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

Paul Barry

TL;DR

The paper reframes Fuss-Catalan numbers through the Riordan group, introducing Fuss-Catalan-Riordan arrays built from a family of $d$-orthogonal polynomials and linking them to lattice-path models. It derives a closed-form for the pre-Fuss-Catalan-Riordan entries via Lagrange inversion, $\tau_{n,k}=\frac{rk+1}{(r-1)n+k+1}\binom{rn}{n-k}$, and shows that rectification yields the Fuss-Catalan matrix with $FC(n,k;r)=\frac{rk+1}{rn+k+1}\binom{rn+k+1}{n}$. The work further expresses these objects as products with the transpose of the binomial matrix, highlights their production matrices (often banded, signaling $d$-orthogonality), and provides lattice-path interpretations. It also discusses parameterized generalizations and potential extensions to Qi-number frameworks, suggesting a unifying approach to combinatorial sequences, polynomial families, and path counting within a Riordan-theoretic setting.

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.

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

TL;DR

The paper reframes Fuss-Catalan numbers through the Riordan group, introducing Fuss-Catalan-Riordan arrays built from a family of -orthogonal polynomials and linking them to lattice-path models. It derives a closed-form for the pre-Fuss-Catalan-Riordan entries via Lagrange inversion, , and shows that rectification yields the Fuss-Catalan matrix with . The work further expresses these objects as products with the transpose of the binomial matrix, highlights their production matrices (often banded, signaling -orthogonality), and provides lattice-path interpretations. It also discusses parameterized generalizations and potential extensions to Qi-number frameworks, suggesting a unifying approach to combinatorial sequences, polynomial families, and path counting within a Riordan-theoretic setting.

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 -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.

Paper Structure

This paper contains 10 sections, 14 theorems, 170 equations, 2 figures.

Table of Contents

  1. Preliminaries on generalized Catalan sequences and their generating function
  2. Preliminaries on Riordan arrays
  3. Preliminaries on constant coefficient $d$-orthogonal polynomials
  4. Special $d$-orthogonal polynomials
  5. The general pre Fuss-Catalan-Riordan matrix
  6. The Fuss-Catalan numbers: definition via Riordan arrays and examples
  7. More on the general case
  8. Lattice paths
  9. Further directions
  10. Conclusions

Key Result

Proposition 1

The number of lattice paths from $(0,0)$ to $(rn,0)$ consisting of $(r-1)n$ up steps $(1,1)$ and $n$ down steps $(1,1-r)$ that never go below the $x$-axis is the Fuss-Catalan number

Figures (2)

  • Figure 1: The Fuss-Catalan-Riordan array (for $r=3$) $(t(x), xt(x))$ as a path matrix
  • Figure 2: The Fuss-Catalan matrix for $r=3$$(t(x), t(x))$ as a path matrix

Theorems & Definitions (34)

  • Proposition 1: Raney
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • Proposition 5
  • Example 6
  • Example 7
  • proof
  • ...and 24 more