Degree-uniform regions for Gautschi's conjecture on subrange Jacobi polynomials

Abstract

Let $\pi_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\leq 1$, with respect to the Jacobi weight $(1-x)^\alpha(1+x)^\beta$, where $-1<\alpha<\beta$. Gautschi conjectured that $$ \left[ \frac{\pi_n(-c)}{\pi_n(c)} \right]^2 \left(\frac{1-c}{1+c}\right)^{\beta-\alpha}<1. $$ By his variation formula, this inequality is sufficient for every positive zero of $\pi_n$ to move to the right as $c$ increases. For $0<c<1$, we prove the conjecture, uniformly in the degree, throughout $0<\alpha<\beta$. Combined with the region $\alpha\leq 0\leq\beta$, recorded by Gautschi on the basis of an unpublished communication from Milovanovi'c, and with Milovanovi'c's published criterion, this settles the full admissible range $\beta\geq 0$. In the remaining negative wedge, writing $\alpha=-r-\lambda$ and $\beta=-r+\lambda$, we prove the conjecture whenever $$ c^2\leq\frac{3}{3+r}. $$ Consequently, it holds for every admissible pair of parameters when $0<c\leq\sqrt{3}/2$. For arbitrary admissible parameters, we also establish the degree-one case and eventual validity as $n\to\infty$. The proof combines an exact boundary identity, a first-crossing argument based on Pearson and root-motion identities, and, in the negative wedge, a Ward identity with positive association for an MTP$_2$ orthogonal polynomial ensemble. The case $c=1$ is immediate.