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 $m_n^{a,c;\F}$, $n\in σ_\F$, which are eigenfunctions of a second order difference operator, where $σ_\F$ is certain infinite set of nonnegative integers, $σ_\F \varsubsetneq \NN$. When $c$ and $\F$ satisfy a suitable admissibility condition, we prove that the polynomials $m_n^{a,c;\F}$, $n\in σ_\F$, 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.

arXiv admin note: substantial text overlap with arXiv:1309.1175

References in corpus (2)

• Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials
• Constructing bispectral orthogonal polynomials from the classical discrete families of Charlier, Meixner and Krawtchouk

Cited by in corpus (3)

• Multi-indexed Meixner and Little $q$-Jacobi (Laguerre) Polynomials
• Recurrence Relations for Wronskian Hermite Polynomials
• Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials