I am interested in modeling a distribution on a sphere with a series expansion in terms of the spherical harmonics $Y_l^m (x)$ where $x \in \mathbb{S}^2$ is a point on the unit sphere. From the paper Harmonic analysis and distribution-free inference for spherical distributions by Jammalamadaka and Terdik I learned that there are multiple ways of achieving this. The following formulation appears to be suitable in my setting:
An exponential family of distributions for directional data was introduced in [4] and on p. 82 of [36]. Apart from the normalizing constant, the density has the form
$$f_e(x) \propto \exp \sum_{l=0}^{\infty} \sum_{m=-l}^{l} c_l^m Y_l^m (x) $$
where $c_l^{m \ast} = (− 1)^m c_l^{−m}$. The normalizing constant corresponds to $c_0^0$ and depends on the rest of the parameters $c_l^m$ as well since the integral of $f_e$ must be 1.
I am interested in this normalization constant $c_0^0$. Following the references in the paper, I could not find an expression for $c_0^0$. Can anybody point me to a paper or book in which an analytical form (or approximation) is given?