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

Abstract

We show how to obtain linear combinations of polynomials in an orthogonal sequence $\{P_n{\}}_{n\ge 0}$, such as $Q_{n,k}(x)=\sum _{i=0}^k{a}_{n,i}{P}_{n-i}(x)$, $a_{n,0}{a}_{n,k}\ne 0$, that characterize quasi-orthogonal polynomials of order $k\le n-1$. The polynomials in the sequence $\{Q_{n,k}{\}}_{n\ge 0}$ 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 $\{P_n{\}}_{n\ge 0}$, where possible.