Poisson–Mellin–Newton cycle

Summary

The Poisson–Mellin–Newton cycle is a chain of three transforms that links a sequence $(f_n)$, its Poisson generating function, and the Mellin transform of that function. Going round the cycle recovers the sequence from the Mellin transform by a Nörlund–Rice integral, which is a contour integral over a Newton-type kernel. It explains why alternating binomial sums and the Mellin transform have the same kind of asymptotic analysis. It is used in the average-case analysis of algorithms such as digital search trees and tries.