Stirling transform explained

In combinatorial mathematics, the Stirling transform of a sequence of numbers is the sequence given by

b_n=\sum

	n

	k=1

\left\{\begin{matrix}n\ k\end{matrix}\right\}a_k

where

\left\{\begin{matrix}n\ k\end{matrix}\right\}

is the Stirling number of the second kind, which is the number of partitions of a set of size

into

parts. This is a linear sequence transformation.

The inverse transform is

a_n=\sum

	n

	k=1

(-1)^n-k\left[{n\atopk}\right]b_k

where $(-1)^ \left[{n\atop k}\right]$ is a signed Stirling number of the first kind, where the unsigned

\left[{n\atopk}\right]

can be defined as the number of permutations on

elements with

cycles.

Berstein and Sloane (cited below) state "If a_n is the number of objects in some class with points labeled 1, 2, ..., n (with all labels distinct, i.e. ordinary labeled structures), then b_n is the number of objects with points labeled 1, 2, ..., n (with repetitions allowed)."

f(x)=

	infty
\sum
	n=1

{a_n\overn!}xⁿ

is a formal power series, and

g(x)=

	infty
\sum
	n=1

{b_n\overn!}xⁿ

with a_n and b_n as above, then

g(x)=f(e^x-1)

Likewise, the inverse transform leads to the generating function identity

f(x)=g(log(1+x))

References

M. . Bernstein . N. J. A. . Sloane. Some canonical sequences of integers . Linear Algebra and Its Applications. 226/228 . 1995 . 57–72 . 10.1016/0024-3795(94)00245-9. math/0205301 . 14672360 . .
Khristo N. Boyadzhiev, Notes on the Binomial Transform, Theory and Table, with Appendix on the Stirling Transform (2018), World Scientific.

Stirling transform explained

See also

References