In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions are a broad extension of the notion of zonal spherical harmonics to allow for a more general symmetry group.
On the two-dimensional sphere, the unique zonal spherical harmonic of degree ℓ invariant under rotations fixing the north pole is represented in spherical coordinates by
In n-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the (n−1)-sphere. Define to be the dual representation of the linear functional
The zonal harmonics appear naturally as coefficients of the Poisson kernel for the unit ball in Rn: for x and y unit vectors,
The coefficients of the Taylor series of the Newton kernel (with suitable normalization) are precisely the ultraspherical polynomials. Thus, the zonal spherical harmonics can be expressed as follows. If α = (n−2)/2, then