Special functions associated to a certain fourth order differential equation

TL;DR

This paper develops a theory of special functions linked to a specific fourth order differential operator, revealing their properties and connections to minimal representations of orthogonal groups.

Contribution

It introduces a new class of special functions associated with a fourth order operator, including explicit formulas, orthogonality, and their role in representation theory.

Findings

Derived generating functions for eigenfunctions.

Established orthogonality and completeness properties.

Connected functions to minimal representations of O(p,q).

Abstract

We develop a theory of "special functions" associated to a certain fourth order differential operator $\mathcal{D}_{\mu,\nu}$ on $\mathbb{R}$ depending on two parameters $\mu,\nu$. For integers $\mu,\nu\geq-1$ with $\mu+\nu\in2\mathbb{N}_0$ this operator extends to a self-adjoint operator on $L^2(\mathbb{R}_+,x^{\mu+\nu+1}dx)$ with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, $L^2$-norms, integral representations and various recurrence relations. This fourth order differential operator $\mathcal{D}_{\mu,\nu}$ arises as the radial part of the Casimir action in the Schr\"odinger model of the minimal representation of the group $O(p,q)$, and our "special functions" give $K$-finite vectors.