Special functions associated to a certain fourth order differential equation

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 $\mu ,\nu \ge -1$ with $\mu +\nu \in 2{\mathbb{N}}_0$ 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.