Opposite power series

TL;DR

This paper explores the relationship between the singularities of power series and the oscillation of their coefficient ratios, introducing the space of opposite power series to analyze this resonance, with applications to the modular group.

Contribution

It introduces the space of opposite power series and establishes a link between singularities and coefficient oscillations in tame power series.

Findings

Resonance between singularities and oscillation behavior of coefficient ratios.

Introduction of the space of opposite power series as a new analytical tool.

Application to the growth series of the modular group PSL(2,Z).

Abstract

Let $\gamma_n$ ($n\in \mathbb{Z}_{\ge0}$) be a sequence of complex numbers, which is tame: $0<\exists u\le \gamma_{n-1}/\gamma_n \le \exists v<\infty$ for all $n>0$. We show a resonance between the singularities of the function of the power series $P(t):=\sum_{n=0}^\infty \gamma_n t^n$ on its boundary of the disc of convergence and the oscillation behavior of the sequences $\gamma_{n-k}/\gamma_n$ ($n\in \mathbb{Z}_{>>0}$) for $k>0$. The resonance is proven by introducing the space of opposite power series, which is the compact subspace of the space of all formal power series in the opposite variable $s=1/t$ and is defined as the accumulating set of the sequence $X_n(s):=\sum_{k=0}^n\frac{\gamma_{n-k}}{\gamma_n}t^k$ ($n\in \mathbb{Z}_{\ge0}$). We analyze in details an example of the growth series $P(t)$ for the modular group $PSL(2,Z)$ due to Machi.