Stirling transform

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 ${\textstyle (-1)^{n-k}\left[{n \atop k}\right]}$ 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

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

See also

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

References

• Bernstein, M.; Sloane, N. J. A. (1995). "Some canonical sequences of integers". Linear Algebra and Its Applications. 226/228: 57–72. arXiv:math/0205301. doi:10.1016/0024-3795(94)00245-9. S2CID 14672360..
• Khristo N. Boyadzhiev, Notes on the Binomial Transform, Theory and Table, with Appendix on the Stirling Transform (2018), World Scientific.