A proof of Gautschi's conjecture on subrange Jacobi polynomials

Abstract

Let $\pi_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\leq1$, 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 $0<c<1$, a first-crossing argument proves the conjecture throughout $0<\alpha<\beta$. Together with the earlier regions recorded by Gautschi and established by Milovanovi\'c, this settles $\beta\geq0$. In the negative wedge, writing $\alpha=-r-\lambda$ and $\beta=-r+\lambda$, an ensemble Ward bound yields the region $c^2\leq3/(3+r)$. A strengthening of the same crossing lemma, using an orthogonal expansion and Markov's theorem, removes this restriction. Consequently, the conjecture holds for every $n\geq1$, $-1<\alpha<\beta$, and $0<c\leq1$. We also give a direct degree-one proof and an explicit asymptotic limit. The case $c=1$ is immediate.