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In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis, which are closely related to the Stirling numbers, the Bernoulli numbers, and the generalized Bernoulli polynomials. There are multiple variants of the Stirling polynomial sequence considered below most notably including the Sheffer sequence form of the sequence, , defined characteristically through the special form of its exponential generating function, and the Stirling (convolution) polynomials, , which also satisfy a characteristic ordinary generating function and that are of use in generalizing the Stirling numbers (of both kinds) to arbitrary complex-valued inputs. We consider the "convolution polynomial" variant of this sequence and its properties second in the last subsection of the article. Still other variants of the Stirling polynomials are studied in the supplementary links to the articles given in the references.
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In mathematics, the Stirlingpolynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis...
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition...
In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in...
especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind...
mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers...
{\displaystyle k>2} Bernoulli numbers Bernoulli polynomials of the second kind StirlingpolynomialPolynomials calculating sums of powers of arithmetic progressions...
to define the multidimensional polynomials. Like the other classical orthogonal polynomials, the Hermite polynomials can be defined from several different...
two xj are the same, the Newton interpolation polynomial is a linear combination of Newton basis polynomials N ( x ) := ∑ j = 0 k a j n j ( x ) {\displaystyle...
be zero after he had converted his formulas for Σ nm from polynomials in N to polynomials in n." In the above Knuth meant B 1 − {\displaystyle B_{1}^{-}}...
polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials. The original use of interpolation polynomials was...
Touchard polynomials, studied by Jacques Touchard (1939), also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence...
difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's polynomials, and...
theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane. These polynomials are named after...
The Bernoulli polynomials of the second kind ψn(x), also known as the Fontana-Bessel polynomials, are the polynomials defined by the following generating...
homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in complete...
numbers of Stirling permutations with a fixed number of descents) are non-negative. They chose the name because of a connection to certain polynomials defined...
_{0}^{y}x(1-x)(2-x)\cdots (n-1-x)\,dx} and therefore |Gn| = Pn+1(1). Stirlingpolynomials Bernoulli polynomials of the second kind Ch. Jordan. The Calculus of Finite...
Quasi-polynomial time algorithms are algorithms whose running time exhibits quasi-polynomial growth, a type of behavior that may be slower than polynomial time...
general graphs in 1932. In 1968, Ronald C. Read asked which polynomials are the chromatic polynomials of some graph, a question that remains open, and introduced...
and rising factorials are closely related to Stirling numbers. Indeed, expanding the product reveals Stirling numbers of the first kind ( x ) n = ∑ k = 0...
generalized α-factorial polynomials, σ(α) n(x) where σ(1) n(x) ≡ σn(x), which generalize the Stirling convolution polynomials from the single factorial...
not satisfy a polynomial equation whose coefficients are themselves polynomials Transcendental number theory, the branch of mathematics dealing with...
Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients...
postulate Sierpinski triangle Star of David theorem Stirling number Stirling transform Stirling's approximation Subfactorial Table of Newtonian series...
include Peirce (1880) and Aitken (1933). Touchard polynomials Catalan number Stirling number Stirling numbers of the first kind Gardner 1978. Halmos, Paul...
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