A proof of Gautschi's conjecture on subrange Jacobi polynomials

Abstract

Let ${\pi }_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\le 1$, with respect to the Jacobi weight $(1-x{)}^{\alpha }(1+x{)}^{\beta }$, where $-1<\alpha <\beta$. Gautschi conjectured that \[ \left[ \frac{π_n(-c)}{π_n(c)} \right]^2 \left(\frac{1-c}{1+c}\right)^{β-α} <1. \] 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 Milovanović, this settles $\beta \ge 0$. In the negative wedge, writing $\alpha =-r-\lambda$ and $\beta =-r+\lambda$, an ensemble Ward bound yields the region $c^2\le 3/(3+r)$. A strengthening of the same crossing lemma, using an orthogonal expansion and Markov's theorem, removes this restriction. Consequently, the conjecture holds for every $n\ge 1$, $-1<\alpha <\beta$, and $0<c\le 1$. We also give a direct degree-one proof and an explicit asymptotic limit. The case $c=1$ is immediate.

Accepted version. To appear in Comp. Appl. Math