Stirling Transform

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

:$b_n=\sum_^n \left\ a_k,$

where $\left\$ is the Stirling number of the second kind, also denoted ''S''(''n'',''k'') (with a capital ''S''), which is the number of partitions of a set of size ''n'' into ''k'' parts.

The inverse transform is

:$a_n=\sum_^n s(n,k) b_k,$

where ''s''(''n'',''k'') (with a lower-case ''s'') is a Stirling number of the first kind.

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)."

If

:$f(x) = \sum_^\infty x^n$

is a formal power series, and

:$g(x) = \sum_^\infty x^n$

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

:$f(x) = g(\log(1+x)).$

See also

* Binomial transform
* Generating function transformation
* List of factorial and binomial topics

References

* {{cite journal, first1=M. , last1=Bernstein , first2=N. J. A. , last2=Sloane
, title=Some canonical sequences of integers , journal=Linear Algebra and its Applications
, volume=226/228 , year=1995 , pages=57–72 , doi=10.1016/0024-3795(94)00245-9, arxiv=math/0205301 .
* Khristo N. Boyadzhiev, ''Notes on the Binomial Transform, Theory and Table, with Appendix on the Stirling Transform'' (2018), World Scientific.

Factorial and binomial topics
Transforms