Calculation of the spherical harmonic sum in the propagator of the particle on a sphere I am calculating the propagator of the free particle on a sphere : $K(\theta_f \phi_f t_f; \theta_i \phi_i t_i)$. The wavefunctions in this case are the spherical harmonics $Y_{lm}(\theta, \phi)=\frac{e^{im \phi}}{\sqrt{2 \pi}}{\Omega(\theta)}$. So the kernel is 
$$K=\sum e^{\frac{-i E_n (t_f-t_i)}{\hbar}} \frac{e^{im (\phi_f - \phi_i)}}{{2 \pi}}{\Omega(\theta_f)}\Omega(\theta_i)$$
The sum is over $m=-l, -l+1, ....., l$, and $l=0,1,2.....$
How do I calculate this sum? 
 A: The summation over the magnetic quantum number can be achieved by using the spherical Harmonics addition formula
$\sum_{m=-l}^{l}Y_{lm}(\hat{\mathbb{n_i}}) Y_{lm}(\hat{\mathbb{n_f}}) = \frac{2l+1}{4 \pi} P_l(\hat{\mathbb{n_i}}.\hat{\mathbb{n_f}})$
Where $\hat{\mathbb{n_i}}$, $\hat{\mathbb{n_f}}$ are the initial and final unit vectors on the sphere and $P_l$ are the Legendre polynomials. The remaining sum has the form:
$ K(\theta, t_f-t_i) =  \sum_{l=0}^{\infty}\frac{2l+1}{4 \pi} e^{i l(l+1)(t_f-t_i)}P_l(cos(\theta))$
Where $\theta = \arccos(\hat{\mathbb{n_i}}.\hat{\mathbb{n_f}})$
(It is assumed that the energy is that of a free particle on the sphere):
$E_n =l(l+1)$. 
The "closest' form of the remainig sum is by means of a fractional derivative of the propagator on the circle which can be expressed by means of the Jacobi's theta function. Please see http:Camporesi's review  (equations 8.38 and 6.35) for the full proof:
$ K(\theta, t_f-t_i) = e^{i/4 (t_f-t_i)} (\frac{1}{2\pi}  \frac{\partial}{\partial (\cos \theta + 1)})^{\frac{1}{2}} \frac{1}{2\pi} \Theta_3(\frac{\theta}{2}, -\frac{t_f-t_i}{\pi})$
