On Erdélyi-Magnus-Nevai conjecture for Jacobi polynomials

Abstract

T. Erdélyi, A.P. Magnus and P. Nevai conjectured that for $α, β\ge - {1/2} ,$ the orthonormal Jacobi polynomials ${\bf P}_k^{(α, β)} (x)$ satisfy the inequality \begin{equation*} \max_{x \in [-1,1]}(1-x)^{α+{1/2}}(1+x)^{β+{1/2}}({\bf P}_k^{(α, β)} (x) )^2 =O (\max \left\{1,(α^2+β^2)^{1/4} \right\}), \end{equation*} [Erdélyi et al.,Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614]. Here we will confirm this conjecture in the ultraspherical case $α= β\ge \frac{1+ \sqrt{2}}{4},$ even in a stronger form by giving very explicit upper bounds. We also show that \begin{equation*} \sqrt{δ^2-x^2} (1-x^2)^α({\bf P}_{2k}^{(α, α)} (x))^2 < \frac{2}π (1+ \frac{1}{8(2k+ α)^2} ) \end{equation*} for a certain choice of $δ,$ such that the interval $(- δ, δ)$ contains all the zeros of ${\bf P}_{2k}^{(α, α)} (x).$ Slightly weaker bounds are given for polynomials of odd degree.