Abstract
We introduce orthogonal polynomials \(M_{j}^{\mu,\ell}(x)\) as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters μ∈ℂ and ℓ∈ℕ0.
These polynomials arise as K-finite vectors in the L 2-model of the minimal unitary representations of indefinite orthogonal groups, and reduce to the classical Laguerre polynomials \(L_{j}^{\mu}(x)\) for ℓ=0.
We establish various recurrence relations and integral representations for our polynomials, as well as a closed formula for the L 2-norm. Further we show that they are uniquely determined as polynomial eigenfunctions.
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T. Kobayashi was partially supported by Grant-in-Aid for Scientific Research (B) (18340037, 22340026), Japan Society for the Promotion of Science, and the Alexander Humboldt Foundation.
J. Möllers was partially supported by the International Research Training Group 1133 “Geometry and Analysis of Symmetries”, and the GCOE program of the University of Tokyo.
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Hilgert, J., Kobayashi, T., Mano, G. et al. Orthogonal polynomials associated to a certain fourth order differential equation. Ramanujan J 26, 295–310 (2011). https://doi.org/10.1007/s11139-011-9338-6
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DOI: https://doi.org/10.1007/s11139-011-9338-6
Keywords
- Orthogonal polynomials
- Generating functions
- Bessel functions
- Laguerre polynomials
- Fourth order
- Recurrence formulas
- Minimal representation
- Meijer’s G-function