The Erd\'elyi--Magnus--Nevai and Krasikov Conjectures for Jacobi Polynomials

Abstract

Let $p_n^{(\alpha,\beta)}$ denote the Jacobi polynomial orthonormal for the weight $(1-x)^\alpha(1+x)^\beta$ on $[-1,1]$, where $\alpha,\beta\ge-1/2$, and put $S=\alpha+\beta+1$. We prove the uniform degree--parameter estimate $$ (1-x)^{\alpha+1/2}(1+x)^{\beta+1/2} \bigl|p_n^{(\alpha,\beta)}(x)\bigr|^2 \le C\max\bigl\{1,S^{1/3},S^{1/2}(n+1)^{-1/6}\bigr\}. $$ This proves, in an equivalent symmetric parametrisation, the stronger degree-sensitive conjecture proposed by Krasikov and implies the Erd\'elyi--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.