Special functions associated with a certain fourth-order differential equation

Hilgert J, Kobayashi T, Mano G, Möllers J (2011)

Publication Type: Journal article

Publication year: 2011

Journal

Ramanujan Journal Springer Verlag (Germany)

Publisher: Springer Verlag (Germany)

Book Volume: 26

Pages Range: 1-34

Journal Issue: 1

DOI: 10.1007/s11139-011-9315-0

Abstract

We develop a theory of “special functions” associated with a certain fourth-order differential operator $D_{μ, ν}$

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 $D_{μ, ν}$

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.

Authors with CRIS profile

Jan Frahm Juniorprofessur für Mathematik (Darstellungstheorie und Operatoralgebren)

Involved external institutions

University of Tokyo

Japan (JP)

How to cite

APA:

Hilgert, J., Kobayashi, T., Mano, G., & Möllers, J. (2011). Special functions associated with a certain fourth-order differential equation. Ramanujan Journal, 26(1), 1-34. https://dx.doi.org/10.1007/s11139-011-9315-0

MLA:

Hilgert, Joachim, et al. "Special functions associated with a certain fourth-order differential equation." Ramanujan Journal 26.1 (2011): 1-34.

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