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

Daniel D. Tcheutia   ; Alta S. Jooste   ; Wolfram Koepf

Symmetry, integrability and geometry: methods and applications, Tome 14 (2018)

Résumé

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.

Détail

Keywords: classical orthogonal polynomials; quasi-orthogonal polynomials; interlacing of zeros.

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