Orthogonal polynomials associated to a certain fourth order differential equation

Joachim Hilgert, Toshiyuki Kobayashi, Gen Mano, Jan Möllers

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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 $\mu\in\mathbb{C}$ 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 languageEnglish
JournalRamanujan Journal
Volume26
Issue3
Pages (from-to)295-310
Number of pages16
ISSN1382-4090
DOIs
Publication statusPublished - 2011
Externally publishedYes

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