Opposite power series

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.