(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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, ... .

2. Relevant equations

Generating function for the sequence of Legendre polynomials:

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

3. 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Legendre Polynomial (anti)symmetry proof

**Physics Forums | Science Articles, Homework Help, Discussion**