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

Abstract

Let ${\pi }_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\le 1$, 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 Combined with the region $\alpha \le 0\le \beta$, recorded by Gautschi on the basis of an unpublished communication from Milovanovi'c, and with Milovanovi'c's published criterion, this settles the full admissible range $\beta \ge 0$. In the remaining negative wedge, writing $\alpha =-r-\lambda$ and $\beta =-r+\lambda$, we prove the conjecture whenever $$c^2\le \frac{3}{3+r}.$$ Consequently, it holds for every admissible pair of parameters when $0<c\le \sqrt{3}/2$. 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.