This article is a short elementary review 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. At the end of the paper one application is described in details—certain Fourier integrals involving and are evaluated in terms of Stirling numbers.

1. Introduction

We review the exponential polynomials and present a list of properties for easy reference. Exponential polynomials in analysis appear, for instance, in the rule for computing derivatives like and the related Mellin derivatives:

Namely, we have or, after the substitution ,

We also include in this review two properties relating exponential polynomials to Bernoulli numbers, . One is the semiorthogonality

where the right-hand side is zero if is odd. The other property is (2.25).

At the end we give one application. Using exponential polynomials we evaluate the integrals

for , in terms of Stirling numbers.

2. Exponential Polynomials

The evaluation of the series

has a long and interesting history. Clearly, , with the agreement that . Several reference books (e.g., [1]) provide the following numbers:

As noted by Gould in [2, page 93], the problem of evaluating appeared in the Russian journal Matematicheskii Sbornik, 3 (1868), page 62, with solution ibid, 4 (1868-9), page 39. Evaluations are presented also in two papers by Dobinski and Ligowski. In 1877 Dobinski [3] evaluated the first eight series by regrouping and continuing like that to . For large this method is not convenient. However, later that year Ligowski [4] 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 [5] of Grunert (1797–1872), professor at Greifswalde. Among other things, Grunert obtained formula (2.9) from which the evaluation of (2.1) follows immediately.

The structure of the series hints at the exponential function. Differentiating the expansion and multiplying both sides by we get

which, for , gives . Repeating the procedure, we find from

and continuing like that, for every we find the relation

where are polynomials of degree . Thus,

The polynomials deserve a closer look. From the defining relation (2.9) we obtain

that is, which helps to find explicitly starting from :

and so on. Another interesting relation, easily proved by induction, is From (2.12) and (2.14) one finds immediately

Obviously, is a zero for all , . It can be seen that all the zeros of are real, simple, and nonpositive. The nice and short induction argument belongs to Harper [6].

The assertion is true for . Suppose that for some the polynomial has distinct real nonpositive zeros (including ). Then the same is true for the function

Moreover, is zero at and by Rolle's theorem its derivative

has distinct real negative zeros. It follows that the function

has distinct real nonpositive zeros (adding here ).

The polynomials can be defined also by the exponential generating function (extending Ligowski's formula)

It is not obvious, however, that the polynomials defined by (2.9) and (2.19) are the same, so we need the following simple statement.

Proposition 2.1. The polynomials defined by (2.9) are exactly the partial derivatives evaluated at .

Equation (2.19) follows from (2.9) after expanding the exponential in double series and changing the order of summation. A different proof will be given later.

Setting , in the generating function (2.19) one finds

which shows that the exponential polynomials are linearly dependent:

In particular, are not orthogonal for any scalar product on polynomials. (However, they have the semiorthogonality property mentioned in Section 1 and proved in Section 4.)

Comparing coefficient for in the equation

yields the binomial identity

With this implies the interesting “orthogonality’’ relation for :

Next, let , be the Bernoulli numbers. Then for we have For proof see Example  4 in [7, page 51], or [9].

Some Historical Notes
As already mentioned, formula (2.9) appears in the work of Grunert [5, page 260], where he gives also the representation (3.4) and computes explicitly the first six exponential polynomials. The polynomials were studied more systematically (and independently) by S. Ramanujan in his unpublished notebooks. Ramanujan's work is presented and discussed by Berndt in [7, Part 1, Chapter 3]. Ramanujan, for example, obtained (2.19) from (2.9) and also proved (2.14), (2.15), and (2.25). Later, these polynomials were studied by Bell [10] and Touchard [11, 12]. Both Bell and Touchard called them “exponential’’ polynomials, because of their relation to the exponential function, for example, (1.2), (1.3), (2.9), and (2.19). This name was used also by Rota [13]. As a matter of fact, Bell introduced in [10] 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 [1417]. These polynomials are also a special case of the actuarial polynomials introduced by Toscano [18] which, on their part, belong to the more general class of Sheffer polynomials [19]. The exponential polynomials appear in a number of papers and in different applications—see [9, 13, 2024] and the references therein. In [25] they appear on page 524 as the horizontal generating functions of the Stirling numbers of the second kind (see (3.4)).

The numbers

are sometimes called exponential numbers, but a more established name is Bell numbers. They have interesting combinatorial and analytical applications [15, 16, 18, 2632]. An extensive list of 202 references for Bell numbers is given in [33].

We note that (2.9) can be used to extend to for any complex number by the formula

(Butzer et al. [34, 35]). The function appearing here is an interesting entire function in both variables, and . Another possibility is to study the polyexponential function

where . When is a negative integer, the polyexponential can be written as a finite linear combination of exponential polynomials (see [9]).

