(a) (l + 1)Pl+1 = (2l + 1)x Pl − l Pl−1 (b) Pl(x) + 2x P'l(x) = P'l+1(x) + P'l−1(x), Where prime means differentiation with respect to x.

Question

For Legendre polynomials Pl(x) the generating function is given by:

Use the generating function to show: 

(a) (l + 1)P_l+1 = (2l + 1)x P_l − l P_l−1 

(b) P_l(x) + 2x P'_l(x) = P'_l+1(x) + P'_l−1(x), Where prime means differentiation with respect to x.

Answer 1

(b) Differentiate with respect to x