Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 051, 26 pages      arXiv:1805.08954      https://doi.org/10.3842/SIGMA.2018.051
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

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

Daniel D. Tcheutia ^a, Alta S. Jooste ^b and Wolfram Koepf ^a
a) Institute of Mathematics, University of Kassel, Heinrich-Plett Str. 40, 34132 Kassel, Germany
^b) Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa

Received January 26, 2018, in final form May 17, 2018; Published online May 23, 2018

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.

Key words: classical orthogonal polynomials; quasi-orthogonal polynomials; interlacing of zeros.

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