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 formula_1 where is a Legendre polynomial of degree . 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 formula_5 in the finite-dimensional Hilbert space H of spherical harmonics of degree ℓ. In other words, the following reproducing property holds: formula_6 for all . 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: for x and y unit vectors, formula_7 where and the constants are given by formula_8 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 , then formula_9 where are the constants above and formula_10 is the ultraspherical polynomial of degree ℓ.