# Department of Mathematics

## Orthogonal polynomials associated to a certain fourth order differential equation

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• Joachim Hilgert, Universität Paderborn, Germany
• Toshiyuki Kobayashi, University of Tokyo, Japan
• Gen Mano, University of Tokyo, Japan
• Jan Möllers
We introduce orthogonal polynomials $M_j^{\mu,\ell}(x)$ as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters $\mu\in\mathbb{C}$ and $\ell\in\mathbb{N}_0$.

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 $\ell=0$.

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.
Original language English Ramanujan Journal 26 3 295-310 16 1382-4090 https://doi.org/10.1007/s11139-011-9338-6 Published - 2011 Yes

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ID: 48397590