A combinatorial equivalence relation for formal power series

In this paper we examine an equivalence relation on the set of formal power series with nonzero constant term. This is done both in terms of functional equations and also by interlacing two concepts from Riordan group theory, the A-sequence and the Bell subgroup. The best known example gives an equivalence class{1+z,1/(1−z),C(z),T(z),Q(z),…}{1+z,1/(1−z),C(z),T(z),Q(z),…} where C(z),T(z)C(z),T(z) and Q(z)Q(z) are generating functions of the Catalan numbers, ternary numbers and quaternary numbers, respectively. A power series for one member of an equivalence class can be transformed into power series for the rest of members in the equivalence class and interpretations in terms of weighted lattice paths can also be given.