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 Pℓ is the normalized Legendre polynomial of degree ℓ,. The generic zonal spherical harmonic of degree ℓ is denoted by, 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 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 in the finite-dimensional Hilbert space Hℓ of spherical harmonics of degree ℓ with respect to the Haar measure on the sphere with total mass A_{n-1} (see Unit sphere). In other words, the following reproducing property holds: for all Y ∈ Hℓ where \Omega is the Haar measure from above.

Relationship with harmonic potentials

The zonal harmonics appear naturally as coefficients of the Poisson kernel for the unit ball in Rn: for x and y unit vectors, where is the surface area of the (n-1)-dimensional sphere. They are also related to the Newton kernel via where x,y ∈ Rn and the constants cn,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 cn,ℓ are the constants above and is the ultraspherical polynomial of degree ℓ.

Properties

f(x,y) on Sn−1×Sn−1 that is a spherical harmonic in y for each fixed x, and that satisfies this invariance property, is a constant multiple of the degree ℓ zonal harmonic. Hℓ , then x = y gives