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

**Academic Editor:**Lance Littlejohn

#### Abstract

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 detailsβcertain 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 (1797β1872), 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 βorthogonalityββ relation for :

Next, let , be the Bernoulli numbers. Then for we have For proof see Exampleββ4 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 βexponentialββ 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 [14β17]. 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 applicationsβsee [9, 13, 20β24] 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, 26β32]. 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 )βsee [25, 28, 36β39]. 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 oldβit 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 Eulerβsee [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 βdiscoveredββ 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 [41β43].

*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 signβsee [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 βbreak of symmetry,ββ 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].

#### References

- A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev,
*Integrals and Series. Vol. 1. Elementary Functions*, Gordon & Breach Science, New York, NY, USA, 1986. View at: MathSciNet - H. W. Gould,
*Topics in Combinatorics*, H. W. Gould, 2nd edition, 2000. - G. Dobinski, βSummerung der Reihe $\sum {n}^{m}}/n!$, für $m=1,2,3,4,5,\dots $,β
*Archiv der Mathematik und Physik*, vol. 61, pp. 333β336, 1877. View at: Google Scholar - W. Ligowski, βZur summerung der Reihe...,β
*Archiv der Mathematik und Physik*, vol. 62, pp. 334β335, 1878. View at: Google Scholar - J. A. Grunert, βUber die Summerung der Reihen...,β
*Journal für die reine und angewandte Mathematik*, vol. 25, pp. 240β279, 1843. View at: Google Scholar - L. H. Harper, βStirling behavior is asymptotically normal,β
*Annals of Mathematical Statistics*, vol. 38, pp. 410β414, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. C. Berndt,
*Ramanujan's Notebooks. Part I*, Springer, New York, NY, USA, 1985. View at: Zentralblatt MATH - B. C. Berndt,
*Ramanujan's Notebooks. Part II*, Springer, New York, NY, USA, 1989. View at: Zentralblatt MATH - K. N. Boyadzhiev, βPolyexponentials,β http://www.arxiv.org/pdf/0710.1332. View at: Google Scholar
- E. T. Bell, βExponential polynomials,β
*Annals of Mathematics*, vol. 35, no. 2, pp. 258β277, 1934. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. Touchard, βNombres exponentiels et nombres de Bernoulli,β
*Canadian Journal of Mathematics*, vol. 8, pp. 305β320, 1956. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. Touchard, βProprietes arithmetiques de certains nombres recurrents,β
*Annales de la Société Scientifique de Bruxelles A*, vol. 53, pp. 21β31, 1933. View at: Google Scholar - G.-C. Rota,
*Finite Operator Calculus*, Academic Press, New York, NY, USA, 1975. View at: MathSciNet - L. Carlitz, βSingle variable Bell polynomials,β
*Collectanea Mathematica*, vol. 14, pp. 13β25, 1962. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. W. Gould and J. Quaintance, βA linear binomial recurrence and the Bell numbers and polynomials,β
*Applicable Analysis and Discrete Mathematics*, vol. 1, no. 2, pp. 371β385, 2007. View at: Publisher Site | Google Scholar | MathSciNet - H. W. Gould and J. Quaintance, βImplications of Spivey's Bell number formula,β
*Journal of Integer Sequences*, vol. 11, no. 3, article 08.3.7, 2008. View at: Google Scholar | MathSciNet - Y.-Q. Zhao, βA uniform asymptotic expansion of the single variable Bell polynomials,β
*Journal of Computational and Applied Mathematics*, vol. 150, no. 2, pp. 329β355, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Toscano, βUna classe di polinomi della matematica attuariale,β
*Rivista di Matematica della Università di Parma*, vol. 1, pp. 459β470, 1950. View at: Google Scholar | Zentralblatt MATH | MathSciNet - R. P. Boas Jr. and R. C. Buck,
*Polynomial Expansions of Analytic Functions*, Academic Press, New York, NY, USA, 1964. View at: MathSciNet - K. N. Boyadzhiev, βA series transformation formula and related polynomials,β
