Originally Posted by

**Deadstar** I have two Legendre questions, can anyone help me with either of them?

Suppose $\displaystyle 0 \le k < n$. Let p be a polynomial. Show that

$\displaystyle (\frac{d}{dx})^k ((x^2 -1)^n p(x)) = (x^2 -1)^{n-k} q(x)$

for some polynomial q.

Taking p(x) = 1 this implies

$\displaystyle (\frac{d}{dx})^k ((x^2 -1)^n) = (x^2 -1) q(x)$

for some polynomial q.

and the other one is...

Use integration by parts to show that

$\displaystyle (L_n , x^k) = 0$ Mr F says: The integration is over the interval (-1, 1).

for all $\displaystyle 0 \le k < n$. Deduce that $\displaystyle (L_0, L_1, ... ,L_n)$ is an orthogonal basis of $\displaystyle P_n$. Mr F says: On the interval (-1, 1).

$\displaystyle L_n$ may be $\displaystyle L_n(x) = \frac{(2n)!}{{2^n}(n!)^2}x^n +...$ or it may be $\displaystyle L_n(x) = \frac{1}{2^n n!}(\frac{d}{dx})^n ((x^2 - 1)^n)$... Im not sure really which is why im stuck! Both are used in my handout

Mr F says: They are equivalent expressions. The former is the series solution to Legendre's Equation. The latter is Rodrigues' Formula.