##
J. Hilgert, T. Kobayashi, G. Mano, and J. Möllers,
*Special functions associated to a certain fourth order differential
equation*, Ramanujan Journal **26** (2011), no.1, 1-34. (published online August 31, 2011).
DOI: 10.1007/s11139-011-9315-0. arXiv:0907.2608
[math.CA]..

We develop a theory of 'special functions' associated to 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 ∈ 2**N**_{0} this operator extends to a self-adjoint operator on *L*^{2}(**R**_{+}, *x*^{μ+ν+1}d*x*) 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.

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© Toshiyuki Kobayashi