Identities induced by Riordan arrays

Historically, there exist two versions of the Riordan array concept. The older one (better known as recursive matrix) consists of bi-infinite matrices (dn,k)n,k∈Z (k>n implies dn,k=0), deals with formal Laurent series and has been mainly used to study algebraic properties of such matrices. The more recent version consists of infinite, lower triangular arrays (dn,k)n,k∈N (k>n implies dn,k=0), deals with formal power series and has been used to study combinatorial problems. Here we show that every Riordan array induces two characteristic combinatorial sums in three parameters n,k,m∈Z. These parameters can be specialized and generate an indefinite number of other combinatorial identities which are valid in the bi-infinite realm of recursive matrices.