If X is a random variable, the moments sn and cumulants (same as the Ursell functions) un are functions of X related by the exponential formula:

(where
is the expectation).
The Ursell functions for multivariate random variables are defined analogously to the above, and in the same way as multivariate cumulants.[1]

The Ursell functions of a single random variable X are obtained from these by setting X = X1 = … = Xn.
The first few are given by
