Show that the following Legendre polynomials are equal

Question

$P_{n}\left( -x\right) =\left( -1\right) ^{n}P_{n}\left( x\right)$

I understand that we could use the generating function and using the fact that

$\left( 1+2rx+r^{2}\right) ^{-1/2}$

We could equate like terms, however I can't seen to show they are similar.

Thanks

Answer 1

Hint: $$(1+2rx+r^2)^{-1/2} = \sum P_n(x)r^n$$ $$\sum P_n(-x)r^n = (1-2rx+r^2)^{-1/2} = \sum P_n(x)(-r)^n.$$