Special functions associated with a certain fourth-order differential equation

Abstract

We develop a theory of “special functions” associated with a certain fourth-order differential operator $\mathcal{D}_{\mu,\nu}$ on ℝ depending on two parameters μ,ν. For integers μ,ν≥−1 with μ+ν∈2ℕ₀, this operator extends to a self-adjoint operator on L ²(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 $\mathcal{D}_{\mu,\nu}$ 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.