Some applications of the (p,r,s) halves of the Riordan arrays

A Riordan array is provided with (hn,k)n,k≥0. Its horizontal half (h2n,n+k)n,k≥0, vertical half (h2n−k,n)n,k≥0, (p,r) horizontal half (hpn+r,(p−1)n+k+r)n,k≥0, (p,r) vertical half (hpn−k+r,(p−1)n+r)n,k≥0, skew (r,s) half (h2n+(s−2)k+r,n+(s−1)k+r)n,k≥0, and (p,r,s) half h(p+1)n+(ps−p−1)k+r,pn+(ps−p)k+r have previously been considered. In this paper, we take into consideration the (p,r,s) halves of the Riordan arrays by replacing p with p−1 and s with [Formula presented] where q is a nonnegative integer. The generic term for this half of H is hpn+(q−2)k+r,(p−1)n+(q−1)k+r. Furthermore, the A and Z-sequences of them are given. In addition, some combinatorial identities are found by utilizing this half of the Riordan arrays. We also investigate the Riordan antecedent of a given Riordan array.