Opposite power series

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 between Ω ( P ) and the highest order poles of P ( t ) on the boundary of its convergent disc.