Orthogonal polynomials associated to a certain fourth order differential equation

We introduce orthogonal polynomials $M_j^{\mu ,\mathrm{\ell }}(x)$ as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters $\mu \in \mathbb{C}$ and $\mathrm{\ell }\in {\mathbb{N}}_0$. 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 $\mathrm{\ell }=0$. 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.