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Dobiński's Formula

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A formula for the Bell polynomial and Bell numbers. The general formula states that

B_n(x)=e^(-x)sum_(k=0)^infty(k^n)/(k!)x^k,
(1)

where B_n(x) is a Bell polynomial (Roman 1984, p. 66). Setting x=1 gives the special case of the nth Bell number,

B_n=1/esum_(k=0)^infty(k^n)/(k!).
(2)

It can be derived by dividing the generating function formula for a Stirling number of the second kind S(n,k) by m!, yielding

(m^n)/(m!)=sum_(k=1)^n(S(n,k))/((m-k)!).
(3)

Then

sum_(m=1)^infty(m^n)/(m!)lambda^m=(sum_(k=1)^nS(n,k)lambda^k)(sum_(j=0)^infty(lambda^j)/(j!)),
(4)

and

sum_(k=1)^nS(n,k)lambda^k=e^(-lambda)sum_(m=1)^infty(m^n)/(m!)lambda^m.
(5)

Now setting lambda=1 gives the identity (Dobiński 1877; Rota 1964; Berge 1971, p. 44; Comtet 1974, p. 211; Roman 1984, p. 66; Lupas 1988; Wilf 1994, p. 106; Chen and Yeh 1994; Pitman 1997).

Dobinski also published a curious infinite product sometimes also known as Dobiński's formula.

See also

Bell Number, Bell Polynomial, Infinite Product

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References

Berge, C. Principles of Combinatorics. New York: Academic Press, 1971.Blasiak, P.; Penson, K. A.; and Solomon, A. I. "Dobiński-Type Relations and the Log-Normal Distribution." J. Phys. A: Math. Gen. 36, L273-278, 2003.Chen, B. and Yeh, Y.-N. "Some Explanations of Dobinski's Formula." Studies Appl. Math. 92, 191-199, 1994.Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974.Dobiński, G. "Summierung der Reihe sumn^m/n! für m=1, 2, 3, 4, 5, ...." Grunert Archiv (Arch. Math. Phys.) 61, 333-336, 1877.Foata, D. La série génératrice exponentielle dans les problèmes d'énumération. Montréal, Canada: Presses de l'Université de Montréal, 1974.Lupas, A. "Dobiński-Type Formula for Binomial Polynomials." Stud. Univ. Babes-Bolyai Math. 33, 30-44, 1988.Penson, K. A.; Blasiak, P.; Duchamp, G.; Horzela, A.; and Solomon, A. I. "Hierarchical Dobiński-Type Relations via Substitution and the Moment Problem." 26 Dec 2003. http://www.arxiv.org/abs/quant-ph/0312202/.Pitman, J. "Some Probabilistic Aspects of Set Partitions." Amer. Math. Monthly 104, 201-209, 1997.Roman, S. The Umbral Calculus. New York: Academic Press, p. 66, 1984.Rota, G.-C. "The Number of Partitions of a Set." Amer. Math. Monthly 71, 498-504, 1964.Wilf, H. Generatingfunctionology, 2nd ed. New York: Academic Press, 1994.

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Dobiński's Formula

Cite this as:

Weisstein, Eric W. "Dobiński's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DobinskisFormula.html

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