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Cumulant-moment relations through determinants

The traditional approach to expressing cumulants in terms of moments is by expansion of the cumulant generating function which is represented as an embedded power series of the moments. The moments are then obtained in terms of cumulants through successive reverse substitutions. In this note we demonstrate how cumulant-moment relations are expressible in a more compact and elegant manner through determinant representations. The results will be of interest to university instructors and should be of pedagogic value as well.