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

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 details—certain Fourier integrals involving Γ(𝑎+𝑖𝑡) and Γ(𝑎+𝑖𝑡)Γ(𝑏−𝑖𝑡) are evaluated in terms of Stirling numbers.

1. Introduction

We review the exponential polynomials 𝜙_𝑛(𝑥) and present a list of properties for easy reference. Exponential polynomials in analysis appear, for instance, in the rule for computing derivatives like (𝑑/𝑑𝑡)^𝑛𝑒^𝑎𝑒𝑡 and the related Mellin derivatives:

𝑑(𝑥)𝑑𝑥^𝑛𝑑𝑓(𝑥),(𝑑𝑥𝑥)^𝑛𝑓(𝑥). (1.1) Namely, we have (𝑑)𝑑𝑡^𝑛𝑒^𝑎𝑒𝑡=𝜙_𝑛(𝑎𝑒^𝑡)𝑒^𝑎𝑒𝑡, (1.2) or, after the substitution 𝑥=𝑒^𝑡,

𝑑(𝑥)𝑑𝑥^𝑛𝑒^𝑎𝑥=𝜙_𝑛(𝑎𝑥)𝑒^𝑎𝑥. (1.3)

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

⁰_−∞𝜙_𝑛(𝑥)𝜙_𝑚(𝑥)𝑒^2𝑥𝑑𝑥𝑥=(−1)^𝑛2^𝑛+𝑚−1𝐵𝑛+𝑚^𝑛+𝑚, (1.4) where the right-hand side is zero if 𝑛+𝑚 is odd. The other property is (2.25).

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

_ℝ𝑒^{−𝑖𝑡𝜆}𝑡^𝑛Γ(𝑎+𝑖𝑡)𝑑𝑡,_ℝ𝑒^{−𝑖𝑡𝜆}𝑡^𝑛Γ(𝑎+𝑖𝑡)Γ(𝑏−𝑖𝑡)𝑑𝑡 (1.5) for 𝑛=0,1,…, in terms of Stirling numbers.

2. Exponential Polynomials

The evaluation of the series

𝑆_𝑛=^∞_𝑘=0𝑘^𝑛𝑘!,𝑛=0,1,2,… (2.1) has a long and interesting history. Clearly, 𝑆₀=𝑒, with the agreement that 0⁰=1. Several reference books (e.g., [1]) provide the following numbers:

𝑆₁=𝑒,𝑆₂=2𝑒,𝑆₃=5𝑒,𝑆₄=15𝑒,𝑆₅𝑆=52𝑒,₆=203𝑒,𝑆₇=877𝑒,𝑆₈=4140𝑒 (2.2) As noted by Gould in [2, page 93], the problem of evaluating 𝑆_𝑛 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 𝑆₁,…,𝑆₈ by regrouping 𝑆₁=^∞₁𝑘2𝑘!=1++32!13!+⋯=1++11!𝑆2!+⋯=𝑒,₂=^∞₁𝑘²2𝑘!=1+²+32!²23!+⋯=1++31!+42!13!+⋯={1++11!+12!13!+⋯}+{+21!+32!3!+⋯}=𝑒+𝑆₁=2𝑒, (2.3) and continuing like that to 𝑆₈. For large 𝑛 this method is not convenient. However, later that year Ligowski [4] suggested a better method, providing a generating function for the numbers 𝑆_𝑛:

𝑒^𝑒𝑧=^∞_𝑘=0𝑒^𝑘𝑧=𝑘!^∞∞_𝑘=0𝑛=0𝑘^𝑛𝑧𝑘!^𝑛=𝑛!^∞_𝑛=0𝑆_𝑛𝑧^𝑛.𝑛! (2.4) Further, an effective iteration formula was found

𝑆_𝑛=^𝑛−1_𝑗=0𝑗𝑆𝑛−1_𝑗 (2.5) by which every 𝑆_𝑛 can be evaluated starting from 𝑆₁.

