Zonal spherical harmonics

The zonal harmonics appear naturally as coefficients of the Poisson kernel for the unit ball in Rn: for x and y unit vectors, \frac{1}{\omega_{n-1}}\frac{1-r^2}^{(k)}(\mathbf{y}/|\mathbf{y}|) where and the constants are given by 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 , then Z^{(\ell)}_{\mathbf{x}}(\mathbf{y}) = \frac{n+2\ell-2}{n-2}C_\ell^{(\alpha)}(\mathbf{x}\cdot\mathbf{y}) where c_{n, \ell} are the constants above and C_\ell^{(\alpha)} is the ultraspherical polynomial of degree \ell. The 2-dimensional caseZ^{(\ell)}(\theta,\phi) = \frac{2\ell + 1}{4 \pi} P_\ell(\cos\theta)is a special case of that, since the Legendre polynomials are the special case of the ultraspherical polynomial when \alpha = 1/2. ==Properties==

• The zonal spherical harmonics are rotationally invariant, meaning that Z^{(\ell)}_{R\mathbf{x}}(R\mathbf{y}) = Z^{(\ell)}_{\mathbf{x}}(\mathbf{y}) for every orthogonal transformation R. Conversely, any function on that is a spherical harmonic in y for each fixed x, and that satisfies this invariance property, is a constant multiple of the degree zonal harmonic. • If Y1, ..., Yd is an orthonormal basis of , then Z^{(\ell)}_{\mathbf{x}}(\mathbf{y}) = \sum_{k=1}^d Y_k(\mathbf{x})\overline{Y_k(\mathbf{y})}. • Evaluating at gives Z^{(\ell)}_{\mathbf{x}}(\mathbf{x}) = \omega_{n-1}^{-1} \dim \mathbf{H}_\ell. ==References==