Ladder operators and differential equations for multiple orthogonal polynomials

TL;DR

This paper develops ladder operators and compatibility conditions for multiple orthogonal polynomials, extending classical results and enabling derivation of their differential equations using Riemann-Hilbert problems and recurrence relations.

Contribution

It introduces new ladder operators and compatibility conditions for multiple orthogonal polynomials, expanding the theoretical framework beyond classical orthogonal polynomials.

Findings

Derived ladder operators for multiple orthogonal polynomials

Established compatibility conditions and differential equations

Provided explicit examples with Hermite and Laguerre polynomials

Abstract

In this paper, we obtain the ladder operators and associated compatibility conditions for the type I and the type II multiple orthogonal polynomials. These ladder equations extend known results for orthogonal polynomials and can be used to derive the differential equations satisfied by multiple orthogonal polynomials. Our approach is based on Riemann-Hilbert problems and the Christoffel-Darboux formula for multiple orthogonal polynomials, and the nearest-neighbor recurrence relations. As an illustration, we give several explicit examples involving multiple Hermite and Laguerre polynomials, and multiple orthogonal polynomials with exponential weights and cubic potentials.