- #1

- 6

- 0

## Homework Statement

Let [itex]P_{n}(x)[/itex] denote the Legendre polynomial of degree n, n = 0, 1, 2, ... . Using the formula for the generating function for the sequence of Legendre polynomials, show that:

[itex]P_{n}(-x) = (-1)^{n}P_{n}(x)[/itex]

for any x [itex]\in[/itex] [-1, 1], n = 0, 1, 2, ... .

## Homework Equations

Generating function for the sequence of Legendre polynomials:

[itex]\sum P_{n}(x)r^{n} = (1 - 2rx + r^{2})^{-\frac{1}{2}}[/itex]

## The Attempt at a Solution

I guess I don't really know where to begin with this. I tried differentiating both sides of the generating function with respect to r to obtain:

[itex]\sum nP_{n}(x)r^{n-1} = (x - r)\sum P_{n}(x)r^{n}[/itex]

but I don't see how this might move closer to the desired result. I imagine differentiating with respect to x would lead to similar difficulties.

I then tried just directly substituting -x into the generating function, giving

[itex](1 + 2rx + r^{2})^{-\frac{1}{2}}[/itex]

as a generating function for [itex]P_{n}(-x)[/itex], but again I don't really see where this might lead me.

Perhaps there's a general method I'm missing; I'm still very new to the concept of Legendre polynomials and generating functions in general. My attempts at scouring the Internet for a proof of this specific identity have turned up a sole reference to a textbook I have no access to. I'm not really looking for a complete proof to be given to me on a plate, but a nudge in the right direction would certainly be appreciated.