An Exploration of Lacunary Orthogonal Polynomials Associated with the Geronimus Transformation

Abstract

In this note, we present two perturbations of sequences of orthogonal polynomials related to two iterations of the Geronimus transformation, with the feature that the resulting sequences of polynomials are missing a single degree. We relate these sequences to quasi-orthogonal polynomials of order 2 and investigate properties including recurrence relations, completeness, and associated differential equations.