*International Journal of Mathematics and Mathematical Sciences*, vol. 2005, no. 23, pp. 3849β3866, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. N. Boyadzhiev, βOn Taylor's coefficients of the Hurwitz zeta function,β
*JP Journal of Algebra, Number Theory and Applications*, vol. 12, no. 1, pp. 103β112, 2008. View at: Google Scholar | MathSciNet - A. Papoulis,
*Probability, Random Variables, and Stochastic Processes*, McGraw-Hill, New York, NY, USA, 1991. - J. Riordan,
*Combinatorial Identities*, John Wiley & Sons, New York, NY, USA, 1968. View at: MathSciNet - J. Riordan,
*An Introduction to Combinatorial Analysis*, John Wiley & Sons, New York, NY, USA, 1967. - J. Sandor and B. Crstici,
*Handbook of Number Theory, Part II*, Springer/Kluwer Academic Publishers, Dordrecht, The Netherlands, 2004. - E. T. Bell, βExponential numbers,β
*The American Mathematical Monthly*, vol. 41, no. 7, pp. 411β419, 1934. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - N. G. de Bruijn,
*Asymptotic Methods in Analysis*, Dover, New York, NY, USA, 3rd edition, 1981. View at: MathSciNet - M. E. Dasef and S. M. Kautz, βSome sums of some significance,β
*The College Mathematics Journal*, vol. 28, pp. 52β55, 1997. View at: Google Scholar - L. F. Epstein, βA function related to the series for ${e}^{{e}^{x}}$,β
*Journal of Mathematics and Physics*, vol. 18, pp. 153β173, 1939. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Klazar, βBell numbers, their relatives, and algebraic differential equations,β
*Journal of Combinatorial Theory. Series A*, vol. 102, no. 1, pp. 63β87, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Pitman, βSome probabilistic aspects of set partitions,β
*The American Mathematical Monthly*, vol. 104, no. 3, pp. 201β209, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Z. Spivey, βA generalized recurrence for Bell numbers,β
*Journal of Integer Sequences*, vol. 11, pp. 1β3, 2008. View at: Google Scholar - H. W. Gould,
*Catalan and Bell Numbers: Research Bibliography of Two Special Number Sequences*, H. W. Gould, 5th edition, 1979. - P. L. Butzer, M. Hauss, and M. Schmidt, βFactorial functions and Stirling numbers of fractional orders,β
*Results in Mathematics*, vol. 16, no. 1-2, pp. 16β48, 1989. View at: Google Scholar | Zentralblatt MATH | MathSciNet - P. L. Butzer and M. Hauss, βOn Stirling functions of the second kind,β
*Studies in Applied Mathematics*, vol. 84, no. 1, pp. 71β91, 1991. View at: Google Scholar | Zentralblatt MATH | MathSciNet - L. Comtet,
*Advanced Combinatorics: The Art of Finite and Infinite Expansions*, D. Reidel, Dordrecht, The Netherlands, 1974. View at: MathSciNet - H. W. Gould, βEuler's formula for $n$th differences of powers,β
*The American Mathematical Monthly*, vol. 85, no. 6, pp. 450β467, 1978. View at: Publisher Site | Google Scholar | MathSciNet - R. L. Graham, D. E. Knuth, and O. Patashnik,
*Concrete Mathematics: A Foundation for Computer Science*, Addison-Wesley, Reading, Mass, USA, 2nd edition, 1994. View at: MathSciNet - D. E. Knuth, βTwo notes on notation,β
*The American Mathematical Monthly*, vol. 99, no. 5, pp. 403β422, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. J. Schwatt,
*An Introduction to the Operations with Series*, Chelsea, New York, NY, USA, 1962. - P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, βFractional calculus in the Mellin setting and Hadamard-type fractional integrals,β
*Journal of Mathematical Analysis and Applications*, vol. 269, no. 1, pp. 1β27, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, βCompositions of Hadamard-type fractional integration operators and the semigroup property,β
*Journal of Mathematical Analysis and Applications*, vol. 269, no. 2, pp. 387β400, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, βMellin transform analysis and integration by parts for Hadamard-type fractional integrals,β
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 1, pp. 1β15, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Oberhettinger,
*Tables of Mellin Transforms*, Springer, New York, NY, USA, 1974. View at: MathSciNet

#### Copyright

Copyright © 2009 Khristo N. Boyadzhiev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.