We develop a theory of 'special functions' associated to a certain fourth order differential operator Dμ,ν 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 L2(R+, xμ+ν+1dx) 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, L2-norms, integral representations and various recurrence relations. This fourth order differential operator Dμ,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.
© Toshiyuki Kobayashi