Orthogonal polynomials associated to a certain fourth order differential equation

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.