Some classical multiple orthogonal polynomials

Abstract

A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. We study multiple orthogonal polynomials with respect to p > 1 weights satisfying Pearson's equation. In particular, we give a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators. We show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order p + 1. We also obtain explicit formulas and recurrence relations for these polynomials.

Asymptotic zero distribution for a class of multiple orthogonal polynomials

“…Recently they also appeared in random matrix theory for matrix ensembles with external source [3,7,8] and Wishard ensembles [6]. The particular limiting behavior which we are considering appears in the examples Jacobi-Piñeiro, Laguerre I [29] and the example associated with modified Bessel functions [30].…”

“…If all the multi-indices are normal, then the system of measures is called perfect. Some famous classes of perfect systems are the Angelesco systems, Nikishin systems (for r = 2) and AT systems; see, e.g., [25,29]. Multiple orthogonal polynomials of type II satisfy a recurrence relation of order r+1.…”

“…These are a generalization of orthogonal polynomials which arises naturally in Hermite-Padé approximation of a system of (Markov) functions [9,10,22]. In particular, they satisfy orthogonality conditions with respect to several positive measures [2,25,29]. Some of their applications are situated in diophantine number theory, rational approximation, spectral and scattering problems for higher-order difference equations and some associated dynamical systems; see, e.g., [5,11,18,27].…”

Structure Relations of Classical Multiple Orthogonal Polynomials by a Generating Function

“…More precisely, they are multiple Hermite polynomials, multiple Laguerre I polynomials, multiple Laguerre II polynomials, Jacobi-Piñeiro polynomials, and multiple Bessel polynomials (see [2,3,16,21] and references therein).…”

Matrix interpretation of multiple orthogonality

“…In this way, multiple orthogonal polynomials are intimately related to Hermite-Padé approximation. In the literature we can find a lot of examples of multiple orthogonal polynomials (see [1,2,3,13,17,19,24,25]). …”