Proving Orthogonality of Legendre Polynomials

[Logarythmic]

Problem:

Show that

$$\int_{-1}^{1} x P_n(x) P_m(x) dx = \frac{2(n+1)}{(2n+1)(2n+3)}\delta_{m,n+1} + \frac{2n}{(2n+1)(2n-1)}\delta_{m,n-1}$$

I guess I should use orthogonality with the Legendre polynomials, but if I integrate by parts to get rid of the x my integral equals zero.
Any tip on how to start working with this?

 

[Astronuc]

First thought would be to use one of the recursion relationships on xP_n(x).

For example -

$$(l+1)P_{l+1}(x)\,-\,(2l+1)xP_l(x)\,+\,lP_{l-1}(x)\,=\,0$$

BTW, has one shown -

$$\int_{-1}^{1} P_n(x) P_m(x) dx = \frac{2}{2n+1}\delta_{m,n}$$

That was demonstrated here on PF recently.

 

[Logarythmic]

Yes, I've got the last equation and I'll try with the recursion, thank you. =)

 

[StatMechGuy]

Another thing I would recommend is to try using the Rodriguez formula for the Legendre polynomials, then play games with integration by parts.

 

[Logarythmic]

And why is that? I solved the problem by the way. Pretty simple when you know about the recursion relationships.