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 𝑆_𝑛 hints at the exponential function. Differentiating the expansion 𝑒^𝑥=^∞_𝑘=0𝑥^𝑘𝑘! (2.6) and multiplying both sides by 𝑥 we get

𝑥𝑒^𝑥=^∞_𝑘=0𝑘𝑥^𝑘,𝑘! (2.7) which, for 𝑥=1, gives 𝑆₁=𝑒. Repeating the procedure, we find 𝑆₂=2𝑒 from

𝑥(𝑥𝑒^𝑥)^′=(𝑥+𝑥²)𝑒^𝑥=^∞_𝑘=0𝑘²𝑥^𝑘𝑘!, (2.8) and continuing like that, for every 𝑛=0,1,2,…, we find the relation

𝑑(𝑥)𝑑𝑥^𝑛𝑒^𝑥=𝜙_𝑛(𝑥)𝑒^𝑥=^∞_𝑘=0𝑘^𝑛𝑥^𝑘𝑘!, (2.9) where 𝜙_𝑛 are polynomials of degree 𝑛. Thus,

𝑆_𝑛=𝜙_𝑛(1)𝑒,∀𝑛≥0. (2.10) The polynomials 𝜙_𝑛 deserve a closer look. From the defining relation (2.9) we obtain

𝑥(𝜙_𝑛𝑒^𝑥)^′=𝑥(𝜙^′_𝑛+𝜙_𝑛)𝑒^𝑥=𝜙_𝑛+1𝑒^𝑥, (2.11) that is, 𝜙_𝑛+1=𝑥(𝜙^′_𝑛+𝜙_𝑛), (2.12) which helps to find 𝜙_𝑛 explicitly starting from 𝜙₀:

𝜙₀𝜙(𝑥)=1,₁𝜙(𝑥)=𝑥,₂(𝑥)=𝑥²𝜙+𝑥,₃(𝑥)=𝑥³+3𝑥²𝜙+𝑥,₄(𝑥)=𝑥⁴+6𝑥³+7𝑥²𝜙+𝑥,₅(𝑥)=𝑥⁵+10𝑥⁴+25𝑥³+15𝑥²+𝑥, (2.13) and so on. Another interesting relation, easily proved by induction, is 𝜙_𝑛+1(𝑥)=𝑥^𝑛_𝑘=0𝑛𝑘𝜙_𝑘(𝑥). (2.14) From (2.12) and (2.14) one finds immediately

𝜙^′_𝑛(𝑥)=^𝑛−1_𝑘=0𝑛𝑘𝜙_𝑘(𝑥). (2.15)

Obviously, 𝑥=0 is a zero for all 𝜙_𝑛, 𝑛>0. It can be seen that all the zeros of 𝜙_𝑛 are real, simple, and nonpositive. The nice and short induction argument belongs to Harper [6].

The assertion is true for 𝑛=1. Suppose that for some 𝑛 the polynomial 𝜙_𝑛 has 𝑛 distinct real nonpositive zeros (including 𝑥=0). Then the same is true for the function

𝑓_𝑛(𝑥)=𝜙_𝑛(𝑥)𝑒^𝑥. (2.16) Moreover, 𝑓_𝑛 is zero at −∞ and by Rolle's theorem its derivative

𝑑𝑓𝑑𝑥^𝑛=𝑑𝑑𝑥(𝜙_𝑛(𝑥)𝑒^𝑥) (2.17) has 𝑛 distinct real negative zeros. It follows that the function

𝜙_𝑛+1(𝑥)𝑒^𝑥𝑑=𝑥𝑑𝑥(𝜙_𝑛(𝑥)𝑒^𝑥) (2.18) has 𝑛+1 distinct real nonpositive zeros (adding here 𝑥=0).

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

𝑒^{𝑥(𝑒𝑧−1)}=^∞_𝑛=0𝜙_𝑛𝑧(𝑥)^𝑛𝑛!. (2.19) 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 𝜙_𝑛(𝑥) defined by (2.9) are exactly the partial derivatives (𝜕/𝜕𝑧)^𝑛𝑒^{𝑥(𝑒𝑧−1)} evaluated at 𝑧=0.

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

Setting 𝑧=2𝑘𝜋𝑖, 𝑘=±1,±2,…, in the generating function (2.19) one finds

𝑒^{2𝑘𝜋𝑖}=1,𝑒^{𝑥(𝑒𝑧−1)}=𝑒⁰=1, (2.20) which shows that the exponential polynomials are linearly dependent:

1=^∞_𝑛=0𝜙_𝑛(𝑥)(2𝑘𝜋𝑖)^𝑛𝑛!or0=^∞_𝑛=1𝜙_𝑛(𝑥)(2𝑘𝜋𝑖)^𝑛𝑛!,𝑘=±1,±2,…. (2.21) In particular, 𝜙_𝑛 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 𝑧 in the equation

