Zonal spherical harmonics

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
where "P"ℓ is a Legendre polynomial of degree ℓ. The general zonal spherical harmonic of degree ℓ is denoted by formula_2, where x is a point on the sphere representing the fixed axis, and y is the variable of the function. This can be obtained by rotation of the basic zonal harmonic formula_3
In "n"-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the ("n"−1)-sphere. Define formula_4 to be the dual representation of the linear functional
in the finite-dimensional Hilbert space Hℓ of spherical harmonics of degree ℓ. In other words, the following reproducing property holds:
for all "Y" ∈ Hℓ. The integral is taken with respect to the invariant probability measure.
Relationship with harmonic potentials.
The zonal harmonics appear naturally as coefficients of the Poisson kernel for the unit ball in R"n": for x and y unit vectors,
where x,y ∈ R"n" and the constants "c""n","k" are given by
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
where "c""n",ℓ are the constants above and formula_10 is the ultraspherical polynomial of degree ℓ.