CumulantGeneratingFunction

CumulantGeneratingFunction[dist, t] gives the cumulant-generating function for the distribution dist as a function of the variable t.

CumulantGeneratingFunction[dist, {t1, t2, ...}] gives the cumulant-generating function for the multivariate distribution dist as a function of the variables t1, t2, ....

Examples

Cumulant generating function of normal distribution:

CumulantGeneratingFunction[NormalDistribution[μ, σ], t]
(* μ t + (σ^2 t^2)/2 *)

For Poisson distribution:

CumulantGeneratingFunction[PoissonDistribution[λ], t]
(* λ (E^t - 1) *)

Please visit the official Wolfram Language Reference for more details.