𝑒^{(𝑥+𝑦)𝑒𝑧}=𝑒^𝑥𝑒𝑧𝑒^𝑦𝑒𝑧 (2.22) yields the binomial identity

𝜙_𝑛(𝑥+𝑦)=^𝑛_𝑘=0𝑛𝑘𝜙_𝑘(𝑥)𝜙_𝑛−𝑘(𝑦). (2.23) With 𝑦=−𝑥 this implies the interesting “orthogonality’’ relation for 𝑛≥1:

^𝑛_𝑘=0𝑛𝑘𝜙_𝑘(𝑥)𝜙_𝑛−𝑘(−𝑥)=0. (2.24)

Next, let 𝐵_𝑛, 𝑛=0,1,…, be the Bernoulli numbers. Then for 𝑝=0,1,…, we have ^𝑥₀𝜙_𝑝1(𝑡)𝑑𝑡=𝑝+1^𝑝+1_𝑘=1𝑘𝐵𝑝+1_{𝑝+1−𝑘}𝜙_𝑘(𝑥). (2.25) 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 𝜙_𝑛 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, Y_𝑛,𝑘, including 𝜙_𝑛 as a particular case. For this reason 𝜙_𝑛 are known also as the single-variable Bell polynomials [14–17]. 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, 20–24] 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

𝑏_𝑛=𝜙_𝑛1(1)=𝑒𝑆_𝑛 (2.26) are sometimes called exponential numbers, but a more established name is Bell numbers. They have interesting combinatorial and analytical applications [15, 16, 18, 26–32]. An extensive list of 202 references for Bell numbers is given in [33].

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

𝜙_𝑧(𝑥)=𝑒^∞−𝑥_𝑘=0𝑘^𝑧𝑥^𝑘𝑘! (2.27) (Butzer et al. [34, 35]). The function appearing here is an interesting entire function in both variables, 𝑥 and 𝑧. Another possibility is to study the polyexponential function

𝑒_𝑠(𝑥,𝜆)=^∞_𝑛=0𝑥^𝑛𝑛!(𝑛+𝜆)^𝑠, (2.28) where Re𝜆>0. When 𝑠 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 𝜙_𝑛 have positive integer coefficients. These coefficients are the Stirling numbers of the second kind {^𝑛_𝑘} (or 𝑆(𝑛,𝑘))—see [25, 28, 36–39]. Given a set of 𝑛 elements, {^𝑛_𝑘} represents the number of ways by which this set can be partitioned into 𝑘 nonempty subsets (0≤𝑘≤𝑛). Obviously, {^𝑛₁}=1, {^𝑛_𝑛}=1 and a short computation gives {^𝑛₂}=2^𝑛−1−1. For symmetry one sets {⁰₀}=1, {^𝑛₀}=0. The definition of {^𝑛_𝑘} implies the property 𝑘𝑛𝑘+𝑛+1𝑛=𝑘𝑘−1 (3.1) (see [38, page 259]) which helps to compute all {^𝑛_𝑘} by iteration. For instance,

𝑛3=3^𝑛−1−2^𝑛+12. (3.2) A general formula for the Stirling numbers of the second kind is

𝑛𝑘=1𝑘!^𝑘_𝑗=1(−1)^𝑘−𝑗𝑘𝑗𝑗^𝑛. (3.3)

Proposition 3.1. For every 𝑛=0,1,2,…𝜙_𝑛𝑛0+𝑛1𝑛2(𝑥)=𝑥+𝑥²𝑛𝑛+⋯+𝑥^𝑛=^𝑛_𝑘=0𝑛𝑘𝑥^𝑘. (3.4)

The proof is by induction and is left to the reader. Setting here 𝑥=1 we come to the well-known representation for the numbers 𝑆_𝑛

𝑆_𝑛𝑛0+𝑛1+𝑛2𝑛𝑛=𝑒(+⋯+). (3.5)

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

𝑛𝑘𝑘! (3.6) appear in the computations of Euler—see [37].

Next we turn to some special differentiation formulas. Let 𝐷=𝑑/𝑑𝑥.

