On Kernel polynomials and self-perturbation of orthogonal polynomials

Abstract

Given an orthogonal polynomial system {Q n (x)} n=0 ∞, define another polynomial system by $P_{n}(x)=Q_{n}(x)-\alpha_{n}Q_{n-t}(x),~~ n\geq 0, $ where α n are complex numbers and t is a positive integer. We find conditions for {P n (x)} n=0 ∞ to be an orthogonal polynomial system. When t=1 and α1≠0, it turns out that {Q n (x)} n=0 ∞ must be kernel polynomials for {P n (x)} n=0 ∞ for which we study, in detail, the location of zeros and semi-classical character.