On the Asymptotic Expansion of the $q$-Dilogarithm

TL;DR

This paper investigates the asymptotic behavior of the $q$-dilogarithm function near $q=1$ and $q=0$ using Mellin transforms and Lerch's functional equation, providing detailed expansions.

Contribution

It introduces new asymptotic expansions of the $q$-dilogarithm at critical points $q=1$ and $q=0$ employing Mellin transform techniques and Lerch's functional equation.

Findings

Derived asymptotic expansions at $q=1$ and $q=0$

Applied Mellin transform to analyze the $q$-dilogarithm

Utilized Lerch's functional equation for decomposition

Abstract

In this work, we study some asymptotic expansion of the $q$-dilogarithm at $q=1$ and $q=0$ by using Mellin transform and adequate decomposition allowed by Lerch functional equation.