Inequalities for zeros of Jacobi polynomials via Sturm’s theorem: Gautschi’s conjectures

Abstract

Let $x_{n,k}^{(\alpha ,\beta )}$, $k=1,\ldots ,n$, be the zeros of Jacobi polynomials $P_{n}^{(\alpha ,\beta )}(x)$ arranged in decreasing order on $(-1,1)$, where $\alpha ,\beta >-1$, and $\theta _{n,k}^{(\alpha ,\beta )}=\arccos x_{n,k}^{(\alpha ,\beta )}$. Gautschi, in a series of recent papers, conjectured that the inequalities

$$n\theta_{n,k}^{(\alpha,\beta)}<(n+1)\theta_{n+1,k}^{(\alpha,\beta)} $$

and

$$(n+(\alpha+\beta+3)/2)\theta_{n+1,k}^{(\alpha,\beta)}<(n+(\alpha+\beta+1)/2)\theta_{n,k}^{(\alpha,\beta)}, $$

hold for all $n\geq 1$, $k=1,\ldots ,n$, and certain values of the parameters $\alpha$ and $\beta$. We establish these conjectures for large domains of the $(\alpha ,\beta )$-plane by using a Sturmian approach.