Orthogonal polynomials associated to a certain fourth order differential equation

TL;DR

This paper introduces a new family of orthogonal polynomials linked to a specific fourth order differential operator, extending classical Laguerre polynomials, with detailed properties and applications in representation theory.

Contribution

The paper defines and analyzes a novel class of orthogonal polynomials associated with a fourth order differential operator, connecting them to minimal unitary representations of orthogonal groups.

Findings

Polynomials are eigenfunctions of a self-adjoint fourth order differential operator.

They generalize classical Laguerre polynomials for certain parameters.

Explicit recurrence relations, integral representations, and norm formulas are established.

Abstract

We introduce orthogonal polynomials $M_j^{\mu,\ell}(x)$ as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters $\mu\in\mathbb{C}$ and $\ell\in\mathbb{N}_0$. These polynomials arise as $K$-finite vectors in the $L^2$-model of the minimal unitary representations of indefinite orthogonal groups, and reduce to the classical Laguerre polynomials $L_j^\mu(x)$ for $\ell=0$. We establish various recurrence relations and integral representations for our polynomials, as well as a closed formula for the $L^2$-norm. Further we show that they are uniquely determined as polynomial eigenfunctions.