One-$p$th Riordan Arrays in the Construction of Identities

Abstract:

      For an integer $p\geq 2$ we construct vertical and horizontal one-$p$th Riordan arrays from a Riordan array. When $p=2$ one-$p$th Riordan arrays are reduced to well known half Riordan arrays. The generating functions of the $A$-sequences of vertical and horizontal one-$p$th Riordan arrays are found. The vertical and horizontal one-$p$th Riordan arrays provide an approach to construct many identities. They can also be used to verify some well known identities readily.