Identities over set partitions
The author and T. H. Savits have used characteristic functions of compound nonhomogeneous Poisson processes to derive certain partition identities; see \textit{G. M. Constantine} and \textit{T. H. Savits} [SIAM J. Discrete Math. 7, No. 2, 194-202 (1994; Zbl 0804.05005)]. Here these results are generalized from the exponential case to arbitrary series by means of Faa di Bruno's formula, allowing the computation of higher-order derivatives of composite functions. Several special cases are demonstrated and, in particular, the case of the geometric series interpretable as probabilities of events associated with the gambler's ruin problem and related problems in queueing theory.
scientific article; zbMATH DE number 1331470