The Erdélyi--Magnus--Nevai and Krasikov Conjectures for Jacobi Polynomials

Abstract

Let $p_n^{(α,β)}$ denote the Jacobi polynomial orthonormal for the weight $(1-x)^α(1+x)^β$ on $[-1,1]$, where $α,β\ge-1/2$, and put $S=α+β+1$. We prove the uniform degree--parameter estimate $$ (1-x)^{α+1/2}(1+x)^{β+1/2} \bigl|p_n^{(α,β)}(x)\bigr|^2 \le C\max\bigl\{1,S^{1/3},S^{1/2}(n+1)^{-1/6}\bigr\}. $$ This proves, in an equivalent symmetric parametrisation, the stronger degree-sensitive conjecture proposed by Krasikov and implies the Erdélyi--Magnus--Nevai conjecture. The proof starts from Krasikov's estimate in the high-parameter quadrant and transports it to the hard edges through weighted contiguous relations whose singular endpoint terms cancel; direct hypergeometric estimates control the remaining endpoint caps. A Bessel turning-point argument shows that the intermediate factor $S^{1/3}$ in the squared estimate cannot be omitted. We also derive degree-sensitive lower bounds for Jacobi Christoffel functions and Gauss--Jacobi quadrature weights.