combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...

mathematics, the Stirling transform of a

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

of numbers is the sequence given by :

b_n=\sum_^n \left\ a_k,

where

\left\

is the

Stirling number In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book ''Methodus differentialis'' (1730). They were redisc ...

of the second kind, also denoted ''S''(''n'',''k'') (with a capital ''S''), which is the number of

partitions Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...

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 In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...

, 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 In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary ser ...

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 . * 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

See also

References