A property of the zeros of the Legendre polynomials

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Abstract

In 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{lunate} {2, 3,...,n - 1}, ∀nε{lunate} {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)]1 2Pn(1 - x), 0 ≤ x ≤ 2. © 1987.

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Grosjean, C. C. (1987). A property of the zeros of the Legendre polynomials. Journal of Approximation Theory, 50(1), 84–88. https://doi.org/10.1016/0021-9045(87)90068-2

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