Special functions associated with a certain fourth-order differential equation - Hilgert, Joachim; Kobayashi, Toshiyuki; Mano, Gen; Möllers, Jan (Ramanujan Journal, 2011)
We develop a theory of “special functions” associated with 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 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 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.
Keywords: Fourth-order differential equations; Generating functions; Bessel functions; Orthogonal polynomials; Laguerre polynomials; Recurrence relations; Meijer’s G-function; Minimal representation; Indefinite orthogonal group