The Koornwinder--Kostenko--Teschl Conjecture for Jacobi Polynomials and the Discrete Laguerre Phase Transition

We prove the refined Koornwinder--Kostenko--Teschl conjecture. For the normalized weighted Jacobi function and all $n\in\mathbb N\_0$, $α,β\ge0$, and $-1\le x\le1$, $$ |g\_n^{(α,β)}(x)| \le \left[ \frac{(n+1)(n+α+β+1)} {(n+α+1)(n+β+1)} \right]^{1/4}\le1. $$ The proof combines a central contour estim