# Close Encounters with the Stirling Numbers of the Second Kind

@article{Boyadzhiev2012CloseEW, title={Close Encounters with the Stirling Numbers of the Second Kind}, author={Khristo N. Boyadzhiev}, journal={Mathematics Magazine}, year={2012}, volume={85}, pages={252 - 266} }

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 Grünert. The paper demonstrates the usefulness of these numbers in analysis. In particular…

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## References

SHOWING 1-10 OF 40 REFERENCES

James Stirling's Methodus differentialis

- Mathematics
- 2003

Contains not only the results and ideas for which Stirling is chiefly remembered, for example, Stirling numbers and Stirling's asymptotic formula for factorials, but also a wealth of material on…

Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals

- Mathematics
- 2009

This article is a short elementary review of the exponential polynomials (also
called single-variable Bell polynomials) from the point of view of analysis. Some new properties
are included, and…

A Primer on Bernoulli Numbers and Polynomials

- Mathematics
- 2008

Bernoulli numbers and polynomials are named after the Swiss mathematician Jakob Bernoulli (1654–1705), who introduced them in his book Ars Conjectandi, published posthumously (Basel, 1713). They…

A series transformation formula and related polynomials

- MathematicsInt. J. Math. Math. Sci.
- 2005

A formula that turns power series into series of functions, based on the geometric series, which reduces to a series transformation formula which appears in the works of Ramanujan and is related to the exponential (or Bell) polynomials.

James Stirling’s Methodus Differentialis: An annotated translation of Stirling’s text, by Ian Tweddle. Pp. 295. €129.95, £ 75.00, sFr 210.00, $ 129.00. 2003. ISBN 1 85233 723 0 (Springer).

- HistoryThe Mathematical Gazette
- 2005

classroom as well as Anthony Ferzola's account of Euler's use of differentials as 'absolute zeros', a non-rigorous intuitive approach which was exonerated in retrospect by Abraham Robinson's…

Calculus of finite differences

- Mathematics
- 1939

On operations Functions important in the calculus of finite differences Inverse operation of differences and means. Sums Stirling's numbers Bernoulli polynomials and numbers Euler's and Boole's…

Derivative Polynomials for tanh, tan, sech and sec in Explicit Form

- Mathematics
- 2007

The derivative polynomials for the hyperbolic and trigonometric tangent, cotangent and secant are found in explicit form, where the coecients are given in terms of the Stirling numbers of the second…

MATH

- Biology
- 1992

It is still unknown whether there are families of tight knots whose lengths grow faster than linearly with crossing numbers, but the largest power has been reduced to 3/z, and some other physical models of knots as flattened ropes or strips which exhibit similar length versus complexity power laws are surveyed.

An Algebraic Identity Leading to Wilson Theorem

- Mathematics
- 2004

In most text books on number theory Wilson Theorem is proved by applying Lagrange theorem concerning polynomial congruences.Hardy and Wright also give a proof using cuadratic residues. In this…

Ueber die Summirung der Reihen von der Form Aφ(0), A1φ(1)x, A2φ(2)x2, .... Anφ(n)xn, ...., wo A eine beliebige constante Größe, An eine beliebige und φ(n) eine ganze rationale algebraische Function der positiven ganzen Zahl n bezeichnet.

- Mathematics
- 1843

t) *i> ^2} ^35 ^49 · · ' ' In 9 · · · · ? deren Glieder Funciionen von und, so wie (wenn nicht ausdrücklich etwas Anderes bemerkt wird) alle im Folgenden vorkommenden Gröfsen reelle Gröfsen sein…