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Inversion relations and the Stirling transform

For any pair of sequences, { f n } {\displaystyle \{f_{n}\}} and { g n } {\displaystyle \{g_{n}\}}, related by a finite sum Stirling number formula given by

for all integers n ≥ 0 {\displaystyle n\geq 0}, we have a corresponding inversion formula for f n {\displaystyle f_{n}} given by

The lower indices could be any integer between 0 {\textstyle 0} and n {\textstyle n}.

These inversion relations between the two sequences translate into functional equations between the sequence exponential generating functions given by the Stirling (generating function) transform as

and

For D = d / d x {\displaystyle D=d/dx}, the differential operators x n D n {\displaystyle x^{n}D^{n}} and (x D) n {\displaystyle (xD)^{n}} are related by the following formulas for all integers n ≥ 0 {\displaystyle n\geq 0}:

Another pair of " inversion " relations involving the Stirling numbers relate the forward differences and the ordinary n t h {\displaystyle n^{th}} derivatives of a function, f (x) {\displaystyle f(x)}, which is analytic for all x {\displaystyle x} by the formulas