Mellin Derivatives
It is easy to see that the first equality in (2.9) extends to (1.3), where 𝑎 is an arbitrary complex number, that is, (𝑥𝐷)^𝑛𝑒^𝑎𝑥=𝜙_𝑛(𝑎𝑥)𝑒^𝑎𝑥 (3.7) by the substitution 𝑥→𝑎𝑥. Even further, this extends to (𝑥𝐷)^𝑛𝑒^𝑎𝑥𝑝=𝑝^𝑛𝜙_𝑛(𝑎𝑥^𝑝)𝑒^𝑎𝑥𝑝 (3.8) for any 𝑎, 𝑝 and 𝑛=0,1,… (simple induction and (2.12)). Again by induction, it is easy to prove that (𝑥𝐷)^𝑛𝑓(𝑥)=^𝑛_𝑘=0𝑛𝑘𝑥^𝑘𝐷^𝑘𝑓(𝑥) (3.9) for any 𝑛-times differentiable function 𝑓. 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 𝑥𝐷 on exponentials, formula (3.9) can be “discovered’’ by using Fourier transform. Let 𝐹[𝑓](𝑡)=_ℝ𝑒^{−𝑖𝑥𝑡}𝑓(𝑥)𝑑𝑥 (3.10) be the Fourier transform of some function 𝑓. Then 1𝑓(𝑥)=2𝜋_ℝ𝑒^𝑖𝑥𝑡𝐹[𝑓](𝑡)𝑑𝑥,(𝑥𝐷)^𝑛1𝑓(𝑥)=2𝜋_ℝ𝑒^𝑖𝑥𝑡𝜙_𝑛=(𝑖𝑥𝑡)𝐹[𝑓](𝑡)𝑑𝑥_𝑛𝑘=0𝑛𝑘𝑥^𝑘𝐹⁻¹[(𝑖𝑡)^𝑘𝐹[𝑓]](𝑥)=^𝑛_𝑘=0𝑛𝑘𝑥^𝑘𝐷^𝑘𝑓(𝑥). (3.11)

Next we turn to formula (1.2) and explain its relation to (1.3). If we set 𝑥=𝑒^𝑡, then for any differentiable function 𝑓

𝑑𝑑𝑑𝑡𝑓=(𝑑𝑥𝑓)𝑑𝑥𝑑𝑑𝑡=(𝑑𝑥𝑓)𝑒^𝑡=(𝑥𝐷)𝑓, (3.12) and we see that (1.2) and (1.3) are equivalent: (𝑑)𝑑𝑡^𝑛𝑒^𝑎𝑒𝑡=(𝑥𝐷)^𝑛𝑒^𝑎𝑥=𝜙_𝑛(𝑎𝑥)𝑒^𝑎𝑥=𝜙_𝑛(𝑎𝑒^𝑡)𝑒^𝑎𝑒𝑡. (3.13)

Proof of Proposition 2.1. We apply (1.2) to the function 𝑓_𝑥(𝑧)=𝑒^{𝑥(𝑒𝑧−1)}=𝑒^𝑥𝑒𝑧𝑒^−𝑥: (𝑑)𝑑𝑧^𝑛𝑓_𝑥(𝑧)=𝜙_𝑛(𝑥𝑒^𝑧)𝑓_𝑥(𝑧). (3.14) From here, with 𝑧=0(𝑑)𝑑𝑧^𝑛𝑓_𝑥(𝑧)|_𝑧=0=𝜙_𝑛(𝑥) (3.15) as needed.

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

(𝑥𝐷)^𝑛𝑥^𝑘=𝑘^𝑛𝑥^𝑘,𝑛=0,1,…,𝑘∈ℝ (3.16) for any function of the form

𝑓(𝑡)=^∞_𝑛=0𝑎_𝑛𝑡^𝑛, (3.17) we define the differential operator

𝑓(𝑥𝐷)=^∞_𝑛=0𝑎_𝑛(𝑥𝐷)^𝑛 (3.18) with action on functions 𝑔(𝑥):

𝑓(𝑥𝐷)𝑔(𝑥)=^∞_𝑛=0𝑎_𝑛(𝑥𝐷)^𝑛𝑔(𝑥). (3.19) When 𝑔(𝑥)=𝑥^𝑘, (3.16) and (3.19) show that

𝑓(𝑥𝐷)𝑥^𝑘=^∞_𝑛=0𝑎_𝑛𝑘^𝑛𝑥^𝑘=𝑓(𝑘)𝑥^𝑘. (3.20) If now 𝑔(𝑥)=^∞_𝑘=0𝑐_𝑘𝑥^𝑘 (3.21) is a function analytical in a neighborhood of zero, the action of 𝑓(𝑥𝐷) on this function is given by 𝑓(𝑥𝐷)𝑔(𝑥)=^∞_𝑘=0𝑐_𝑘𝑓(𝑘)𝑥^𝑘, (3.22) provided that the series on the right side converges. When 𝑓 is a polynomial, formula (3.22) helps to evaluate series like ^∞_𝑘=0𝑐_𝑘𝑓(𝑘)𝑥^𝑘 (3.23) in a closed form. This idea was exploited by Schwatt [40] and more recently by the present author in [20]. For instance, when 𝑔(𝑥)=𝑒^𝑥, (3.22) becomes

^∞_𝑘=0𝑥𝑓(𝑘)^𝑘𝑘!=𝑒^𝑥∞_𝑛=0𝑎_𝑛𝜙_𝑛(𝑥). (3.24) As shown in [20] this series transformation can be used for asymptotic series expansions of certain functions.

