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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.
For nonnegative integersk, the Stirling polynomials, Sk(x), are a Sheffer sequence for [1] defined by the exponential generating function
The Stirling polynomials are a special case of the (or ) [2] each with exponential generating function
given by the relation .
The first 10 Stirling polynomials are given in the following table:
k
Sk(x)
0
1
2
3
4
5
6
7
8
9
Yet another variant of the Stirling polynomials is considered in [3] (see also the subsection on Stirling convolution polynomials below). In particular, the article by I. Gessel and R. P. Stanley defines the modified Stirling polynomial sequences, and where are the unsignedStirling numbers of the first kind, in terms of the two Stirling number triangles for non-negative integers . For fixed , both and are polynomials of the input each of degree and with leading coefficient given by the double factorial term .
By differentiating the generating function it readily follows that
Stirling convolution polynomials[]
Definition and examples[]
Another variant of the Stirling polynomial sequence corresponds to a special case of the convolution polynomials studied by Knuth's article [5]
and in the Concrete Mathematics reference. We first define these polynomials through the Stirling numbers of the first kind as
It follows that these polynomials satisfy the next recurrence relation given by
These Stirling "convolution" polynomials may be used to define the Stirling numbers, and
, for integers and arbitrary complex values of .
The next table provides several special cases of these Stirling polynomials for the first few .
n
σn(x)
0
1
2
3
4
Generating functions[]
This variant of the Stirling polynomial sequence has particularly nice ordinary generating functions of the following forms:
More generally, if is a power series that satisfies , we have that
^Knuth, D. E. (1992). "Convolution Polynomials". Mathematica J. 2: 67–78. arXiv:math/9207221. Bibcode:1992math......7221K.
The article contains definitions and properties of special convolution polynomial families defined by special generating functions of the form for . Special cases of these convolution polynomial sequences include the binomial power series, , so-termed tree polynomials, the Bell numbers, , and the Laguerre polynomials. For , the polynomials are said to be of binomial type, and moreover, satisfy the generating function relation for all , where is implicitly defined by a functional equation of the form . The article also discusses asymptotic approximations and methods applied to polynomial sequences of this type.
This article incorporates material from Stirling polynomial on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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