Combinant

In the mathematical theory of probability, the combinants ''c''_''n'' of a random variable ''X'' are defined via the combinant-generating function ''G''(''t''), which is defined from the moment generating function ''M''(''z'') as

:$G_X(t)=M_X(\log(1+t))$

which can be expressed directly in terms of a random variable ''X'' as

: G_X(t) := E\left 1+t)^X\right \quad t \in \mathbb,

wherever this expectation exists.

The ''n''th combinant can be obtained as the ''n''th derivatives of the logarithm of combinant generating function evaluated at –1 divided by ''n'' factorial:

:$c_n = \frac \frac \log(G (t)) \bigg, _$

Important features in common with the cumulants are:
* the combinants share the additivity property of the cumulants;
* for infinite divisibility (probability) distributions, both sets of moments are strictly positive.

References

*
Google Books

Theory of probability distributions

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