Close Encounters with the Stirling Numbers of the Second Kind

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.