Integral involving product of derivatives of Legendre polynomials

[hanson]

Anyone how to evaluate this integral?

$\int_{-1}^{1} (1-x^2) P_{n}^{'} P_m^{'} dx$, where the primes represent derivative with respect to $x$?

I tried using different recurrence relations for derivatives of the Legendre polynomial, but it didn't get me anywhere...

 

[lurflurf]

Use the facts

$$\left((1-x^2)P_n^\prime \right)^\prime=-n(n+1)P_n$$

and

$$\int_{-1}^1 P_m P_n \text{ dx}=\dfrac{2}{2n+1} \delta_{mn}$$

to integrate by parts

or just use

$$P_n=\frac{1}{(2n)!} \dfrac{d^n}{dx^n} (x^2-1)^n$$

 

[hanson]

Thank you very much!