Question on Rodrigues' equation in Legendre polynomials.

[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?

 

[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 x^2n-2m = 0.

 

[yungman]

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 x^2n-2m = 0.

Thanks.