Mathematics Magazine· 2012Q4
Close Encounters with the Stirling Numbers of the Second Kind
- 105citations
- Q4SCImago
- 2012year
Short summary
Stirling numbers of the second kind, S(m, k), are represented by a binomial transform formula, connecting them to differentiation, summation, Bernoulli numbers, and Euler polynomials.
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Key points
- Stirling numbers of the second kind, S(m, k), are analyzed using a binomial transform formula.
- These numbers originated in James Stirling's 'Methodus Differentialis' (1730) and were further developed by Johann Grünert.
- S(m, k) appear in differentiation and summation formulas.
- Connections are shown between S(m, k), Bernoulli numbers, Euler polynomials, and power sums.
AI-generated from the title and abstract; the full text is not read.
Abstract
SummaryThis 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 Grünert. 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. This article is part of the following collections: Carl B. Allendoerfer Prize: 2010s Additional informationNotes on contributorsKhristo N. BoyadzhievKHRISTO BOYADZHIEV is a Professor of mathematics at Ohio Northern University. He received his masters and Ph.D. degrees in mathematics from the University of Sofia, Bulgaria. His main interests are in classical analysis and geometry.
The authors' abstract, as published at the source. Mathematics Magazine, 2012 · DOI ↗
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Field: Theoretical Computer Science
Theoretical Computer ScienceMathematics