# Question on Rodrigues' equation in Legendre polynomials.

1. Jan 27, 2010

### yungman

I have problem understand in one step of deriving the Legendre polymonial formula. We start with:

$$P_n (x)=\frac{1}{2^n } \sum ^M_{m=0} (-1)^m \frac{2n-2m)}{m!(n-m)(n-2m)}x^n-2m$$

Where M=n/2 for n=even and M=(n-1)/2 for n=odd.

For 0<=m<=M

$$\Rightarrow \frac{d^n}{dx^n}x^2n-2m = \frac{2n-2m)}{m!(n-m)(n-2m)}x^n-2m$$

For M<m<=n

$$\Rightarrow \frac{d^n}{dx^n}x^2n-2m = 0$$

$$P_n (x)=\frac{1}{2^n n!} \sum ^M_{m=0} (-1)^m \frac{n!)}{m!(n-m)}\frac{d^n}{dx^n}x^2n-2m$$(1)

$$\Rightarrow P_n (x)=\frac{1}{2^n n!}\frac{d^n}{dx^n} \sum ^n_{m=0} (-1)^m \frac{n!)}{m!(n-m)}(x^2)^{n-m}$$(2)

Notice the $$\sum^M_{m=0}$$ change to $$\sum^n_{m=0}$$ from (1) to (2). Can anyone explain this to me?

2. Jan 28, 2010

### tiny-tim

Hi yungman!

(in LaTeX, ^ or _ has to be followed by {} unless there's only one character )

It' s because of the line before (1) …

m=M+1n {…blah…} dn/dxn x2n-2m = 0.

3. Jan 29, 2010

Thanks.