Gazeau–Josse-Michaux–Monceau conjecture on extreme Hermite zeros

The conjecture asserts that the product of the two extreme positive zeros of $H_n$ increases when $n$ is replaced by $n+2$, for $n\ge 4$. It arose from numerical experiments in a paper whose first version appeared in 2004.

Known results

• K. Castillo (2024): proved only a restricted generalized-Laguerre monotonicity result, for $-1<\alpha\le 47.9603$, $n\ge 5$, and $t\in[0,0.568774]$; the Hermite conjecture remained conjectural.

September 2026 claimed resolution

On September 8, 2026, the preprint Symmetric products of Laguerre zeros claimed a stronger generalized-Laguerre zero theorem proving the Gazeau–Josse-Michaux–Monceau conjecture and giving a Jacobi-matrix extension. The claim is unverified.

Current status (as of September 2026): A September 2026 preprint claims to prove the extreme-Hermite-zero monotonicity and a Jacobi-matrix extension, but independent verification is absent.