On the half of a Riordan array

The half of an infinite lower triangular matrix G = (g(n,k))(n,k >= 0) is defined to be the infinite lower triangular matrix G((1)) = (g(n,k)((1)))(n,k >= 0) such that g(n,k)((1)) = g(2n-k,n) for all n >= k >= 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.