Bialgebras of recursive sequences and combinatorial identities (Q696861)

An infinite sequence of elements of some fixed ground field satisfying a recursion relation of finite order is called a recursive sequence. Some background material on linearly recursive sequences is presented, and a simple framework for defining recursive sequences to study their algebraic properties is established in this paper. Certain bialgebra structures on linear spaces of recursive sequences are investigated. It is shown that by selecting suitable bases for these bialgebras an explicit formula for the coproduct can lead to interesting combinatorial identities. The paper concludes by providing some simple observations on the calculations of coproduct expansions, and by discussing directions for future research.