The Erdélyi--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 $$(1-x{)}^{\alpha +1/2}(1+x{)}^{\beta +1/2}|p_n^{(\alpha ,\beta )}(x){|}^2\le C\max \{1,S^{1/3},S^{1/2}(n+1{)}^{-1/6}\}.$$ 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.

18 pages