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In the mathematical theory of probability, the combinants cn 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
which can be expressed directly in terms of a random variable X as
![{\displaystyle G_{X}(t):=E\left[(1+t)^{X}\right],\quad t\in \mathbb {R} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d1df6664083a3822fb3dc54a383f4e1e18f1f8)
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:
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
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