Hello,

I'm having trouble integrating Legendre Polynomials in the following expression.

$\displaystyle \int_{S^2}P_n(x\cdot v) P_k(x\cdot w) dx$

where $\displaystyle v$ and $\displaystyle w$ are two vectors from the unit sphere $\displaystyle S^2$ and $\displaystyle x\cdot v$ denotes the standard inner product on $\displaystyle \mathbb{R}^3$.

I'd like this integral to be zero if $\displaystyle n\neq k$ and I'm guessing orthogonality of the legendre polynomials $\displaystyle P_n(x)$ should help, that is

$\displaystyle \int_{-1}^1 P_n(x) P_k(x) dx = \delta_{nk}$.

I appreciate any tips, thank you!