Stirling Transform

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The transformation 👁 S[{a_n}_(n=0)^N]
of a sequence 👁 {a_n}_(n=0)^N
into a sequence 👁 {b_n}_(n=0)^N
by the formula

where 👁 S(n,k)
is a Stirling number of the second kind. The inverse transform is given by

where 👁 s(n,k)
is a Stirling number of the first kind (Sloane and Plouffe 1995, p. 23).

The following table summarized Stirling transforms for some common sequences, where 👁 [S]
denotes the Iverson bracket and 👁 P
denotes the primes.

Here, 👁 S[{1}_(n=0)^N]
gives the Bell numbers.

👁 S[{n}_(n=0)^N]
has the exponential generating function

See also

Binomial Transform, Euler Transform, Exponential Transform, Möbius Transform, Stirling Number of the First Kind, Stirling Number of the Second Kind

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References

Bernstein, M. and Sloane, N. J. A. "Some Canonical Sequences of Integers." Linear Algebra Appl. 226-228, 57-72, 1995.Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Factorial Factors." §4.4 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, p. 252, 1994.Riordan, J. Combinatorial Identities. New York: Wiley, p. 90, 1979.Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, p. 48, 1980.Sloane, N. J. A. Sequences A000110/M1483, A005493/M2851, A024429, A024430, A033999, A052437, and A085507 in "The On-Line Encyclopedia of Integer Sequences."Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

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Stirling Transform

Cite this as:

Weisstein, Eric W. "Stirling Transform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StirlingTransform.html

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