Ursell function

In statistical mechanics, an Ursell function or connected correlation function, is a cumulant of a random variable. It can often be obtained by summing over connected Feynman diagrams.
The Ursell function was named after Harold Ursell, who introduced it in 1927.

Definition

If X is a random variable, the moments s_n and cumulants u_n are functions of X related by the exponential formula:
.
The Ursell functions for multivariate random variables are defined analogously to the above, and in the same way as multivariate cumulants.
The Ursell functions of a single random variable X are obtained from these by setting.
The first few are given by

showed that the Ursell functions, considered as multilinear functions of several random variables, are uniquely determined up to a constant by the fact that they vanish whenever the variables X_i can be divided into two nonempty independent sets.