Generalized Riordan groups and zero generalized Pascal matrices

TL;DR

This paper explores generalized Riordan groups and their connection to zero generalized Pascal matrices, introducing new groups related to points at infinity and extending classical theorems like Lagrange inversion.

Contribution

It introduces new groups associated with matrices at infinity in the space of generalized Pascal matrices and extends the Lagrange inversion theorem to these groups.

Findings

Defined groups similar to generalized Riordan groups at infinity

Provided examples including q-binomial coefficients at q=-1 and Pascal modulo 2

Extended Lagrange inversion theorem to these new groups

Abstract

The generalized Riordan group consists of infinite lower triangular matrices that correspond to certain operators in the space of formal power series. Each such group contains the matrix (generalized Pascal matrix), elements of which are generalized binomial coefficients. Generalized Pascal matrices with non-negative elements form an infinite-dimensional vector space. The paper gives an idea of groups similar to the generalized Riordan groups, but associated with matrices, which in the space of generalized Pascal matrices correspond to the points at infinity; examples of such matrices are the matrix of $q$-binomial coefficients for $q=-1$ and the Pascal triangle modulo $2$. An analog of the Lagrange inversion theorem for these groups is given and the corresponding examples are considered.