$d$-Orthogonal Analogs of Classical Orthogonal Polynomials

TL;DR

This paper explores a new class of polynomial systems that generalize classical orthogonal polynomials by being eigenfunctions of higher-order differential operators and orthogonal with respect to multiple measures, extending their theoretical framework.

Contribution

It introduces and studies $d$-orthogonal analogs of classical polynomials, including their hypergeometric representations, generating functions, and asymptotic formulas.

Findings

Derived hypergeometric representations of $d$-orthogonal polynomials

Established generating functions for these polynomial systems

Obtained Mehler-Heine type formulas for asymptotic analysis

Abstract

Classical orthogonal polynomial systems of Jacobi, Hermite and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they are the only systems on the real line with this property. Similar results hold for the discrete orthogonal polynomials. In a recent paper we introduced a natural class of polynomial systems whose members are the eigenfunctions of a differential operator of higher order and which are orthogonal with respect to d measures, rather than one. These polynomial systems, enjoy a number of properties which make them a natural analog of the classical orthogonal polynomials. In the present paper we continue their study. The most important new properties are their hypergeometric representations which allow us to derive their generating functions and in some cases also Mehler-Heine type formulas.