Combinant

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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

${\displaystyle G_{X}(t)=M_{X}(\log(1+t))}$

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

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

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:

${\displaystyle c_{n}={\frac {1}{n!}}{\frac {\partial ^{n}}{\partial t^{n}}}\log(G(t)){\bigg |}_{t=-1}}$

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.