|\arg(1+z)|\leq \pi$, $a$ and $s$ bounded complex numbers, $\operatorname{Re} a>0$. Let $N$ be a fixed integer with $N>\operatorname{Re} a$. Take $|z|$ large and let $M$ be a positive integer such that $|M-s+1|\approx |(N+1-a)\ln z|$. Then as $z\to\infty$ uniformly with respect to $\arg z\in[-\pi,\pi]$. The $A_n(z,s,a)$ are defined in FL2 and $n=1,2,3,\ldots$.

Theorems & Definitions (1)

  • Theorem 1