Spherical Harmonic Addition Theorem

A Formula also known as the Legendre Addition Theorem which is derived by finding Green's Functions for the Spherical Harmonic expansion and equating them to the generating function for Legendre Polynomials. When $\gamma$ is defined by

$\begin{displaymath} \cos \gamma\equiv \cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2\cos\phi_1-\phi_2, \end{displaymath}$

$P_n(\cos \gamma) = {4\pi\over 2n+1} \sum_{m=-n}^n(-1)^mY_m^n(\theta_1,\phi_1)Y_{-m}^n(\theta_2,\phi_2)$

$= {4\pi\over 2n+1} \sum_{m=-n}^n Y_m^n(\theta_1,\phi_1){Y_m^n}^*(\theta_2,\phi_2)$

$= P_n(\cos\theta_1)P_n(\cos\theta_2)+2\sum_{m=-n}^n{(n-m)!\over(n+m)!}P_m^n(\cos\theta_1)P_m^n(\cos\theta_2)\cos[m(\phi_1-\phi_2)].$

References

Arfken, G. ``The Addition Theorem for Spherical Harmonics.'' §12.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 693-695, 1985.

$P_n(\cos \gamma) = {4\pi\over 2n+1} \sum_{m=-n}^n(-1)^mY_m^n(\theta_1,\phi_1)Y_{-m}^n(\theta_2,\phi_2)$
$= {4\pi\over 2n+1} \sum_{m=-n}^n Y_m^n(\theta_1,\phi_1){Y_m^n}^*(\theta_2,\phi_2)$
$= P_n(\cos\theta_1)P_n(\cos\theta_2)+2\sum_{m=-n}^n{(n-m)!\over(n+m)!}P_m^n(\cos\theta_1)P_m^n(\cos\theta_2)\cos[m(\phi_1-\phi_2)].$