Stirling Transform

# Stirling transform

to get instant updates about 'Stirling Transform' on your MyPage. Meet other similar minded people. Its Free!

X

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

[itex]b_n=sum_^n left a_k,[/itex]

where [itex]left[/itex] 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

[itex]a_n=sum_^n s(n,k) b_k,[/itex]

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

[itex]f(x)=sum_^infty x^n[/itex]

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

[itex]g(x)=sum_^infty x^n[/itex]

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

[itex]g(x)=f(e^x-1).,[/itex]

## References

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

No feeds found

All