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Close Encounters with the Stirling Numbers of the Second Kind

Khristo N. Boyadzhiev
Mathematics Magazine
Vol. 85, No. 4 (October 2012), pp. 252-266
DOI: 10.4169/math.mag.85.4.252
Stable URL: http://www.jstor.org/stable/10.4169/math.mag.85.4.252
Page Count: 15
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Close Encounters with the Stirling Numbers of the Second Kind
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Abstract

Summary This is a short introduction to the theory of Stirling numbers of the second kind S(m, k) from the point of view of analysis. It is written as an historical survey centered on the representation of these numbers by a certain binomial transform formula. We tell the story of their birth in the book Methodus Differentialis (1730) by James Stirling, and show how they mature in the works of Johann Gr¨unert. The paper demonstrates the usefulness of these numbers in analysis. In particular, they appear in several differentiation and summation formulas. The reader can also see the connection of S(m, k) to Bernoulli numbers, to Euler polynomials, and to power sums.

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