The skew halves of a Riordan array

Abstract

Given a Riordan array (gn,k)n,k∈N, its vertical half (g2n−k,n)n,k∈N and horizontal half (g2n,n+k)n,k∈N are studied separately before. In the present paper, we introduce the skew (r,s)-halves of a Riordan array which are infinite lower triangular matrices with generic (n,k)-th entries g2n+(s−2)k+r,n+(s−1)k+r for n≥k≥0. This allows us to discuss the vertical half and horizontal half in a uniform context. We show that the skew halves of a Riordan array are all Riordan arrays. As applications, we find several new identities involving the Pascal matrix, and Catalan triangles by applying the skew halves. We also consider the inversion problems: given a Riordan array G, we can construct its Riordan antecedent H such that the (r,s)-half of H is equal to G.