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

Abstract: We show how to obtain linear combinations of polynomials in an orthogonal sequence ${P_n}{n\geq 0}$, 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.