Stirling transform

In combinatorial mathematics, the Stirling transform of a sequence { an : n = 1, 2, 3, ... } of numbers is the sequence { bn : 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, 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


 * $$a_n=\sum_{k=1}^n s(n,k) b_k,$$

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

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


 * $$g(x) = f(e^x-1).\,$$

Likewise, the inverse transform leads to the generating function identity


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