Generalized Riordan arrays and zero generalized Pascal matrices

TL;DR

This paper explores generalized Riordan arrays and zero generalized Pascal matrices, expanding the concept to include limiting cases and establishing their group properties, bridging different types of binomial coefficient-based matrices.

Contribution

It introduces a new class of matrices as limits of generalized Pascal matrices and extends the Riordan array framework to include these cases.

Findings

Generalized Pascal matrices are part of the generalized Riordan array group.

A special class of matrices is identified as limits of generalized Pascal matrices.

These matrices are shown to belong to a matrix group similar to the generalized Riordan group.

Abstract

Generalized Pascal matrix whose elements are generalized binomial coefficients is included in the group of generalized Riordan arrays. There is a special set of generalized Riordan arrays defined by parameter $q$. If $q=0$, they are ordinary Riordan arrays, if $q=1$, they are exponential Riordan arrays. In other cases, except $q=-1$, they are arrays associated with the $q$-binomial coefficients as well as the exponential Riordan arrays are associated with the ordinary binomial coefficients. Case $q=-1$ does not fit into the concept of generalized Riordan arrays, but it is necessary to expand for it. Introduced a special class of matrices, each of which is a limiting case of a certain set of generalized Pascal matrices. It is shown that every such matrix included in the matrix group similar to the generalized Riordan group.