1. ## Legendre proofs

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

$\displaystyle (L_n , x^k) = 0$

for all $\displaystyle 0 \le k < n$. Deduce that $\displaystyle (L_0, L_1, ... ,L_n)$ is an orthogonal basis of $\displaystyle P_n$.

$\displaystyle L_n$ 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

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.

[snip]
Are you allowed to use proof by induction? Or do you have to derive (ha ha) the result?

3. i thought about that as they had the p(x) = 1 example so thought that would be the base case. I suppose you could yeah, it doesnt specify that you have to derive it.

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.
..

5. right so what im integrating is;

$\displaystyle \int_{-1}^{1} \frac{(2n)!}{2^n(n!)^2} x^n x^k dx$

can the $\displaystyle \frac{(2n)!}{2^n(n!)^2}$ be taken outside the integral as a constant? My guess is no which means im nowhere new. I already knew the integral was between 1 and -1, I just dont know how to integrate that. Do I use integration by parts? Substitution?

right so what im integrating is;

$\displaystyle \int_{-1}^{1} \frac{(2n)!}{2^n(n!)^2} x^n x^k dx$

can the $\displaystyle \frac{(2n)!}{2^n(n!)^2}$ be taken outside the integral as a constant? My guess is no which means im nowhere new. I already knew the integral was between 1 and -1, I just dont know how to integrate that. Do I use integration by parts? Substitution?
I'm not sure what you are doing or if it is right but:

$\displaystyle \int_{-1}^{1} \frac{(2n)!}{2^n(n!)^2} x^n x^k dx= \frac{(2n)!}{2^n(n!)^2}\int_{-1}^{1} x^{n+k} dx$

RonL

7. yeah that was where i'd got too. should be able to solve from there, thanks.