A property of the zeros of the Legendre polynomials

AbstractIn this paper, it is proven that the zeros of the Legendre polynomials Pn(x) satisfy the inequality (1 − xj − 1(n))(1 − xj + 1(n)) < (1 − xj(n))2, ∀jϵ {2, 3,…,n − 1}, ∀nϵ {3, 4,…}. This result is obtained by applying Sturm's comparison theorem to two homogeneous linear differential equations of second order, each of which has a particular solution deduced from the function [x(2 − x)]12Pn(1 − x), 0 ⩽ x ⩽ 2