International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 547459, 14 pages

doi:10.1155/2012/547459

Research Article

The Multiple Gamma-Functions and the Log-Gamma Integrals

X.-H. Wang¹ and Y.-L. Lu²

¹Xi’an International Studies University, Xi’an, Shaanxi 710128, China

²Department Of Mathematics, Weinan Teachers’ College, Shaanxi 714000, China

Received 15 May 2012; Accepted 30 July 2012

Academic Editor: Shigeru Kanemitsu

Copyright © 2012 X.-H. Wang and Y.-L. Lu. 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.

Abstract.

In this paper, which is a companion paper to [W], starting from the Euler integral which appears in a generalization of Jensen’s formula, we shall give a closed form for the integral of log . This enables us to locate the genesis of two new functions and considered by Srivastava and Choi. We consider the closely related function A(a) and the Hurwitz zeta function, which render the task easier than working with the functions themselves. We shall also give a direct proof of Theorem 4.1, which is a consequence of [CKK, Corollary 1.1], though.

1. Introduction

If is analytic in a domain containing the circle and has no zero on the circle, then the Gauss mean value theorem

is true. In [1, page 207] the case is considered where has a zero on the circle, and (1.1) turns out that the Euler integral

which is essential in proving a generalization of Jensen's formula [1, pages 207-208].

Let denote the Catalan constant defined by the absolutely convergent series

where is the nonprincipal Dirichlet character mod 4.

As a next step from (1.2) the relation

holds true. In this connection, in [2] we obtained some results on viewing it as an intrinsic value to the Barnes -function. The Barnes -function (which is in the class of multiple gamma functions) is defined as the solution to the difference equation (cf. (2.3))

with the initial condition

and the asymptotic formula to be satisfied

, where indicates the Euler gamma function (cf., e.g., [3]).

Invoking the reciprocity relation for the gamma function

it is natural to consider the integrals of or of multiple gamma functions (cf., e.g., [4, 5]). Barnes’ theorem [6, page 283] reads

valid for nonintegral values of .

In this paper, motivated by the above, we proceed in another direction to developing some generalizations of the above integrals considered by Srivastava and Choi [7]. For -analogues of the results, compare the recent book of the same authors [8]. Our main result is Theorem 2.1 which gives a closed form for and locates its genesis. A slight modification of Theorem 2.1 gives the counterpart of Barnes’ formula (1.9) which reads.

Corollary 1.1. Except for integral values of , one has

Srivastava and Choi introduced two functions and by (2.9) and (2.9) with formal replacement of by , respectively. They state , which is rather ambiguous as to how we interpret the meaning because (2.9) is defined for [7, page 347, l.11]. They use this function to express the integral , without giving proof. This being the case, it may be of interest to locate the integral of [7, (13), page 349], thereby [7, page 347].

For this purpose we use a more fundamental function than defined by

where is the Hurwitz zeta-function

in the first instance. For its theory, compare, for instance, [3], [9, Chapter 3].

We shall prove the following corollary which gives the right interpretation of the function .

Corollary 1.2. For ,

or

2. Barnes Formula

There is a generalization of (1.4) as well as (1.2) in the form [7, equation (28), page 31]:

Equation (2.1) is Barnes’ formula [6, page 279] which is equivalent to Kinkelin's 1860 result [10] [7, equation (26), page 30]:

Since (1.5) is equivalent to

it follows that

Putting , we obtain

which is (1.2).

The counterpart of (2.1) follows from the reciprocity relation (1.8), known as Alexeievsky's Theorem [7, equation (42), page 32].

which in turn is a special case of (1.9).

Indeed, in [7, page 207], only (1.9) and the integral of are in closed form and the integral of is not. A general formula is given by Barnes [4] with constants to be worked out. We shall state a concrete form for this integral in Section 3, using the relation [7, equation (455), page 210] between and the integral of and appealing to a closed form for the latter in [11].

Formula (2.6) is stated in the following form [7, equation (12), page 349]:

where is the Glaisher-Kinkelin constant defined by [7, equation (2), page 25]

and is defined by [7, equation (9), page 347]

for .

Comparing (2.6) and (2.7), we immediately obtain

on using the difference relation .

Thus, in a sense we have located the genesis of the function . although they prove (2.7) by an elementary method [7, page 348].

Indeed, and are almost the same:

a proof being given below. However, is more directly connected with for which we have rich resources of information as given in [9, Chapter 3].

