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!