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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 -sphere. Define Z x {\displaystyle Z_{\mathbf {x} }^{}} to be the dual representation of the linear functional