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

Let $p_n^{(\alpha ,\beta )}$ denote the Jacobi polynomial orthonormal for the weight $(1-x{)}^{\alpha }(1+x{)}^{\beta }$ on $[-1,1]$, where $\alpha ,\beta \ge -1\mathrm{/}2$, and put $S=\alpha +\beta +1$. We prove the uniform degree–parameter estimate $$(1-x{)}^{\alpha +1\mathrm{/}2}(1+x{)}^{\beta +1\mathrm{/}2}\mid p_n^{(\alpha ,\beta )}(x){\mid }^2\le C\max \{1,S^{1\mathrm{/}3},S^{1\mathrm{/}2}(n+1{)}^{-1\mathrm{/}6}\}\mathrm{.}$$ 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 $S^{1\mathrm{/}3}$ in the squared estimate cannot be omitted. We also derive degree-sensitive lower bounds for Jacobi Christoffel functions and Gauss–Jacobi quadrature weights.