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  "body": "\n## Summary\n\nThe Poisson–Mellin–Newton cycle is a chain of three transforms that links a sequence $(f_n)$, its Poisson generating function, and the [[Mellin transform|N5PVGRJE1T]] of that function.\nGoing 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.\nIt explains why alternating binomial sums and the Mellin transform have the same kind of asymptotic analysis.\nIt is used in the average-case analysis of algorithms such as digital search trees and tries.\n\n## Explanations\n\n### L1 human\n\nStart with a sequence $f_0, f_1, f_2, \\dots$ of numbers.\nMix it with a Poisson weight to get a smooth function $g(t) = e^{-t} \\sum_n f_n t^n / n!$, which is the mean of $f_N$ when $N$ is a Poisson random number with mean $t$.\nTake the Mellin transform of $g$; its poles describe how $g(t)$ behaves for small and large $t$.\nA contour integral then turns the Mellin transform back into the numbers $f_n$, and this closes the cycle.\nFor example, for the constant sequence $f_n = 1$ one gets $g(t) = 1$ for all $t$.\n",
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