On the $f$-Matrices of Pascal-like Triangles Defined by Riordan Arrays

TL;DR

This paper introduces and characterizes $f$-matrices linked to Pascal-like matrices derived from Riordan arrays, revealing their generating functions through continued fractions and their relation to $ ext{gamma}$- and $h$-matrices.

Contribution

It generalizes face matrices of simplices and hypercubes using Riordan arrays, providing explicit generating function expressions and transformation relationships.

Findings

Generated functions expressed via continued fractions.

Established transformations between $f$-matrices and $ ext{gamma}$-/$h$-matrices.

Unified framework for Pascal-like matrices and their face matrices.

Abstract

We define and characterize the $f$-matrices associated to Pascal-like matrices that are defined by ordinary and exponential Riordan arrays. These generalize the face matrices of simplices and hypercubes. Their generating functions can be expressed simply in terms of continued fractions, which are shown to be transformations of the generating functions of the corresponding $\gamma$- and $h$-matrices.