Exceptional Hahn and Jacobi orthogonal polynomials

• Published in: Journal of approximation theory Vol. 214; pp. 9 - 48
• Main Author: Durán, Antonio J.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-02-2017
• Subjects: Difference and differential operators; Exceptional polynomial; Hahn polynomials; Jacobi polynomials; Orthogonal polynomials; Jacobi polynomials; 33C45; 33E30; Exceptional polynomial; Orthogonal polynomials; Difference and differential operators; 42C05; Hahn polynomials
• ISSN: 0021-9045; 1096-0430
• DOI: 10.1016/j.jat.2016.11.003

Using Casorati determinants of Hahn polynomials (hnα,β,N)n, we construct for each pair F=(F1,F2) of finite sets of positive integers polynomials hnα,β,N;F, n∈σF, which are eigenfunctions of a second order difference operator, where σF is certain set of nonnegative integers, σF⊊︀N. When N∈N and α, β, N and F satisfy a suitable admissibility condition, we prove that the polynomials hnα,β,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 (Pnα,β)n. Under suitable conditions for α, β and F, these Wronskian type determinants turn out to be exceptional Jacobi polynomials.