Opposite power series

Abstract

In order to analyze the singularities of a power series function P(t) on the boundary of its convergent disc, we introduced the space Ω(P) of opposite power series in the opposite variable s=1/t, where P(t) was, mainly, the growth function (Poincaré series) for a finitely generated group or a monoid Saito (2010) [10]. In the present paper, forgetting about that geometric or combinatorial background, we study the space Ω(P) abstractly for any suitably tame power series P(t)∈C{t}. For the case when Ω(P) is a finite set and P(t) is meromorphic in a neighborhood of the closure of its convergent disc, we show a duality betweenΩ(P)and the highest order poles ofP(t)on the boundary of its convergent disc.