we may obtain properties of (fn(x)) from those of G, which generates this sequence and is called a generating function of the sequence.
(a) Show that
is a generating function of the Legendre polynomials. Start from the binomial expansion of 1/1-v, then set v = 2xu - u2, multiply the powers of 2xu - u2 out, collect all the terms involving un, and verify that the sum of these terms is pn(x)un.
(b) Let A1 and A2 be two points in space (Fig. 108, r2 > 0). Using (12), show that
This formula has applications in potential theory. (Q/r is the electrostatic potential at A2 due to a charge Q located at A1. And the series expresses 1/r in terms of the distances of A1 and A2 from any origin O and the angle Î¸ between the segments OA1 and OA2.)
(c) Show that Pn(1) = 1, Pn(-1) = (-1)n, P2n+1(0) = 0, and P2n(0) = (-1)n · 1 · 3 . . . (2n - 1)/[2 · 4 . . .(2n)].