Generating functions play a significant role in modern applied mathematics. The idea is simple. If we want to study a certain sequence (f_n(x)) and can find a function

we may obtain properties of (f_n(x)) from those of G, which €œgenerates€ 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/ˆš1-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 p_n(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 θ between the segments OA₁ and OA₂.)

(c) Show that P_n(1) = 1, P_n(-1) = (-1)ⁿ, P_2n+1(0) = 0, and P_2n(0) = (-1)ⁿ · 1 · 3 . . . (2n - 1)/[2 · 4 . . .(2n)].

Gu, ) 3D 2 fal", G(u, ) %3D 1-2+ u .u" n-0