A vertical recursive relational approach to Riordan arrays is derived, and horizontal recursive relations are represented by the $A$ and $Z$ sequences. This vertically recursive approach provides a way to represent the entries of the Riordan array $(g,f)$ with a recursive linear combination of the coefficients of $g$. A matrix representation of the vertical recurrence relation is also given. The set of all these matrices form a group called the quasi-Riordan group. Extensions of the horizontal and vertical recurrence relations for the $c$- and $C$- Riordan arrays are defined graphically using Luke’s and Laguerre triangles. These extensions describe how to study the nonlinear recursive relations of the entries of some triangular matrices from the linear recursive relations of the Riordan array entries. In addition, the matrix representation of the vertically recursive relation of Riordan arrays provides transformations between low-order and high-order finite Riordan arrays. where $m$ order Riordan array is $(g,f)_m=(d_{n ,k})_{m\geq n,k\geq 0}$. provides a unified approach to constructing identities.
