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
G X ( t ) = M X ( log ( 1 + t ) ) {\displaystyle 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 [ ( 1 + t ) X ] , t ∈ R , {\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:
c n = 1 n ! ∂ n ∂ t n log ( G ( t ) ) | t = − 1 {\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.
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