Leibniz Rule
The higher-order Mellin derivative (𝑥𝐷)^𝑛 satisfies the Leibniz rule (𝑥𝐷)^𝑛(𝑓𝑔)=^𝑛_𝑘=0𝑛𝑘[(𝑥𝐷)^𝑛−𝑘𝑓][(𝑥𝐷)^𝑘𝑔]. (3.25) 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 𝑛,𝑚=0,1,2,…𝜙_𝑛+𝑚(𝑥)=^𝑛𝑚_𝑘=0𝑗=0𝑛𝑘𝑚𝑗𝑗^𝑛−𝑘𝑥^𝑗𝜙_𝑘(𝑥). (3.26)

Proof. One has 𝜙_𝑛+𝑚(𝑥)=(𝑥𝐷)^𝑛+𝑚𝑒^𝑥=(𝑥𝐷)^𝑛(𝑥𝐷)^𝑚𝑒^𝑥=(𝑥𝐷)^𝑛(𝜙_𝑚(𝑥)𝑒^𝑥), (3.27) which by the Leibniz rule (3.25) equals ^𝑛_𝑘=0𝑛𝑘[(𝑥𝐷)^𝑛−𝑘𝜙_𝑚(𝑥)][(𝑥𝐷)^𝑘𝑒^𝑥]. (3.28) Using (3.3) and (3.16) we write (𝑥𝐷)^𝑛−𝑘𝜙_𝑚(𝑥)=^𝑚_𝑗=0𝑚𝑗𝑗^𝑛−𝑘𝑥^𝑗, (3.29) and since also (𝑥𝐷)^𝑘𝑒^𝑥=𝜙_𝑘(𝑥)𝑒^𝑥, (3.30) we obtain (3.26) from (3.27). The proof is completed.

Setting 𝑥=1 in (3.25) yields an identity for the Bell numbers: 𝑏_𝑛+𝑚=^𝑛𝑚_𝑘=0𝑗=0𝑛𝑘𝑚𝑗𝑗^𝑛−𝑘𝑏_𝑘. (3.31) 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 𝑥𝐷 we can prove also the following extension of property (2.24).

Proposition 3.3. For any two integers 𝑛,𝑚≥0(𝑥𝐷)^𝑛𝜙_𝑚(𝑥)=^𝑚_𝑘=0𝑚𝑘𝑘^𝑛𝑥^𝑘=^𝑛_𝑘=0𝑛𝑘𝜙_𝑚+𝑘(𝑥)𝜙_𝑛−𝑘(−𝑥). (3.32)

The proof is simple. Just compute

(𝑥𝐷)^𝑛𝜙_𝑚(𝑥)=(𝑥𝐷)^𝑛[(𝑒^−𝑥)(𝜙_𝑚(𝑥)𝑒^𝑥=)]_𝑛𝑘=0𝑛𝑘[(𝑥𝐷)^𝑛−𝑘𝑒^−𝑥][(𝑥𝐷)^𝑘(𝜙_𝑚(𝑥)𝑒^𝑥)] (3.33) and (3.32) follows from (1.3).

For completeness we mention also the following three properties involving the operator 𝐷𝑥. Proofs and details are left to the reader: (𝐷𝑥)^𝑛𝑒^𝑎𝑥=𝜙_𝑛+1(𝑎𝑥)𝑒𝑎𝑥^𝑎𝑥,(𝐷𝑥)^𝑛𝑓(𝑥)=^𝑛_𝑘=0𝑥𝑛+1𝑘+1^𝑘𝐷^𝑘𝑓(𝑥),𝑓(𝐷𝑥)𝑔(𝑥)=^∞_𝑘=0𝑐_𝑘𝑓(𝑘+1)𝑥^𝑘, (3.34) analogous to (1.3), (3.9), and (3.22) correspondingly.

For a comprehensive study of the Mellin derivative we refer to [41–43].

