Tauberian Theorem: Square-root singularity and Ramified covering of degree two

TL;DR

This paper proves a Tauberian theorem for power series with square root singularities, providing detailed asymptotic expansions of their coefficients, which is part of a broader study on random walks on free groups.

Contribution

It introduces a new Tauberian theorem specifically for power series with square root singularities, advancing asymptotic analysis techniques.

Findings

Provides asymptotic expansions to all orders for coefficients of such power series.

Establishes a foundation for analyzing passage probabilities of random walks on free groups.

Part of a triptych aiming to fully characterize these asymptotics.

Abstract

We prove a Tauberian theorem concerning power series admitting square root singularities. More precisely we give an asymptotic expansion to any order of the coefficients of a power series admitting square-root type singularities. This article is one of a triptych with [Che25b] and [Che25c] that aims at proving an asympototic expansion to any order of the passage probability of an irreducible finite range random walk on free groups.