Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials

Abstract

We show how to obtain linear combinations of polynomials in an orthogonal sequence {Pn}n≥0, that characterize quasi-orthogonal polynomials of order k ≤ n-1. The polynomials in the sequence {Qn,k}n≥0 are obtained from Pn, by making use of parameter shifts. We use an algorithmic approach to find these linear combinations for each family applicable, and these equations are used to prove the quasi-orthogonality of order k. We also determine the location of the extreme zeros of the quasi-orthogonal polynomials with respect to the endpoints of the interval of orthogonality of the sequence {Pn}n≥0, where possible.