# Cumulant generating function

Let $\mathbb{T}$ be a time scale with $0 \in \mathbb{T}$ and $\sup \mathbb{T}=\infty$. Let $f \colon \mathbb{T} \rightarrow \mathbb{R}$ be a probability density function and let $M_f$ be its associated moment generating function. The cumulant generating function of $f$ is defined to be $$C_f(z) = \log M_f(z).$$