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 Γ 1± t . This enables us to locate the genesis of two new functions A1/a and C1/a 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 A1/a functions themselves. We shall also give a direct proof of Theorem 4.1, which is a consequence of CKK, Corollary 1.1 , though.


Introduction
and the asymptotic formula to be satisfied N → ∞, where Γ s 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 log Γ α t or of multiple gamma functions Γ r cf., e.g., 4, 5 .Barnes' theorem 6, page 283 reads valid for nonintegral values of a.

International Journal of Mathematics and Mathematical Sciences 3
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 qanalogues 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 a 0 log Γ 1 − t dt and locates its genesis.A slight modification of Theorem 2.1 gives the counterpart of Barnes' formula 1.9 which reads.

1.10
Srivastava and Choi introduced two functions log A 1/a and log C 1/a by 2.9 and 2.9 with formal replacement of 1/a by −1/a, respectively.They state C 1/a A −1/a , which is rather ambiguous as to how we interpret the meaning because 2.9 is defined for a > 0 7, page 347, l.11 .They use this C 1/a function to express the integral a 0 log Γ 1 − t dt, without giving proof.This being the case, it may be of interest to locate the integral of log Γ 1 − t 7, 13 , page 349 , thereby log C 1/a 7, page 347 .
For this purpose we use a more fundamental function A a than A 1/a defined by where ζ s, a 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 C 1/a . 1.14

Barnes Formula
There is a generalization of 1.4 as well as Putting a 1/2, we obtain 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 log G t α are in closed form and the integral of log Γ 3 t α 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 log Γ 3 t α and the integral of ψ 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 : International Journal of Mathematics and Mathematical Sciences 5 where log A is the Glaisher-Kinkelin constant defined by 7, equation 2 , page 25 and log A 1/a is defined by 7, equation 9 , page 347 2.9 for a > 0.
Comparing 2.6 and 2.7 , we immediately obtain 2.10 on using the difference relation Γ a 1 aΓ a .Thus, in a sense we have located the genesis of the function log A 1/a .although they prove 2.7 by an elementary method 7, page 348 .
Indeed, A 1/a and A a are almost the same: a proof being given below.However, log A a is more directly connected with ζ −1, a 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 a 0 log Γ 1−t dt, thereby giving the genesis of the constant C 1/a .

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Proof.We evaluate the integral in two ways.First, On the other hand, noting that I is the sum of 2.1 and 2.7 , we deduce that

2.17
The first two terms on the right of 2.17 become while the 3rd and the 4th terms give, in view of 2.11 , 1/4 a 2 a log a.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 C 1/a and A 1/a is 1.14 , that is, one between C 1/a and A 1−1/a rather than C 1/a A −1/a 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.
9, 3.15 , page 59 , where the last integral may be also expressed as and where B k t is the kth periodic Bernoulli polynomial.Then 2.23 whence in particular, we have the generic formula for ζ −1, a and consequently for log A a through 1.11 :

2.24
This may be slightly modified in the form log A a lim a 2 a log a.

International Journal of Mathematics and Mathematical Sciences
The merit of using A a is that by way of ζ −1, a , we have a closed form for it:

2.26
In the same way, via another important relation 7, equation 23 , page 94 , log G a

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

2.28
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 γ 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 A is connected with A 1 and A 1 as follows: This can also be seen from Vardi's formula 7, 31 , page 97 : which is 1.11 with a 1.
We may also give another direct proof of Corollary 1.2.

Proof of Corollary 1.2 (another proof)
. log C 1/a is the limit of the expression

2.32
International Journal of Mathematics and Mathematical Sciences 9 where α 1 − a.Let N M 1.Then

2.33
Hence, simplifying, we find that

2.34
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 .

2.36
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 On exponentiating, 2.37 leads to 2.36 .
International Journal of Mathematics and Mathematical Sciences

Polygamma Function of Negative Order
In this section we introduce the function A k q 13 : which is closely related to the polygamma function of negative order and states some simple applications.We recall some properties of A k q : 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:  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 .

The Triple Gamma Function
For general material, we refer to 7, page 42 .As can been seen on 7, page 207 , the important integral z 0 log Γ 3 t a dt is not in closed form.Recently, Chakraborty-Kanemitsu-Kuzumaki 5, Corollary 1.1 have given a general expressions for all the integrals in log Γ r , 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 λ 3 .The first reads

4.5
This theorem enables us to put many formulas in 7 in closed form including, for instance, 7, 698 , page 245 .Compare 5 .

∞ n 1 −1 n− 1 2n − 1 2
If f z is analytic in a domain D containing the circle C : |z| r and has no zero on the circle, then the Gauss mean value theorem log f In 1, page 207 the case is considered where f z has a zero re iθ 0 on the circle, and 1.1 turns out that the Euler integral is essential in proving a generalization of Jensen's formula 1, pages 207-208 . 2 International Journal of Mathematics and Mathematical Sciences Let G denote the Catalan constant defined by the absolutely convergent series G L 2, χ 4 , 1.3 where χ 4 is the nonprincipal Dirichlet character mod 4. As a next step from 1.2 the relation true.In this connection, in 2 we obtained some results on G viewing it as an intrinsic value to the Barnes G-function.The Barnes G-function which is Γ 2 −1 in the class of multiple gamma functions is defined as the solution to the difference equation cf.2.3 log G z 1 − log G z log Γ z 1 where S m, n are the Stirling numbers of the second kind 7, page 58 .To express the values of ζ l − 3 , we appeal to 7 i ζ 0 − 1/2 log 2π 7, 20 , page 92 , ii ζ −2 log B ζ 3 /4π 3 log A 3 log B z.
1.2 in the form 7, equation 28 , page 31 : 27, pages 99-100 and 2.31 .After some elementary but long calculations, we arrive at Theorem 4.1 see 5, Example 2.3 .Except for the singularities of the multiple gamma function, one has