  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.
