A Set of Mathematical Constants Arising Naturally in the Theory of the Multiple Gamma Functions
We introduce a set of mathematical constants which is involved naturally in the theory of multiple Gamma functions. Then we present general asymptotic inequalities for these constants whose special cases are seen to contain all results very recently given in Chen 2011.
1. Introduction and Preliminaries
The double Gamma function and the multiple Gamma functions 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 , Alexeiewsky  and Kinkelin . In about the middle of the 1980s, these functions were revived in the study of the determinants of the Laplacians on the -dimensional unit sphere (see [8–13]). 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 given in (1.11) involves naturally in the theory of the multiple Gamma functions (see [14–20] and references therein). For example, for sufficiently large real and , we have the Stirling formula for the -function (see ; see also [21, page 26, equation (7)]): where is the Glaisher-Kinkelin constant (see [7, 22–24]) given in (1.16) below. The Glaisher-Kinkelin constant , the constants and 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 , , and are significant. Very recently Chen  presented nice asymptotic inequalities for these constants , , and by mainly using the Euler-Maclaurin summation formulas. Here, we aim at presenting asymptotic inequalities for a set of the mathematical constants given in (1.11) some of whose special cases are seen to yield all results in .
For this purpose, we begin by summarizing some differential and integral formulas of the function in (1.2).
Lemma 1.1. Differentiating the function times, we obtain where is a polynomial of degree in satisfying the following recurrence relation: In fact, by mathematical induction on , we can give an explicit expression for as follows: Setting in (1.3) and (1.5), respectively, we get where are the harmonic numbers defined by Differentiating in (1.6) times, we obtain Integrating the function in (1.2) from to , we get
For each , define a sequence by where are Bernoulli numbers given in (1.12), are given in (1.5), and denotes (as usual) the greatest integer ≦. Define a set of mathematical constants by The Bernoulli numbers 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 is the Riemann Zeta function defined by It is easy to observe from (1.13) that
Remark 1.2. We find that the constants , and correspond with the Glaisher-Kinkelin constant , the constants and introduced by Choi and Srivastava, respectively: where denotes the Glaisher-Kinkelin constant whose numerical value is Here and are constants whose approximate numerical values are given by The constants and were considered recently by Choi and Srivastava [16, 18]. See also Adamchik [27, page 199]. Bendersky  presented a set of constants including and .
2. Euler-Maclaurin Summation Formula
We begin by recalling the Euler-Maclaurin summation formula (cf. Hardy ([29, 30], page 318)): where is an arbitrary constant to be determined in each special case and are the Bernoulli numbers given in (1.12). For another useful summation formula, see Edwards [31, page 117].
Let be a function of class , and let the interval be partitioned into subintervals of the same length . Then we have another useful form of the Euler-Maclaurin summation formula (see, e.g., ): There exists such that where . Under the same conditions in (2.2), setting , , , and in (2.2), we obtain a simple summation formula (see ): where, for convenience, the remainder term is given by which is seen to be bounded by
Lemma 2.1. Let and let have its first derivatives on an interval such that and (or and ) and . Then the following results hold true: (i)The sequence converges. Let . (ii)For and , we have For and , we have (iii)There exists such that
3. Asymptotic Formulas and Inequalities for
Applying the function in (1.2) to the Euler-Maclaurin summation formula (2.1) with and using the results presented in Lemma 1.1, we obtain an asymptotic formula for the sequence as in the following theorem.
Theorem 3.1. The following asymptotic formulas for the constants and hold true: where 's are constants dependent on each and an empty sum is understood (as usual) to be nil. And
Proof. We only note that(i)(ii)
Theorem 3.2. The following inequalities hold true:
Proof. Setting the function in (1.2) in the formula (2.9) with , and using the results presented in Lemma 1.1, we get
for some .
Replacing by and in (3.6), respectively, we obtain In view of (1.15), we find the following inequalities: Finally it is easily seen that the two-sided inequalities can be expressed in a single form (3.5).
Theorem 3.4. The following inequalities hold true: where, for convenience,
Proof. Setting the function in (1.2) in the formula (2.3), and using the results presented in Lemma 1.1, we have, for some , Replacing by and in (3.12), respectively, we obtain In view of (1.15), we find from that Now, taking the limit on each side of the inequalities in as , we obtain the results in Theorem 3.4.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2012-0002957).
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