Combinant

In the mathematical theory of probability, the combinants c_n of a random variableX are defined via the combinant-generating function G(t), which is defined from the moment generating function M(z) as

[math]\displaystyle{ G_X(t)=M_X(\log(1+t)) }[/math]

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

[math]\displaystyle{ G_X(t) := E\left[(1+t)^X\right], \quad t \in \mathbb{R}, }[/math]

wherever this expectation exists.

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

[math]\displaystyle{ c_n = \frac{1}{n!} \frac{\partial ^n}{\partial t^n} \log(G (t)) \bigg|_{t=-1} }[/math]

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

• Kittel, W.; De Wolf, E. A.. Soft Multihadron Dynamics. pp. 306 ff. ISBN 978-9812562951.  Google Books