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.2. Start front Pg [3:] = 1 and P1 [x] = 1*. use the first recursion relation we have pros-en in class {_ Boas page STD. (5.8] a) and write down P5 [17} and Fight) . Use Mathematical to check your answers and plot them between -1 and 1. {You don't need to show the plots in the homework. Just try to get a feeling of what the}; look like.) (20 points). 3. Use series solution to solve the following equation. Then use elementary method. Compare your results. {211] points]. y\"=4y