More Stirling Numbers
The polynomials 𝜙_𝑛, 𝑛=0,1,…, 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 1,𝑥,𝑥²,…,𝑥^𝑛,…. We can solve for 𝑥^𝑘 in (3.4) and express the standard basis in terms of the exponential polynomials 1=𝜙₀,𝑥=𝜙₁,𝑥²=−𝜙₁+𝜙₂,𝑥³=2𝜙₁−3𝜙₂+𝜙₃,𝑥⁴=−6𝜙₁+11𝜙₂−6𝜙₃+𝜙₄, (3.35) and so forth. The coefficients here are also special numbers. If we write 𝑥^𝑛=^𝑛_𝑘=0(−1)^𝑛−𝑘𝑛𝑘𝜙_𝑘, (3.36) then [^𝑛_𝑘] are the (absolute) Stirling numbers of first kind, as defined in [38]. (The numbers [^𝑛_𝑘] are nonnegative. The symbol 𝑠(𝑛,𝑘)=(−1)^𝑛−𝑘[^𝑛_𝑘] is used for Stirling numbers of the first kind with changing sign—see [28, 33, 39] for more details.) [^𝑛_𝑘] is the number of ways to arrange 𝑛 objects into 𝑘 cycles. According to this interpretation, 𝑛𝑘𝑘=(𝑛−1)+𝑛−1𝑛−1𝑘−1,𝑛≥1. (3.37)

4. Semiorthogonality of 𝜙_𝑛

Proposition 4.1. For every 𝑛,𝑚=1,2,…, one has ^∞₀𝜙_𝑛(−𝑥)𝜙_𝑚(−𝑥)𝑒^−2𝑥𝑑𝑥𝑥=(−1)^𝑛−12^𝑛+𝑚−1𝐵𝑛+𝑚^𝑛+𝑚. (4.1) Here 𝐵_𝑘 are the Bernoulli numbers. Note that the right-hand side is zero when 𝑘+𝑚 is odd, as all Bernoulli numbers with odd indices >1 are zeros.

Using the representation (3.4) in (4.1) and integrating termwise one obtains an equivalent form of (4.1): ^𝑛𝑚_𝑘=0𝑗=0(−1)^𝑘+𝑗𝑛𝑘𝑚𝑗(𝑘+𝑗−1)!2^𝑘+𝑗=(−1)^𝑛−12^𝑛+𝑚−1𝐵𝑛+𝑚^𝑛+𝑚. (4.2) This (double sum) identity extends the known identity [38, page 317, Problem 6.76] ^𝑚_𝑗=0(−1)^𝑗+1𝑚𝑗𝑗!2^𝑗+1=2^𝑚+1−1𝐵𝑚+1^𝑚+1. (4.3) Namely, (4.3) results from (4.2) for 𝑛=1. The presence of (−1)^𝑛−1 at the right-hand side in (4.1) is not a “break of symmetry,’’ because when 𝑛+𝑚 is even, then 𝑛 and 𝑚 are both even or both odd.

Proof. Starting from Γ(𝑧)=^∞₀𝑥^𝑧−1𝑒^−𝑥𝑑𝑥, (4.4) we set 𝑥=𝑒^𝜆, 𝑧=𝑎+𝑖𝑡, to obtain the representation Γ(𝑎+𝑖𝑡)=^+∞_−∞𝑒^𝑖𝜆𝑡𝑒^𝑎𝜆𝑒^−𝑒𝜆𝑑𝜆, (4.5) which is a Fourier transform integral. The inverse transform is 𝑒^𝑎𝜆𝑒^−𝑒𝜆=12𝜋_ℝ𝑒^{−𝑖𝜆𝑡}Γ(𝑎+𝑖𝑡)𝑑𝑡. (4.6) When 𝑎=1, this is −𝑒^𝜆𝑒^−𝑒𝜆=𝑑𝑒𝑑𝜆^−𝑒𝜆=−12𝜋_ℝ𝑒^{−𝑖𝜆𝑡}Γ(1+𝑖𝑡)𝑑𝑡. (4.7) Differentiating (4.7) 𝑛−1 times for 𝜆 we find (𝑑)𝑑𝜆^𝑛𝑒^−𝑒𝜆=𝜙_𝑛(−𝑒^𝜆)𝑒^−𝑒𝜆=−12𝜋_ℝ𝑒^{−𝑖𝜆𝑡}(−𝑖𝑡)^𝑛−1Γ(1+𝑖𝑡)𝑑𝑡, (4.8) and Parceval's formula yields the equation _ℝ𝜙_𝑛(−𝑒^𝜆)𝜙_𝑚(−𝑒^𝜆)𝑒^−2𝑒𝜆1𝑑𝜆=2𝜋_ℝ(−𝑖𝑡)^𝑛−1(𝑖𝑡)^𝑚−1|Γ(1+𝑖𝑡)|²𝑑𝑡, (4.9) or, with 𝑥=𝑒^𝜆∞₀𝜙_𝑛(−𝑥)𝜙_𝑚(−𝑥)𝑒^−2𝑥𝑑𝑥𝑥=(−1)^𝑛𝑖^𝑛+𝑚2𝜋_ℝ𝑡^{𝑛+𝑚−2}𝜋𝑡sinh(𝜋𝑡)𝑖𝑡. (4.10) The right-hand side is 0 when 𝑛+𝑚 is odd. When 𝑛+𝑚 is even, we use the integral [1, page 351] ^∞₀𝑡^2𝑝−12sinh(𝜋𝑡)𝑑𝑡=^2𝑝−12𝑝(−1)^𝑝=1𝐵_2𝑝 (4.11) to finish the proof.

