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, formula_1, defined characteristically through the special form of its exponential generating function, and the "Stirling (convolution) polynomials", formula_2, 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 integers "k", the Stirling polynomials, "S""k"("x"), are a Sheffer sequence for formula_3 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 formula_6.
The first 10 Stirling polynomials are given in the following table:
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, formula_7 and formula_8 where formula_9 are the "unsigned" Stirling numbers of the first kind, in terms of the two Stirling number triangles for non-negative integers formula_10. For fixed formula_11, both formula_12 and formula_13 are polynomials of the input formula_14 each of degree formula_15 and with leading coefficient given by the double factorial term formula_16.
Properties.
Below formula_17 denote the Bernoulli polynomials and formula_18 the Bernoulli numbers under the convention formula_19 formula_20 denotes a Stirling number of the first kind; and formula_21 denotes Stirling numbers of the second kind.
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 
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, formula_37 and 
formula_38, for integers formula_39 and "arbitrary" complex values of formula_40.
The next table provides several special cases of these Stirling polynomials for the first few formula_39.
Generating functions.
This variant of the Stirling polynomial sequence has particularly nice ordinary generating functions of the following forms:
More generally, if formula_43 is a power series that satisfies formula_44, we have that
We also have the related series identity 
and the Stirling (Sheffer) polynomial related generating functions given by
Properties and relations.
For integers formula_49 and formula_50, these polynomials satisfy the two Stirling convolution formulas given by
and
When formula_53, we also have that the polynomials, formula_54, are defined through their relations to the Stirling numbers
and their relations to the Bernoulli numbers given by