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

SIGMA 12 (2016), 042, 13 pages      arXiv:1510.08599
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Zeros of Quasi-Orthogonal Jacobi Polynomials

Kathy Driver and Kerstin Jordaan
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa

Received October 30, 2015, in final form April 20, 2016; Published online April 27, 2016

We consider interlacing properties satisfied by the zeros of Jacobi polynomials in quasi-orthogonal sequences characterised by $\alpha\gt-1$, $-2\lt\beta\lt-1$. We give necessary and sufficient conditions under which a conjecture by Askey, that the zeros of Jacobi polynomials $P_n^{(\alpha, \beta)}$ and $P_{n}^{(\alpha,\beta+2)}$ are interlacing, holds when the parameters $\alpha$ and $\beta$ are in the range $\alpha\gt-1$ and $-2\lt\beta\lt-1$. We prove that the zeros of $P_n^{(\alpha, \beta)}$ and $P_{n+1}^{(\alpha,\beta)}$ do not interlace for any $n\in\mathbb{N}$, $n\geq2$ and any fixed $\alpha$, $\beta$ with $\alpha\gt-1$, $-2\lt\beta\lt-1$. The interlacing of zeros of $P_n^{(\alpha,\beta)}$ and $P_m^{(\alpha,\beta+t)}$ for $m,n\in\mathbb{N}$ is discussed for $\alpha$ and $\beta$ in this range, $t\geq 1$, and new upper and lower bounds are derived for the zero of $P_n^{(\alpha,\beta)}$ that is less than $-1$.

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

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