Say we have take vectors $(x_1,..,x_d) \in S^{d-1}$ and we look at vectors $(a_1,..,a_d) \in (\mathbb{Z^+ \cup \{0\}})^d$ such that $\sum_{i=1}^da_i =k$ for some positive integer $k$.
Is there any known parameterization of $S^{d-1}$ which is particularly suited to understand the behaviour of the function $\prod_{i=1}^d \left ( x_i ^{a_i}/ a_i ! \right )$ (given that these are homogeneous monomials on a homogeneous space)?