A Set of Mathematical Constants Arising Naturally in the Theory of the Multiple Gamma Functions

and Applied Analysis 3 Differentiating f q x in 1.6 r times, we obtain f q r x −1 r q! r − 1 ! 1 xr r ∈ N . 1.8 Integrating the function f x in 1.2 from 1 to n, we get ∫n 1 f x dx n 1 q 1 logn 1 − n 1 ( q 1 )2 (q, n ∈ N). 1.9 For each q ∈ N, define a sequence {Aq n }n 1 by logAq n : n ∑ k 1 k log k − ⎛ ⎝ n 1 q 1 n 2 q 1 /2 ∑ r 1 B2r 2r ! · 2r−1 ∏ j 1 ( q − j 1) · n 1−2r ⎞ ⎠ logn n 1 ( q 1 )2 − q 1 /2 −1 −1 /2 ∑ r 1 B2r 2r ! P2r−1 ( q ) n 1−2r ( n, q ∈ N), 1.10 where Br are Bernoulli numbers given in 1.12 , Pr q are given in 1.5 , and x denotes as usual the greatest integer x. Define a set of mathematical constants Aq q ∈ N by logAq : lim n→∞ logAq n ( q ∈ N). 1.11 The Bernoulli numbers Br are defined by the generating function see 21, Section 1.6 ; see also, 26, Section 1.7 : z ez − 1 ∞ ∑ r 0 Br z r! |z| < 2π . 1.12 We introduce a well-known formula see 21, Section 2.3 : B2p −1 p 1 2 ( 2p ) ! 2π 2p ζ ( 2p ) ( p ∈ N0 : N ∪ {0} ) , 1.13 where ζ s is the Riemann Zeta function defined by ζ s : ⎪⎪⎪⎨ ⎪⎪⎩ ∞ ∑ n 1 1 ns 1 1 − 2−s ∞ ∑ n 1 1 2n − 1 s s > 1 1 1 − 21−s ∞ ∑ n 1 −1 n−1 ns s > 0; s / 1 . 1.14 4 Abstract and Applied Analysis It is easy to observe from 1.13 that B4p < 0, B4p 2 > 0 ( p ∈ N0 ) . 1.15 Remark 1.2. We find that the constants A1, A2 and A3 correspond with the Glaisher-Kinkelin constant A, the constants B and C introduced by Choi and Srivastava, respectively: logA1 lim n→∞ { n ∑ k 1 k log k − ( n2 2 n 2 1 12 ) log n n2 4 } logA, 1.16 where A denotes the Glaisher-Kinkelin constant whose numerical value is A ∼ 1.282427130 · · · , logA2 lim n→∞ { n ∑ k 1 k2 log k − ( n3 3 n2 2 n 6 ) logn n3 9 − n 12 } logB, logA3 lim n→∞ { n ∑ k 1 k3 log k − ( n4 4 n3 2 n2 4 − 1 120 ) logn n4 16 − n 2 12 } logC. 1.17 Here B and C are constants whose approximate numerical values are given by B ∼ 1.03091 675 · · · , C ∼ 0.97955 746 · · · . 1.18 The constants B and C were considered recently by Choi and Srivastava 16, 18 . See also Adamchik 27, page 199 . Bendersky 28 presented a set of constants including B and C. 2. Euler-Maclaurin Summation Formula We begin by recalling the Euler-Maclaurin summation formula cf. Hardy 29, 30 , page 318 :


Introduction and Preliminaries
The double Gamma function Γ 2 1/G and the multiple Gamma functions Γ n were defined and studied systematically by Barnes 1-4 in about 1900. Before their investigation by Barnes, these functions had been introduced in a different form by, for example, Hölder 5 , Alexeiewsky 6 and Kinkelin 7 . In about the middle of the 1980s, these functions were revived in the study of the determinants of the Laplacians on the n-dimensional unit sphere S n see [8][9][10][11][12][13] . Since then the multiple Gamma functions have attracted many authors' concern and have been used in various ways. It is seen that a set of constants {A q | q ∈ N : {1, 2, 3, . . .}} given in 1.11 involves naturally in the theory of the multiple Gamma functions Γ n see 14-20 and references therein . For example, for sufficiently large real x and a ∈ C, we have the Stirling formula for the G-function see 1 ; see also 21, page 26, equation 7 : where A is the Glaisher-Kinkelin constant see 7, 22-24 given in 1.16 below. The Glaisher-Kinkelin constant A, the constants B and C below introduced by Choi and Srivastava have been used, among other things, in the closed-form evaluation of certain series involving zeta functions and in calculation of some integrals of multiple Gamma functions. So trying to give asymptotic formulas for these constants A, B, and C are significant. Very recently Chen 25 presented nice asymptotic inequalities for these constants A, B, and C by mainly using the Euler-Maclaurin summation formulas. Here, we aim at presenting asymptotic inequalities for a set of the mathematical constants A q q ∈ N given in 1.11 some of whose special cases are seen to yield all results in 25 .
For this purpose, we begin by summarizing some differential and integral formulas of the function f x in 1.2 .

