# Evaluating an integral with derivatives of Associated Legendre polynomials

I came across the following integral

$$\int_{-1}^{+1} (1-x^{2}) \frac{\partial P_{lm}(x)}{\partial x} \frac{\partial P_{km}(x)}{\partial x} dx$$ where $$P_{lm}(x)$$ is an associated Legendre polynomial, while trying to derive an expression in a physics paper. I am not entirely sure if this is solvable, though. I have tried to integrate the expression using the recurrence and orthogonality relations listed here. However, no matter which recurrence relation I use, I am unable to exactly evaluate this integral. Is there an way to find this integral exactly?

• Do you mean $P_l^m$ and $P_k^m$?
– Gary
Commented Apr 6, 2022 at 8:02
• yes, but the paper I am studying uses subscripts on the $Y_{lm}$ (spherical harmonics)
According to the definitions of the associated Legendre polynomials and assuming $$m+1\leq l,k$$ we have that
$$\int_{-1}^1 (1-x^2)\partial_x P_l^m \partial_x P_k^m \:dx = \int_{-1}^1 P_l^{m+1}P_k^{m+1}\:dx = \frac{2(l+m+1)!}{(2l+1)(l-m-1)!}\delta_{kl}$$
the orthogonality condition for fixed $$m$$.