# 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 is a generating function of the Legendre polynomials. Start from the binomial expansion of 1/1-v, then set v = 2xu - u 2 , multiply the powers of2xu - u

Chapter 5, PROBLEM SET 5.2 #10

## This problem has been solved!

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

we may obtain properties of (fn(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 - uout, 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 Aand Afrom 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)].

Related Book For