Property (4.1) resembles the semiorthogonal property of the Bernoulli polynomials ¹₀𝐵_𝑛(𝑥)𝐵_𝑚(𝑥)𝑑𝑥=(−1)^𝑛−1𝑛!𝑚!𝐵(𝑛+𝑚)!^𝑛+𝑚, (4.12) 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 Γ(𝑎+𝑖𝑡) and Γ(𝑎+𝑖𝑡)Γ(𝑏−𝑖𝑡).

Proposition 5.1. For every 𝑛=0,1,… and 𝑎,𝑏>0 one has _ℝ𝑒^{−𝑖𝜇𝑡}𝑡^𝑛Γ(𝑎+𝑖𝑡)Γ(𝑏−𝑖𝑡)𝑑𝑡=𝑖^𝑛2𝜋𝑒^{𝑛−𝑏𝜇𝑘}_𝑘=0𝑚=0𝑛𝑘𝑘𝑚(−1)^𝑚𝑎^𝑛−𝑘Γ(𝑎+𝑏+𝑚)(1+𝑒^−𝜇)^{𝑎+𝑏+𝑚}_ℝ,𝑒^{−𝑖𝜆𝑡}𝑡^𝑛Γ(𝑎+𝑖𝑡)𝑑𝑡=𝑖^𝑛2𝜋𝑒^𝑎𝜆𝑒^{−𝑒𝜆𝑛}_𝑘=0𝑛𝑘𝑎^{𝑘𝑛−𝑘}_𝑚=0𝑘𝑚(−1)^𝑚𝑒^𝜆𝑚. (5.1)(5.2) In particular, when 𝜆=𝜇=0, one obtains the moments 𝐺_𝑛(𝑎,𝑏)≡_ℝ𝑡^𝑛Γ(𝑎+𝑖𝑡)Γ(𝑏−𝑖𝑡)𝑑𝑡=𝑖^𝑛𝜋^𝑛𝑘_𝑘=0𝑚=0𝑛𝑘𝑘𝑚(−1)^𝑚𝑎^𝑛−𝑘Γ(𝑎+𝑏+𝑚)2^{𝑎+𝑏+𝑚−1},𝐺_𝑛(𝑎)≡_ℝ𝑡^𝑛Γ(𝑎+𝑖𝑡)𝑑𝑡=2𝜋𝑖^𝑛𝑒^𝑛𝑘_𝑘=0𝑚=0𝑛𝑘𝑘𝑚(−1)^𝑚𝑎^𝑛−𝑘. (5.3)(5.4)
When 𝑛=0 in (5.1) one has the known integral _ℝ𝑒^{−𝑖𝜇𝑡}Γ(𝑎+𝑖𝑡)Γ(𝑏−𝑖𝑡)𝑑𝑡=2𝜋Γ(𝑎+𝑏)𝑒^−𝑏𝜇(1+𝑒^−𝜇)^{−𝑎−𝑏}, (5.5) which can be found in the form of an inverse Mellin transform in [44].

