Stirling transform

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title: "Stirling transform" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["factorial-and-binomial-topics", "transforms"] topic_path: "arts/film" source: "https://en.wikipedia.org/wiki/Stirling_transform" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In combinatorial mathematics, the Stirling transform of a sequence { a**n : n = 1, 2, 3, ... } of numbers is the sequence { b**n : n = 1, 2, 3, ... } given by

:b_n=\sum_{k=1}^n \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 n into k parts. This is a linear sequence transformation.

The inverse transform is

:a_n=\sum_{k=1}^n (-1)^{n-k} \left[{n \atop k}\right] b_k,

where (-1)^{n-k} \left[{n\atop k}\right] is a signed Stirling number of the first kind, where the unsigned \left[{n\atop k}\right] can be defined as the number of permutations on n elements with k 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)."

If

:f(x) = \sum_{n=1}^\infty {a_n \over n!} x^n

is a formal power series, and

:g(x) = \sum_{n=1}^\infty {b_n \over n!} 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)).

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 |s2cid=14672360 }}.
  • Khristo N. Boyadzhiev, Notes on the Binomial Transform, Theory and Table, with Appendix on the Stirling Transform (2018), World Scientific.

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