Quasi-Orthogonality of Some Hypergeometric and $q$-Hypergeometric Polynomials

TL;DR

This paper develops an algorithmic method to construct quasi-orthogonal polynomials from orthogonal polynomial sequences using parameter shifts, and analyzes their zero locations.

Contribution

It introduces a systematic approach to generate quasi-orthogonal polynomials from hypergeometric families and studies their zero distributions.

Findings

Linear combinations characterize quasi-orthogonality of order k

Method applies to hypergeometric and q-hypergeometric polynomials

Zero locations are determined relative to orthogonality interval

Abstract

We show how to obtain linear combinations of polynomials in an orthogonal sequence $\{P_n\}_{n\geq 0}$, such as $Q_{n,k}(x)=\sum\limits_{i=0}^k a_{n,i}P_{n-i}(x)$, $a_{n,0}a_{n,k}\neq0$, that characterize quasi-orthogonal polynomials of order $k\le n-1$. The polynomials in the sequence $\{Q_{n,k}\}_{n\geq 0}$ are obtained from $P_{n}$, 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 quasi-orthogonality of order $k$. We also determine the location of the extreme zeros of the quasi-orthogonal polynomials with respect to the end points of the interval of orthogonality of the sequence $\{P_n\}_{n\geq 0}$, where possible.