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.

References in corpus (1)

• Special functions associated to a certain fourth order differential equation

Cited by in corpus (2)

• Minimal representations via Bessel operators
• Varna Lecture on L^2-Analysis of Minimal Representations