About: 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 where 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 where s(n,k) (with a lower-case s) is a Stirling number of the first kind. If is a formal power series, and with an and bn as above, then Likewise, the inverse transform leads to the generating function identity