Proof of spherical harmonic addition theorem (Reposted from here and will be removed on this site if answered on MSE)
I have been trying to prove that
$$
P_\ell(\cos\gamma)=\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^\ell Y_{\ell m}^{*}(\theta', \phi')Y_{\ell m}(\theta, \phi)
$$
for $\cos\gamma=\cos\theta\cos\theta'+\sin\theta\sin\theta'\cos(\phi'-\phi)$, where $P_\ell$ denote the Legendre polynomials defined as
$$
P_\ell(x)=\frac{1}{2^\ell\ell!}\frac{\partial^\ell}{\partial x^\ell}(x^2-1)^\ell
$$
and $Y_{\ell m}$ are the spherical harmonics:
$$
Y_{\ell m}(\theta, \phi)=\sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell+m)!}{(\ell-m)!}}P_\ell^m(\cos\theta)e^{im\phi}
$$
and, finally, $P_\ell^m$ are the associated Legendre polynomials, written as (for $m>0$)
$$
P_\ell^m(x)=(-1)^m(x^2-1)^{m/2}\frac{\partial^m}{\partial x^m}P_\ell(x).
$$
I cannot find any derivation online that does not make use of a group theoretic argument. Is there any elementary proof of this? Thanks!
 A: Like most such things, this was shown by Ferrers (1877), in Chapter IV, Art. 14, in very elementary (and therefore not very compact, but still readable) fashion.
A: The OP asks for a group theoretic derivation that is also elementary. I have not found one which combines these two properties (unless one considers the rotation operator as "elementary"). Considered separately, I can offer:
• On the Inductive Proof of Legendre Addition Theorem lists a dozen proofs of the spherical harmonic addition theorem, several of which avoid the differential equation and its Green function. No group theory, but the proof by induction does qualify as an "elementary proof", since it only uses the recurrence formula for the Legendre polynomials.
• A less elementary group theoretic derivation is given in Wikipedia, based on the fact that the spherical harmonics are eigenfunctions of the rotation operator and form an irreducible representation of ${\rm SO}(3)$. This is worked out using the bra-ket notation of quantum mechanics in The Addition Theorem of Spherical Harmonics.