3. Stirling Numbers and Mellin Derivatives

The iteration formula (2.12) shows that all polynomials have positive integer coefficients. These coefficients are the Stirling numbers of the second kind (or )—see [25, 28, 3639]. Given a set of elements, represents the number of ways by which this set can be partitioned into nonempty subsets . Obviously, , and a short computation gives . For symmetry one sets , . The definition of implies the property (see [38, page 259]) which helps to compute all by iteration. For instance,

A general formula for the Stirling numbers of the second kind is

Proposition 3.1. For every

The proof is by induction and is left to the reader. Setting here we come to the well-known representation for the numbers

It is interesting that formula (3.4) is very old—it was obtained by Grunert [5, page 260] together with the representation (3.3) for the coefficients which are called now Stirling numbers of the second kind. In fact, coefficients of the form

appear in the computations of Euler—see [37].

Next we turn to some special differentiation formulas. Let .

Mellin Derivatives
It is easy to see that the first equality in (2.9) extends to (1.3), where is an arbitrary complex number, that is, by the substitution . Even further, this extends to for any , and (simple induction and (2.12)). Again by induction, it is easy to prove that for any -times differentiable function . This formula was obtained by Grunert [5, pages 257-258] (see also [2, page 89], where a proof by induction is given).

As we know the action of on exponentials, formula (3.9) can be “discovered’’ by using Fourier transform. Let be the Fourier transform of some function . Then

Next we turn to formula (1.2) and explain its relation to (1.3). If we set , then for any differentiable function

and we see that (1.2) and (1.3) are equivalent:

Proof of Proposition 2.1. We apply (1.2) to the function : From here, with as needed.

Now we list some simple operational formulas. Starting from the obvious relation

for any function of the form

we define the differential operator

with action on functions :

When , (3.16) and (3.19) show that

If now is a function analytical in a neighborhood of zero, the action of on this function is given by provided that the series on the right side converges. When is a polynomial, formula (3.22) helps to evaluate series like in a closed form. This idea was exploited by Schwatt [40] and more recently by the present author in [20]. For instance, when , (3.22) becomes

As shown in [20] this series transformation can be used for asymptotic series expansions of certain functions.

Leibniz Rule
The higher-order Mellin derivative satisfies the Leibniz rule The proof is easy, by induction, and is left to the reader. We shall use this rule to prove the following proposition.

Proposition 3.2. For all

Proof. One has which by the Leibniz rule (3.25) equals Using (3.3) and (3.16) we write and since also we obtain (3.26) from (3.27). The proof is completed.

Setting in (3.25) yields an identity for the Bell numbers: This identity was recently published by Spivey [32], who gave a combinatorial proof. After that Gould and Quaintance [16] obtained the generalization (3.26) together with two equivalent versions. The proof in [16] is different from the one above.

Using the Leibniz rule for we can prove also the following extension of property (2.24).

Proposition 3.3. For any two integers

The proof is simple. Just compute

and (3.32) follows from (1.3).

For completeness we mention also the following three properties involving the operator . Proofs and details are left to the reader: analogous to (1.3), (3.9), and (3.22) correspondingly.

For a comprehensive study of the Mellin derivative we refer to [4143].

More Stirling Numbers
The polynomials , , form a basis in the linear space of all polynomials. Formula (3.4) shows how this basis is expressed in terms of the standard basis We can solve for in (3.4) and express the standard basis in terms of the exponential polynomials and so forth. The coefficients here are also special numbers. If we write then are the (absolute) Stirling numbers of first kind, as defined in [38]. (The numbers are nonnegative. The symbol is used for Stirling numbers of the first kind with changing sign—see [28, 33, 39] for more details.) is the number of ways to arrange objects into cycles. According to this interpretation,

4. Semiorthogonality of

Proposition 4.1. For every , one has Here are the Bernoulli numbers. Note that the right-hand side is zero when is odd, as all Bernoulli numbers with odd indices are zeros.

Using the representation (3.4) in (4.1) and integrating termwise one obtains an equivalent form of (4.1): This (double sum) identity extends the known identity [38, page 317, Problem 6.76] Namely, (4.3) results from (4.2) for . The presence of at the right-hand side in (4.1) is not a “break of symmetry,’’ because when is even, then and are both even or both odd.

Proof. Starting from we set , , to obtain the representation which is a Fourier transform integral. The inverse transform is When this is Differentiating (4.7) times for we find and Parceval's formula yields the equation or, with The right-hand side is when is odd. When is even, we use the integral [1, page 351] to finish the proof.

Property (4.1) resembles the semiorthogonal property of the Bernoulli polynomials see, for instance, [25, page 530].

5. Gamma Integrals

We use the technique in the previous section to compute certain Fourier integrals and evaluate the moments of and .

Proposition 5.1. For every and one has In particular, when , one obtains the moments
When in (5.1) one has the known integral which can be found in the form of an inverse Mellin transform in [44].

Proof. Using again (4.6) we differentiate both side times and then, according to the Leibniz rule and (1.2) the left-hand side becomes Therefore, and (5.2) follows from here.
Replacing by we write (5.6) in the form and then Parceval's formula for Fourier integrals applied to (5.9) and (5.10) yields Returning to the variable we write this in the form which is (5.1). The proof is complete.

Next, we observe that for any polynomial

one can use (5.4) to write the following evaluation:

In particular, when we have

and therefore,

More applications can be found in the recent papers [9, 20, 21].