Exponential Polynomials, Stirling Numbers,and Evaluation of Some Gamma Integrals

This article is a survey of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of Analysis. Some new properties are included and several Analysis-related applications are mentioned.

(1. 2) We also include in this review two properties relating exponential polynomials to Bernoulli numbers, . One is the semi-orthogonality , (1. 3) where the right hand side is zero if is odd. The other property is (2.10).
At the end we give one application. Using exponential polynomials we evaluate the integrals , (1. 4) and (1. 5) for in terms of Stirling numbers.

Exponential polynomials
The evaluation of the series , (2.1) has a long and interesting history. Clearly, , with the agreement that . Several reference books (for instance, [31]) provide the following numbers. .
As noted by H. Gould in [19, p. 93], the problem of evaluating appeared in the Russian journal Matematicheskii Sbornik, 3 (1868), p.62, with solution ibid , 4 (1868-9), p. 39.) Evaluations are presented also in two papers by Dobi½ski and Ligowski. In 1877 G. Dobi½ski [15] evaluated the first eight series by regrouping: , and continuing like that to . For large this method is not convenient. However, later that year Ligowski [27] suggested a better method, providing a generating function for the numbers .
Further, an effective iteration formula was found by which every can be evaluated starting from .
These results were preceded, however, by the work [23]  (2.6). This name was used also by Gian-Carlo Rota [34]. As a matter of fact, Bell introduced in [1] a more general class of polynomials of many variables, , including as a particular case. For this reason are known also as the single-variable Bell polynomials [13], [20], [21], [41]. These polynomials are also a special case of the actuarial polynomials introduced by Toscano [38] which, on their part, belong to the more general class of Sheffer polynomials [7].
We note that equation (2.2) can be used to extend to for any complex number by the formula (2.12) (Butzer et al. [9], [10]). The function appearing here is an interesting entire function in both variables, and . Another possibility is to study the polyexponential function , (2.13) where . When is a negative integer, the polyexponential can be written as a finite linear combination of exponential polynomials (see [6]).

Stirling numbers and Mellin derivatives
The iteration formula (2.3) shows that all polynomials have positive integer coefficients. These coefficients are the Stirling numbers of the second kind (or )see [12], [14], [17], [22], [26], [35]. Given a set of elements, represents the number of ways this set can be partitioned into nonempty subsets (  [19], where a proof by induction is given).
As we know the action of on exponentials, formula (3.5) can be "discovered" by using Fourier transform. Let .
Next we turn to formula (1.1) and explain its relation to (1.2). If we set , then for any differentiable function and we see that (1.1) and (1.2) are equivalent. provided the series on the right side converges. When is a polynomial, formula (3.12) helps to evaluate series like in a closed form. This idea was exploited by Schwatt [36] and more recently by the present author in [4]. For instance, when equation (3.12) becomes . (3.13) As shown in [4] this series transformation can be used for asymptotic series expansions of certain functions.
Leibniz Rule. The higher order Mellin derivative satisfies the Leibniz rule (3.14) The proof is easy, by induction, and is left to the reader. We shall use this rule to prove the following proposition.
Setting in (3.14) yields an identity for the Bell numbers.
. (3.16) This identity was recently published by Spivey [37], who gave a combinatorial proof . After that Gould and Quaintance [21] obtained the generalization (3.15) together with two equivalent versions. The proof in [21] is different from the one above.
Using the Leibniz rule for we can prove also the following extension of property For completeness we mention also the following three properties involving the operator . Proofs and details are left to the reader.
, (3.18) , (3.19) and , (3.20) analogous to (1.2). (3.5), and (3.12) correspondingly For a comprehensive study of the Mellin derivative we refer to [11]. then are the (absolute) Stirling numbers of first kind, as defined in [22]. (The numbers are non-negative. The symbol is used for Stirling numbers of the first kind with changing sign -see [14], [18] and [26] for more details.) is the number of ways to arrange objects into cycles. According to this interpretation, , .

Gamma integrals.
We use the technique in the previous section to compute certain Fourier integrals and evaluate the moments of and . .
Returning to the variable we write this in the form (5.10) which is (5.1). The proof is complete.
Next, we observe that for any polynomial (5.11) one can use (5.4) to write the following evaluation . (5.12) In particular, when we have , (5.13) and therefore, . (5.14) More applications can be found in the recent papers [4], [5] and [6].