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}
}
  • K. Boyadzhiev
  • Published 1 October 2012
  • Mathematics
  • Mathematics Magazine
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… 
The q-Stirling numbers of the second kind and its applications
The study of q-Stirling numbers of the second kind began with Carlitz [L. Carlitz, Duke Math. J., 15 (1948), 987–1000] in 1948. Following Carlitz, we derive some identities and relations related to
On the uniformity of the approximation for r-associated Stirling numbers of the second Kind
TLDR
This work provides a proof of the uniformity of the Hennecart approximation for the associated Stirling numbers, which has been utilized without a proper justification due to the absence of a rigorous proof.
Generalized Stirling numbers and sums of powers of arithmetic progressions
ABSTRACT In this paper, we first focus on the sum of powers of the first n positive odd integers, , and derive in an elementary way a polynomial formula for in terms of a specific type of generalized
Sums of powers of integers and hyperharmonic numbers
  • J. Cereceda
  • Mathematics
    Notes on Number Theory and Discrete Mathematics
  • 2021
In this paper, we obtain a new formula for the sums of k-th powers of the first n positive integers, Sk(n), that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then,
Combinatorial Models of the Distribution of Prime Numbers
This work is divided into two parts. In the first one, the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved
Stirling Functions and a Generalization of Wilson's Theorem
For positive integers m and n, denote S(m,n) as the associated Stirling number of the second kind and let z be a complex variable. In this paper, we introduce the Stirling functions S(m,n,z) which
k-ary Lyndon Words and Necklaces Arising as Rational Arguments of Hurwitz–Lerch Zeta Function and Apostol–Bernoulli Polynomials
The main motivation of this paper was to give finite and infinite generating functions for the numbers of the k-ary Lyndon words and necklaces. In order to construct our new generating functions, we
Stirling Numbers via Combinatorial Sums
In this paper, we have derived a formula to find combinatorial sums of the type $\sum_{r=0}^n r^k {n\choose r}$ for $k \in \mathbb{N}$. The formula is conveniently expressed as a linear combination
Stirling numbers and inverse factorial series
We study inverse factorial series and their relation to Stirling numbers of the first kind. We prove a special representation of the polylogarithm function in terms of series with such numbers. Using
...
...

References

SHOWING 1-10 OF 40 REFERENCES
James Stirling's Methodus differentialis
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
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
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
TLDR
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).
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
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
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
TLDR
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
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.
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
...
...