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.
Definition and examples
For nonnegative
integersk, the Stirling polynomials, Sk(x), are a
Sheffer sequence for [1] defined by the exponential generating function
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
5
6
7
8
9
10
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.
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.
Definition and examples
For nonnegative
integersk, the Stirling polynomials, Sk(x), are a
Sheffer sequence for [1] defined by the exponential generating function
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
5
6
7
8
9
10
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.