Gautschi’s conjecture

Gautschi’s 2018 conjecture asks whether a strict inequality for monic Jacobi orthogonal polynomials holds for every degree when $-1<\alpha<\beta$ and $0<c\leq 1$. The unresolved issue is uniform validity across all admissible parameters and degrees.

Known results

• Gautschi’s recorded mixed-sign result and Milovanović’s published criterion together cover the full range $\beta\geq 0$.

August 2026 partial resolution

The manuscript claims all degrees for $0<\alpha<\beta$, and in the negative wedge claims all degrees when $c^2\leq 3/(3+r)$, hence for every admissible pair when $0<c\leq\sqrt{3}/2$. It also claims degree one generally and eventual validity for each fixed $(\alpha,\beta,c)$. The region $\alpha<\beta<0$ with $c>\sqrt{3}/2$ remains open uniformly in degree.

Current status (as of August 2026): The conjecture has substantial claimed partial coverage, including the full $\beta\geq 0$ range, but the negative-wedge case with $c>\sqrt{3}/2$ is not settled uniformly in degree.