  In this article, we study properties of the exponential Hilbert series of a
$G$-equivariant projective variety, where $G$ is a semisimple, simply-connected
complex linear algebraic group. We prove a relationship between the exponential
Hilbert series and the degree and dimension of the variety. We then prove a
combinatorial identity for the coefficients of the polynomial representing the
exponential Hilbert series. This formula is used in examples to prove further
combinatorial identities involving Stirling numbers of the first and second
kinds.
