Exceptional Meixner and Laguerre orthogonal polynomials

Abstract

Using Casorati determinants of Meixner polynomials $(m_n^{a,c})_n$, we construct for each pair $\F=(F_1,F_2)$ of finite sets of positive integers a sequence of polynomials $m_n^{a,c;\F}$, $n\in \sigma_\F$, which are eigenfunctions of a second order difference operator, where $\sigma_\F$ is certain infinite set of nonnegative integers, $\sigma_\F \varsubsetneq \NN$. When $c$ and $\F$ satisfy a suitable admissibility condition, we prove that the polynomials $m_n^{a,c;\F}$, $n\in \sigma_\F$, are actually exceptional Meixner polynomials; that is, in addition, they are orthogonal and complete with respect to a positive measure. By passing to the limit, we transform the Casorati determinant of Meixner polynomials into a Wronskian type determinant of Laguerre polynomials $(L_n^\alpha)_n$. Under the admissibility conditions for $\F$ and $\alpha$, these Wronskian type determinants turn out to be exceptional Laguerre polynomials.