(a) Find the roots, Pn(t) = 0. of the Legendre polynomials P2, P3 and P4.
(b) Prove that for 0 < j < k, the polynomial Rj,k (t) defined in (5.49) has roots of order k - j at t = ± 1, and j additional simple roots lying between - 1 and 1.
(c) Conclude that all k roots of the Legendre polynomial Pk(t) are real, simple and lie in the interval - 1 < t < 1.