4. Use similar technique I've shown in class to prove the following recursion relation. (20 points). Pi-1(x) = Pi (x) - 2 x PL_, (x) + PL-2(X) Hint: Start with an identity between (x, h) and ax p(x, h). 5. Prove the following recursion relation (Boas page 570 (5.8) c) (20 points). Pi(x) - x PL_1(x) = LP-1(x)Recursion Relations The generating function is useful in deriving the recursion relations (also called recurrence relations) for Legendre polynomials. These recur- sion relations are identities in a and are used (as trigonometric identities are) to simplify work and to help in proofs and derivations. Some examples of recursion relations are: (a) IPi(a) = (21 - 1)x PI-1(x) - (1-1) PL-2(x), (b) xPi(x) - Pli(x) = 1Pi(x), (c) Pi(x) - xPL_1(x) = 1P1-1(x), (5.8) (d) (1 -x2) Pi(x) = 1Pu-1(x) - lxPi(x), (e) (21 + 1) Pl(x) = Pit1(x) - Pi-1(x), (f) (1 - x2) Pl_1(x) = lxP-1(x) - 1Pi(x). We shall now derive (5.8a); the problems outline derivations of the other equations. From (5.1) we get =(1 - 2.ch + h2)-3/2(-2.x + 2h); (5.9) oh = (1 - 2ch + h?)- ah = (x - h)d.Hint: (1) Differentiate (with respect to x) the first recursive relation (Boas page 570 (5.8) a) (2) Use it along with the recursion relation in problem 4 to finish the proof