Lemma 1.1. Differentiating the function
where P q is a polynomial of degree − 1 in q satisfying the following recurrence relation:

1.4
In fact, by mathematical induction on ∈ N, we can give an explicit expression for P q as follows: Setting q in 1.3 and 1.5 , respectively, we get where H n are the harmonic numbers defined by H n : Differentiating f q x in 1.6 r times, we obtain Integrating the function f x in 1.2 from 1 to n, we get For each q ∈ N, define a sequence {A q n } ∞ n 1 by log A q n : where B r are Bernoulli numbers given in 1.12 , P r q are given in 1.5 , and x denotes as usual the greatest integer x. Define a set of mathematical constants A q q ∈ N by The Bernoulli numbers B r are defined by the generating function see 21, Section 1.6 ; see also, 26, Section 1.7 : We introduce a well-known formula see 21, Section 2.3 : where ζ s is the Riemann Zeta function defined by ζ s :

4 Abstract and Applied Analysis
It is easy to observe from 1.13 that Remark 1.2. We find that the constants A 1 , A 2 and A 3 correspond with the Glaisher-Kinkelin constant A, the constants B and C introduced by Choi and Srivastava, respectively: where A denotes the Glaisher-Kinkelin constant whose numerical value is 1.17 Here B and C are constants whose approximate numerical values are given by The constants B and C were considered recently by Choi and Srivastava 16,18 . See also Adamchik 27, page 199 . Bendersky 28 presented a set of constants including B and C.

Euler-Maclaurin Summation Formula
We begin by recalling the Euler-Maclaurin summation formula cf. Hardy 29,30 , page 318 : where C 0 is an arbitrary constant to be determined in each special case and B r are the Bernoulli numbers given in 1.12 . For another useful summation formula, see Edwards 31, page 117 .
Abstract and Applied Analysis 5 Let f be a function of class C 2p 2 a, b , and let the interval a, b be partitioned into m subintervals of the same length h b − a /m. Then we have another useful form of the Euler-Maclaurin summation formula see, e.g., 32 : There exists 0 < θ < 1 such that where m, p ∈ N. Under the same conditions in 2.2 , setting m n − 1, a 1, b n, and h 1 in 2.2 , we obtain a simple summation formula see 25 : where, for convenience, the remainder term R n f, p is given by  i The sequence a n : converges. Let a : lim n → ∞ a n .
ii For f 2p x > 0 and f 2p 2 x > 0, we have For f 2p x < 0 and f 2p 2 x < 0, we have iii There exists θ ∈ 0, 1 such that

Asymptotic Formulas and Inequalities for A q
Applying the function f x in 1.2 to the Euler-Maclaurin summation formula 2.1 with a 1 and using the results presented in Lemma 1.1, we obtain an asymptotic formula for the sequence A q n as in the following theorem.
Theorem 3.1. The following asymptotic formulas for the constants A q n and A q hold true:

3.1
where C q 's are constants dependent on each q and an empty sum is understood (as usual) to be nil. And Proof. We only note that i 1 r q 1 /2 f 2r−1 n n q 1−2r · 2r−1 j 1 q − j 1 · log n n q 1−2r P 2r−1 q .

3.3
Abstract and Applied Analysis Applying the function f x in 1.2 to the formula 2.9 with 1, and using the results presented in Lemma 1.1, we get two sided inequalities for the difference of log A q n and log A q asserted by Theorem 3.2.
Theorem 3.2. The following inequalities hold true:
Setting the function f x in 1.2 in the formula 2.9 with 1, and using the results presented in Lemma 1.1, we get for some θ ∈ 0, 1 .

8 Abstract and Applied Analysis
In view of 1.15 , we find the following inequalities: Finally it is easily seen that the two-sided inequalities 3.8 can be expressed in a single form 3.5 .
Remark 3.3. The special cases of 3.5 when q 1, q 2, and q 3 are easily seen to correspond with Equations 8 , 31 , and 32 in Chen's work 25 , respectively.
Applying the function f x in 1.2 to the formula 2.3 and using the results presented in Lemma 1.1, we get two-sided inequalities for the log A q asserted by Theorem 3.4.
Theorem 3.4. The following inequalities hold true:

3.10
Abstract and Applied Analysis 9 where, for convenience, Proof.
Setting the function f x in 1.2 in the formula 2.3 , and using the results presented in Lemma 1.1, we have, for some θ ∈ 0, 1 ,

3.12
Replacing p by 2p and 2p 1 in 3.12 , respectively, we obtain

3.13
In view of 1.15 , we find from 3.13 that

10
Abstract and Applied Analysis 3.14 Now, taking the limit on each side of the inequalities in 3.14 as n → ∞, we obtain the results in Theorem 3.4.

Remark 3.5.
It is easily seen that the specialized inequalities of 3.9 when q 1 and q 3 and 3.10 when q 2 correspond with those inequalities of Equations 9 , 34 , and 33 in Chen's work 25 , respectively.