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 $\mathcal{D}_{μ,ν}$ on $\mathbb{R}$ depending on two parameters $μ,ν$. For integers $μ,ν\geq-1$ with $μ+ν\in2\mathbb{N}_0$ this operator extends to a self-adjoint operator on $L^2(\mathbb{R}_+,x^{μ+ν+1}dx)$ 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 $\mathcal{D}_{μ,ν}$ arises as the radial part of the Casimir action in the Schrödinger model of the minimal representation of the group $O(p,q)$, and our "special functions" give $K$-finite vectors.

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