We prove the following theorem which gives a closed form for , thereby giving the genesis of the constant .

Theorem 2.1. For , one has

If , then

Proof. We evaluate the integral

in two ways. First,

On the other hand, noting that is the sum of (2.1) and (2.7), we deduce that

Substituting (1.5), we obtain

The first two terms on the right of (2.17) become

while the 3rd and the 4th terms give, in view of (2.11), .

Hence, altogether

Comparing (2.15) and (2.19) proves (2.12), completing the proof.

Comparing (2.13) and [7, equation (13), page 349]

we prove Corollary 1.2.

Hence the relation between and is (1.14), that is, one between and rather than as Srivastava and Choi state.

At this point we shall dwell on the underlying integral representation for (the derivative of) the Hurwitz zeta-function, which makes the argument rather simple and lucid as in [12] and gives some consequences.

Proof of (2.11). Consider that

[9, (3.15), page 59], where the last integral may be also expressed as

and where is the th periodic Bernoulli polynomial. Then

whence in particular, we have the generic formula for and consequently for through (1.11):

This may be slightly modified in the form

Comparing (2.9) and (2.25), we verify (2.11).

The merit of using is that by way of , we have a closed form for it:

In the same way, via another important relation [7, equation (23), page 94],

Equation (2.21) gives a closed form for , too. We also have from (1.11) and (2.27)

There are some known expressions not so handy as given by (2.27). For example, [7, page 25] and [7, equation (440), page 206], one of which reads

with designating the Euler constant. Equation (2.29) is a basis of (2.2) (cf. proof of [2, Lemma 1]).

Remark 2.2. The Glaisher-Kinkelin constant is connected with and as follows:

This can also be seen from Vardi's formula [7, (31), page 97]:

which is (1.11) with .

We may also give another direct proof of Corollary 1.2.

Proof of Corollary 1.2 (another proof). is the limit of the expression

where . Let . Then

Hence, simplifying, we find that

Hence

which is (1.14). This completes the proof.

As an immediate consequence of Corollary 1.2, we prove (2.36) as can be found in [7, pages 350–351].

Proof of (2.36). From (2.28), (1.5), and (1.8), we obtain

On the other hand, by (2.11) and (1.13), we see that the left-hand side of (2.37) is

whence we conclude that

On exponentiating, (2.37) leads to (2.36).

3. Polygamma Function of Negative Order

In this section we introduce the function [13]:

which is closely related to the polygamma function of negative order and states some simple applications. We recall some properties of :

Equation (3.3) is [2, equation (2.31)], which is used in proving [2, Theorem 2] and can be read off from the distribution property [9, equation (3.72), page 76] as follows:

Differentiation gives

Putting , we obtain

which we solve in :

Substituting (3.2) and and simplifying, we conclude that

and that

whence (3.3).

Using these, we deduce from (2.37) the following.

Example 3.1.

Proof. By (1.11) and (3.1), for ,

Since , it follows from (3.3) that the left-hand side of (2.37) is

which is

where we used (2.31).

The right-hand side of (2.37), , becomes , in view of known values of [7, page 30].

Hence, altogether, (2.37) with reads

Invoking (2.11), this becomes (3.10).

We note that (3.14) gives a proof of the third equality in (3.2). Both (2.36) and (3.10) are contained in [14, 1999a] and are given as exercises in [7].

4. The Triple Gamma Function

For general material, we refer to [7, page 42]. As can been seen on [7, page 207], the important integral is not in closed form. Recently, Chakraborty-Kanemitsu-Kuzumaki [5, Corollary 1.1] have given a general expressions for all the integrals in , by appealing to Barnes' original results.

In this section, we shall give a direct derivation of a closed form by combining [7, (455), page 210] and [11, Corollary 3] (with ). The first reads

while the second reads (cf. also [15])

where are defined by

and where are the Stirling numbers of the second kind [7, page 58]. To express the values of , we appeal to [7]

(i) [7, (20), page 92], (ii) [7, pages 99-100]

and (2.31). After some elementary but long calculations, we arrive at

Combining we have the following.

Theorem 4.1 (see [5, Example 2.3]). Except for the singularities of the multiple gamma function, one has

This theorem enables us to put many formulas in [7] in closed form including, for instance, [7, (698), page 245]. Compare [5].

Acknowledgment

The authors would like to express their hearty thanks to Professor S. Kanemitsu for his enlightening supervision and encouragement.

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