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

$${\displaystyle Z^{(\ell )}(\theta ,\phi )=P_{\ell }(\cos \theta )}$$

${\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )}$

${\displaystyle Z^{(\ell )}(\theta ,\phi ).}$

In n-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the (n−1)-sphere. Define ${\displaystyle Z_{\mathbf {x} }^{(\ell )))$ to be the dual representation of the linear functional

$${\displaystyle P\mapsto P(\mathbf {x} )}$$

$${\displaystyle Y(\mathbf {x} )=\int _{S^{n-1))Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )Y(\mathbf {y} )\,d\Omega (y)}$$

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,

$${\displaystyle {\frac {1}{\omega _{n-1))}{\frac {1-r^{2)){|\mathbf {x} -r\mathbf {y} |^{n))}=\sum _{k=0}^{\infty }r^{k}Z_{\mathbf {x} }^{(k)}(\mathbf {y} ),}$$

${\displaystyle \omega _{n-1))$

$${\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {y} |^{n-2))}=\sum _{k=0}^{\infty }c_{n,k}{\frac {|\mathbf {x} |^{k)){|\mathbf {y} |^{n+k-2))}Z_{\mathbf {x} /|\mathbf {x} |}^{(k)}(\mathbf {y} /|\mathbf {y} |)}$$

$${\displaystyle c_{n,k}={\frac {1}{\omega _{n-1))}{\frac {2k+n-2}{(n-2))).}$$

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

$${\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )={\frac {n+2\ell -2}{n-2))C_{\ell }^{(\alpha )}(\mathbf {x} \cdot \mathbf {y} )}$$

${\displaystyle C_{\ell }^{(\alpha )))$

Properties

• The zonal spherical harmonics are rotationally invariant, meaning that
  $${\displaystyle Z_{R\mathbf {x} }^{(\ell )}(R\mathbf {y} )=Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )}$$

  
  for every orthogonal transformation R. Conversely, any function 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.
• If Y₁, ..., Y_d is an orthonormal basis of H_ℓ, then
  $${\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )=\sum _{k=1}^{d}Y_{k}(\mathbf {x} ){\overline {Y_{k}(\mathbf {y} ))).}$$

  
• Evaluating at x = y gives
  $${\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {x} )=\omega _{n-1}^{-1}\dim \mathbf {H} _{\ell }.}$$