Exceptional Hahn and Jacobi orthogonal polynomials

Abstract

Using Casorati determinants of Hahn polynomials $(h_n^{\alpha,\beta,N})_n$, we construct for each pair $\F=(F_1,F_2)$ of finite sets of positive integers polynomials $h_n^{\alpha,\beta,N;\F}$, $n\in \sigma _\F$, which are eigenfunctions of a second order difference operator, where $\sigma _\F$ is certain set of nonnegative integers, $\sigma _\F \varsubsetneq \NN$. When $N\in \NN$ and $\alpha$, $\beta$, $N$ and $\F$ satisfy a suitable admissibility condition, we prove that the polynomials $h_n^{\alpha,\beta,N;\F}$ are also orthogonal and complete with respect to a positive measure (exceptional Hahn polynomials). By passing to the limit, we transform the Casorati determinant of Hahn polynomials into a Wronskian type determinant of Jacobi polynomials $(P_n^{\alpha,\beta})_n$. Under suitable conditions for $\alpha$, $\beta$ and $\F$, these Wronskian type determinants turn out to be exceptional Jacobi polynomials.