Stirling polynomials

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 integers k, the Stirling polynomials, Sk(x), are a Sheffer sequence for defined by the exponential generating function The Stirling polynomials are a special case of the Nørlund polynomials (or generalized Bernoulli polynomials) each with exponential generating function given by the relation . The first 10 Stirling polynomials are given in the following table: {| class="wikitable"

!k !! Sk(x)
0
-
1
-
2
-
3
-
4
-
5
-
6
-
7
-
8
-
9
}
Yet another variant of the Stirling polynomials is considered in (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 unsigned Stirling 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 .

!k !! Sk(x)
0
-
1
-
2
-
3
-
4
-
5
-
6
-
7
-
8
-
9
}
Yet another variant of the Stirling polynomials is considered in (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 unsigned Stirling 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 .