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Bidiagonal matrix factorisations associated with symmetric multiple orthogonal polynomials and lattice paths

Hélder Lima

TL;DR

This work links linear-algebraic structure and combinatorics by showing that infinite banded Hessenberg matrices $\mathrm{H}^{(r;j)}$ with a bidiagonal factorisation act both as recurrence matrices for the components of the $(r+1)$-fold symmetric $r$-orthogonal polynomials and as production matrices for generating polynomials of partial $r$-Dyck paths. By integrating bidiagonal factorizations, multiple-orthogonality, lattice-path theory, and branched continued fractions, the authors obtain a decomposition of $(r+1)$-fold symmetric $r$-OPS on a star-like set and construct corresponding orthogonality measures on the star and on $\mathbb{R}^+$. They provide explicit recurrence coefficients and an Appell-type example with Meijer $G$-density representations, together with total-positivity results for the associated polynomial families. The results reveal a deep, computable bridge between Hessenberg recurrences, generalized Stieltjes/Jacobi-Rogers polynomials, and lattice-path combinatorics, with potential implications for spectral theory and branched continued fractions.

Abstract

The central object of study in this paper are infinite banded Hessenberg matrices admitting factorisations as products of bidiagonal matrices. In the two main novel results of this paper, we show that these Hessenberg matrices are associated with the decomposition of $(r+1)$-fold symmetric $r$-orthogonal polynomials and are the production matrices of the generating polynomials of $r$-Dyck paths. We combine the aforementioned bidiagonal matrix factorisations and the recently found connection of multiple orthogonal polynomials with lattice paths and branched continued fractions to study $(r+1)$-fold symmetric $r$-orthogonal polynomials on a star-like set of the complex plane and their decomposition via multiple orthogonal polynomials on the positive real line. As an explicit example, we give formulas as terminating hypergeometric series for the Appell sequences of $(r+1)$-fold symmetric $r$-orthogonal polynomials on a star-like set and show that the densities of their orthogonality measures can be expressed via Meijer G-functions on the positive real line.

Bidiagonal matrix factorisations associated with symmetric multiple orthogonal polynomials and lattice paths

TL;DR

This work links linear-algebraic structure and combinatorics by showing that infinite banded Hessenberg matrices with a bidiagonal factorisation act both as recurrence matrices for the components of the -fold symmetric -orthogonal polynomials and as production matrices for generating polynomials of partial -Dyck paths. By integrating bidiagonal factorizations, multiple-orthogonality, lattice-path theory, and branched continued fractions, the authors obtain a decomposition of -fold symmetric -OPS on a star-like set and construct corresponding orthogonality measures on the star and on . They provide explicit recurrence coefficients and an Appell-type example with Meijer -density representations, together with total-positivity results for the associated polynomial families. The results reveal a deep, computable bridge between Hessenberg recurrences, generalized Stieltjes/Jacobi-Rogers polynomials, and lattice-path combinatorics, with potential implications for spectral theory and branched continued fractions.

Abstract

The central object of study in this paper are infinite banded Hessenberg matrices admitting factorisations as products of bidiagonal matrices. In the two main novel results of this paper, we show that these Hessenberg matrices are associated with the decomposition of -fold symmetric -orthogonal polynomials and are the production matrices of the generating polynomials of -Dyck paths. We combine the aforementioned bidiagonal matrix factorisations and the recently found connection of multiple orthogonal polynomials with lattice paths and branched continued fractions to study -fold symmetric -orthogonal polynomials on a star-like set of the complex plane and their decomposition via multiple orthogonal polynomials on the positive real line. As an explicit example, we give formulas as terminating hypergeometric series for the Appell sequences of -fold symmetric -orthogonal polynomials on a star-like set and show that the densities of their orthogonality measures can be expressed via Meijer G-functions on the positive real line.
Paper Structure (11 sections, 13 theorems, 75 equations, 2 figures)

This paper contains 11 sections, 13 theorems, 75 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Multiple orthogonality and $m$-fold symmetry
  4. Lattice paths and production matrices
  5. Bidiagonal matrix factorisations for banded Hessenberg matrices
  6. Hessenberg matrices associated with symmetric multiple orthogonal polynomials
  7. Production matrices of generating polynomials of lattice paths
  8. On $(r+1)$-fold symmetric $r$-orthogonal polynomials
  9. Location and interlacing properties of the zeros
  10. Orthogonality measures on a star-like set and on the positive real line
  11. An explicit example involving hypergeometric Appell polynomials

Key Result

Theorem 1

For $r\in\mathbb{Z}^+$ and a sequence $\left(\alpha_n\right)_{n\in\mathbb{N}}$ of nonzero elements in a field $\mathbb{K}$, let: Then, for any $0\leq j\leq r$, the recurrence matrix for $\left(P_n^{[j]}(x)\right)_{n\in\mathbb{N}}$ is the infinite $(r+2)$-banded unit-lower-Hessenberg matrix $\mathrm{H}^{(r;j)}$ admitting the factorisation products of lower bidiagonal matrices with an upper bidiago

Figures (2)

  • Figure 1: $3$-star, $4$-star, and $5$-star (from left to right).
  • Figure 2: A $2$-Dyck path of length $12$ (left) and a $3$-Lukasiewick path of length $13$ (right).

Theorems & Definitions (23)

  • Theorem 1
  • proof
  • Corollary 2
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5
  • Corollary 6
  • Proposition 1
  • ...and 13 more