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, $${\displaystyle {\frac {1}{\omega _{n-1}}}{\frac {1-r^{2}}{|\mathbf {x} -r\mathbf {y} |^{n}}}=\sum _{k=0}^{\infty }r^{k}Z_{\mathbf {x} }^{($$k)}(\mathbf {y} ),}   where ${\displaystyle \omega _{n-1}}$  is the surface area of the (n-1)-dimensional sphere. They are also related to the Newton kernel via $${\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {y} |^{n-2}}}=\sum _{k=0}^{\infty }c_{n,k}{\frac {|\mathbf {x} |^{k}}{|\mathbf {y} |^{n+k-2}}}Z_{\mathbf {x} /|\mathbf {x} |}^{($$k)}(\mathbf {y} /|\mathbf {y} |)}   where x,y ∈ Rⁿ and the constants c_n,k are given by $${\displaystyle c_{n,k}={\frac {1}{\omega _{n-1}}}{\frac {2k+n-2}{(n-2)}}.}$$ 

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 $${\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )={\frac {n+2\ell -2}{n-2}}C_{\ell }^{(\alpha )}(\mathbf {x} \cdot \mathbf {y} )}$$  where c_n,ℓ are the constants above and ${\displaystyle C_{\ell }^{(\alpha )}}$  is the ultraspherical polynomial of degree ℓ.

• 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)onSⁿ⁻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.
• IfY₁, ..., Y_d is an orthonormal basisofH_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 $${\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {x} )=\omega _{n-1}^{-1}\dim \mathbf {H} _{\ell }.}$$ 

• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9.