Estimating the derivative of the Legendre polynomial

AbstractFor the derivative of the Legendre polynomial, the estimates $$ \left| {(1 - x^2 )^\alpha dP_n (x)/dx} \right| < A(n + a)^\delta $$ are available on the interval of orthogonality −1 ≤ x ≤ 1. Here, δ = 2 and A = 1/2 for α = 0, and δ = 1/2, A = $$ \sqrt {2/\pi } $$ = 0.797885 for α = 1. In both cases, the constants δ and A are sharp. For α = 3/4, the estimate δ = 1/2, A = $$ \sqrt {4/\pi } $$ = 1.128379 is available in the literature. In this paper, we give sharp values for the basic constants for α = 3/4, namely: $$ \delta = \frac{1} {2},A = \mathop {\max }\limits_{0 \leqslant t < \infty } \sqrt t J_1 (t) = 0.825031; $$ here, J1 is the Bessel function. The value a = 2/3 is admissible for all n ≥ 0. The infimum of a for which the inequality in question is valid for all sufficiently large n is 1/2; however, the value a = 1/2 is not admissible.