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X.-H. Wang, Y.-L. Lu, "The Multiple Gamma-Functions and the Log-Gamma Integrals", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 547459, 14 pages, 2012. https://doi.org/10.1155/2012/547459
The Multiple Gamma-Functions and the Log-Gamma Integrals
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.
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  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., ).
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 . For -analogues of the results, compare the recent book of the same authors . 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].
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  [7, equation (26), page 30]: Since (1.5) is equivalent to it follows that Putting , we obtain which is (1.2).
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  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 .
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 .
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.
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  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]).
We may also give another direct proof of Corollary 1.2.
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 : which is closely related to the polygamma function of negative order and states some simple applications. We recall some properties of :
Differentiation gives Putting , we obtain which we solve in :
Using these, we deduce from (2.37) the following.
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).
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 ) 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  (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
The authors would like to express their hearty thanks to Professor S. Kanemitsu for his enlightening supervision and encouragement.
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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.