Constructing Krall–Hahn orthogonal polynomials

Abstract

Given a sequence of polynomials (pn)n, an algebra of operators A that acts in the linear space of polynomials and an operator Dp∈A with Dp(pn)=θnpn, where θn is any arbitrary eigenvalue, we construct a new sequence of polynomials (qn)n by considering a linear combination of m+1 consecutive pn: qn=pn+∑j=1mβn,jpn−j. Using the concept of a D-operator, we determine the structure of the sequences βn,j, j=1,…,m, such that the polynomials (qn)n are eigenfunctions of an operator in the algebra A. As an application, from the classical discrete family of Hahn polynomials, we construct orthogonal polynomials (qn)n that are also eigenfunctions of higher-order difference operators.