On the Half of a Riordan Array

Abstract

The half of an infinite lower triangular matrix \(G = (g_{n,k})_{n,k\geq 0}\) is defined to be the infinite lower triangular matrix \(G^{(1)} = (g^{(1)}_{n,k \geq 0})\) such that \(g^{(1)}_{n,k} = g_{2n-k,n}\) for all \(n \geq k \geq 0\). In this paper, we will show that if \(G\) is a Riordan array, then its half \(G^{(1)}\) is also a Riordan array. We use Lagrange inversion theorem to characterize the generating functions of \(G^{(1)}\) in terms of the generating functions of \(G\). Consequently, a tight relation between \(G^{(1)}\) and the initial array \(G\) is given, hence it is possible to invert the process and rebuild the original Riordan array \(G\) from the array \(G^{(1)}\). If the process of taking half of a Riordan array \(G\) is iterated \(r\) times, then we obtain a Riordan array \(G^{(r)}\). The further relation between the result array \(G^{(r)}\) and the initial array \(G\) is also considered. Some examples and applications are presented.