Proof. Using again (4.6) 𝑒^𝑎𝜆𝑒^−𝑒𝜆=12𝜋_ℝ𝑒^{−𝑖𝜆𝑡}Γ(𝑎+𝑖𝑡)𝑑𝑡, (5.6) we differentiate both side 𝑛 times (𝑑)𝑑𝜆^𝑛[𝑒^𝑎𝜆𝑒^−𝑒𝜆1]=2𝜋_ℝ𝑒^{−𝑖𝜆𝑡}(−𝑖𝑡)^𝑛Γ(𝑎+𝑖𝑡)𝑑𝑡, (5.7) and then, according to the Leibniz rule and (1.2) the left-hand side becomes (𝑑)𝑑𝜆^𝑛[𝑒^𝑎𝜆𝑒^−𝑒𝜆]=𝑒^𝑎𝜆𝑒^{−𝑒𝜆𝑛}_𝑘=0𝑛𝑘𝜙_𝑘(−𝑒^𝜆)𝑎^𝑛−𝑘. (5.8) Therefore, 𝑒^𝑎𝜆𝑒^{−𝑒𝜆𝑛}_𝑘=0𝑛𝑘𝜙_𝑘(−𝑒^𝜆)𝑎^𝑛−𝑘=12𝜋_ℝ𝑒^{−𝑖𝜆𝑡}(−𝑖𝑡)^𝑛Γ(𝑎+𝑖𝑡)𝑑𝑡, (5.9) and (5.2) follows from here.
Replacing 𝜆 by 𝜆−𝜇 we write (5.6) in the form 𝑒^𝑏𝜆𝑒^−𝑏𝜇𝑒^{−𝑒𝜆𝑒−𝜇}=12𝜋_ℝ𝑒^{−𝑖𝜆𝑡}𝑒^𝑖𝜇𝑡Γ(𝑏+𝑖𝑡)𝑑𝑡, (5.10) and then Parceval's formula for Fourier integrals applied to (5.9) and (5.10) yields 𝑒^{𝑛−𝑏𝜇}_𝑘=0𝑛𝑘𝑎^𝑛−𝑘_ℝ𝑒^{(𝑎+𝑏)𝜆}𝑒^{−𝑒𝜆(1+𝑒−𝜇)}𝜙_𝑘(−𝑒^𝜆=)𝑑𝜆(−𝑖)^𝑛2𝜋_ℝ𝑒^{−𝑖𝜇𝑡}𝑡^𝑛Γ(𝑎+𝑖𝑡)Γ(𝑏−𝑖𝑡)𝑑𝑡. (5.11) Returning to the variable 𝑥=𝑒^𝜆 we write this in the form 12𝜋_ℝ𝑒^{−𝑖𝜇𝑡}𝑡^𝑛Γ(𝑎+𝑖𝑡)Γ(𝑏−𝑖𝑡)𝑑𝑡=𝑖^𝑛𝑒^{𝑛−𝑏𝜇}_𝑘=0𝑛𝑘𝑎^{𝑛−𝑘∞}₀𝜙_𝑘(−𝑥)𝑥^{𝑎+𝑏−1}𝑒^{−𝑥(1+𝑒−𝜇)}𝑑𝑥=𝑖^𝑛𝑒^{𝑛−𝑏𝜇𝑘}_𝑘=0𝑗=0𝑛𝑘𝑘𝑗𝑎^𝑛−𝑘(−1)^𝑗∞₀𝑥^{𝑎+𝑏+𝑗−1}𝑒^{−𝑥(1+𝑒−𝜇)}𝑑𝑥=𝑖^𝑛𝑒^{𝑛−𝑏𝜇𝑘}_𝑘=0𝑗=0𝑛𝑘𝑘𝑗𝑎^𝑛−𝑘(−1)^𝑗Γ(𝑎+𝑏+𝑗)(1+𝑒^−𝜇)^{𝑎+𝑏+𝑗}, (5.12) which is (5.1). The proof is complete.

Next, we observe that for any polynomial

𝑝(𝑡)=^𝑚_𝑛=0𝑎_𝑛𝑡^𝑛 (5.13) one can use (5.4) to write the following evaluation:

_ℝ𝑝(𝑡)Γ(𝑎+𝑖𝑡)𝑑𝑡=^𝑚_𝑛=0𝑎_𝑛𝐺_𝑛(𝑎). (5.14) In particular, when 𝑎=1 we have

𝐺_𝑛(1)=2𝜋𝑖^𝑛𝑒⁻¹𝜙_𝑛+1(−1), (5.15) and therefore,

_ℝ𝑝(𝑡)Γ(1+𝑖𝑡)𝑑𝑡=2𝜋𝑒^𝑚_𝑛=0𝑎_𝑛𝑖^𝑛𝜙_𝑛+1(−1). (5.16)

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