Stirling transform

Stirling Transform

Stirling transform

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Description:
In combinatorial mathematics, the Stirling transform of a sequence of numbers is the sequence given by

<math>b_n=sum_^n left a_k,</math>


where <math>left</math> 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

<math>a_n=sum_^n s(n,k) b_k,</math>


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<sub>n</sub> 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<sub>n</sub> is the number of objects with points labeled 1, 2, ..., n (with repetitions allowed)."

If

<math>f(x)=sum_^infty x^n</math>


is a formal power series (note that the lower bound of summation is 1, not 0), and

<math>g(x)=sum_^infty x^n</math>


with a<sub>n</sub> and b<sub>n</sub> as above, then

<math>g(x)=f(e^x-1).,</math>


See also



References

  • M. Bernstein and N. J. A. Sloane, "Some canonical sequences of integers", Linear Algebra and Applications, 226/228 (1995), 57-72.



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