On the Half of a Riordan Array

Abstract

The half of an infinite lower triangular matrix $G=(g_{n,k}{)}_{n,k\ge 0}$ is defined to be the infinite lower triangular matrix $G^{(1)}=(g_{n,k\ge 0}^{(1)})$ such that $g_{n,k}^{(1)}=g_{2n-k,n}$ for all $n\ge k\ge 0$. 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.