Zonal spherical harmonics

Summary

Fix a point $y$ of the sphere and a degree $\ell$. The zonal spherical harmonic $Z_y^{(\ell)}$ is the spherical harmonic of degree $\ell$ that reproduces the value at $y$ of every harmonic of degree $\ell$. It is invariant under the rotations that fix $y$, so it depends only on the angle between $x$ and $y$. It is a multiple of a Gegenbauer polynomial in $x \cdot y$, and of the Legendre polynomial $P_\ell(x \cdot y)$ on the sphere in $\R^3$. It is the building block of expansions in spherical harmonics and of the Poisson kernel of the ball.