# Homework Help: Associated Legendre polynomials for negative order

1. Mar 27, 2013

### Rulonegger

1. The problem statement, all variables and given/known data
I just need to deduce the expression for the associated Legendre polynomial $P_{n}^{-m}(x)$ using the Rodrigues' formula

2. Relevant equations
Rodrigues formula reads $$P_{n}(x)=\frac{1}{2^{n}n!}\frac{d^n}{dx^n}(x^2-1)^n$$ and knowing that $$P_{n}^{m}(x)=(-1)^{m}(1-x^2)^{\frac{m}{2}}\frac{d^m}{dx^m}\left[P_n(x)\right]$$

3. The attempt at a solution
Using the expressions above mentioned i get $$P_{n}^{m}(x)=\frac{(-1)^m}{2^{n}n!}(1-x^2)^{\frac{m}{2}}\frac{d^{n+m}}{dx^{n+m}}(x^2-1)^n$$
Then i see that -n≤m≤n, so the last expression must be useful to determine the $P_{n}^{-m}(x)$ polynomial, substituting m by -m, but i cannot find the relationship between the derivatives $\frac{d^{n+m}}{dx^{n+m}}$ and $\frac{d^{n-m}}{dx^{n-m}}$, that i know it must be $$\frac{d^{n-m}}{dx^{n-m}}(x^2-1)^n=(-1)^m\frac{(n-m)!}{(n+m)!}(1-x^2)^{m}\frac{d^{n+m}}{dx^{n+m}}(x^2-1)^n$$
Any help would be greatly appreciated.