Some Polynomially Recursive Sequences and Combinatorial Identities

Abstract

A polynomially recursive sequence satisfies a recursive relation with variable coefficients. The set of these sequences has the structure of a topological bialgebra. If such a sequence is of a combinatorial nature, a formula for its coproduct can (upon appropriate evaluation) be interpreted as a combinatorial identity. Here we give a coproduct formula for each sequence , one for each t ≥ 0, and its interpretation as a combinatorial identity. We also obtain a q-version of this coproduct formula and combinatorial identity.