Rims - 1720 opposite Power Series

In order to analyze the singularities of a power series 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 [S1]. 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 neighbourhood of the closure of its convergent disc, we show a duality between the set Ω(P ) and the set of the highest order poles of P (t) on the boundary of its convergent disc.