$d$-orthogonal polynomials, Fuss-Catalan matrices and lattice paths

TL;DR

This paper introduces Fuss-Catalan-Riordan arrays derived from $d$-orthogonal polynomials, linking them to Fuss-Catalan numbers and lattice paths, and explores their production matrices.

Contribution

It defines a new class of Riordan arrays based on $d$-orthogonal polynomials and connects them to combinatorial structures like Fuss-Catalan numbers and lattice paths.

Findings

Fuss-Catalan-Riordan arrays are constructed from $d$-orthogonal polynomials

These arrays relate to Fuss-Catalan numbers and specific lattice paths

Production matrices play a key role in understanding these arrays

Abstract

In this note, we show how to define certain Riordan arrays, that we call the Fuss-Catalan-Riordan arrays, by means of a special family of $d$-orthogonal polynomials. We relate the Fuss-Catalan Riordan arrays to the Fuss Catalan numbers, and to certain lattice paths. We emphasise the role of the production matrices of the Riordan arrays that we encounter in our study.