Multiple orthogonal polynomials, $d$-orthogonal polynomials, production matrices, and branched continued fractions

Abstract

I analyze an unexpected connection between multiple orthogonal polynomials, $d$-orthogonal polynomials, production matrices and branched continued fractions. This work can be viewed as a partial extension of Viennot's combinatorial theory of orthogonal polynomials to the case where the production matrix is lower-Hessenberg but is not necessarily tridiagonal.

arXiv comment41 pages, LaTeX2e. Version 2 corrects a few small typos, uses the term "coefficient matrix" instead of "representing matrix" for a sequence of monic polynomials, and adds two references. To be published in the Transactions of the AMS

References in corpus (8)

• Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits
• Q-systems, Heaps, Paths and Cluster Positivity
• Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights
• Pade and Hermite-Pade approximation and orthogonality
• What is $\dots$ a multiple orthogonal polynomial?
• Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions
• Total positivity of some polynomial matrices that enumerate labeled trees and forests, I. Forests of rooted labeled trees
• Higher order matching polynomials and d-orthogonality

Cited by in corpus (4)

• Lattice paths and branched continued fractions. III. Generalizations of the Laguerre, rook and Lah polynomials
• A MATLAB package computing simultaneous Gaussian quadrature rules for Multiple Orthogonal Polynomials
• Multiple Laguerre polynomials: Combinatorial model and Stieltjes moment representation
• Bidiagonal matrix factorisations associated with symmetric multiple orthogonal polynomials and lattice paths