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

Abstract

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 Γ₂=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 [8–13]). 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)]): log𝐺(𝑥+𝑎+1)=𝑥+𝑎2()1log2𝜋−log𝐴+−123𝑥²4+𝑥−𝑎𝑥₂2−1+𝑎122²+𝑎𝑥log𝑥+𝑂𝑥⁻¹(𝑥→∞),(1.1) 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 𝑓(𝑥)∶=𝑥^𝑞log𝑥𝑞∈ℕ;𝑥>0(1.2)ℓ times, we obtain 𝑓^(ℓ)(𝑥)=𝑥^{𝑞−ℓℓ}_𝑗=1𝑞−𝑗+1log𝑥+𝑃_ℓ𝑞ℓ∈ℕ;1≦ℓ≦𝑞,(1.3) where 𝑃_ℓ(𝑞) is a polynomial of degree ℓ−1 in 𝑞 satisfying the following recurrence relation: 𝑃_ℓ𝑞=𝑞−ℓ+1𝑃_ℓ−1𝑞+∏^ℓ−1_𝑗=1{1},1().𝑞−𝑗+1ℓ∈ℕ⧵;2≦ℓ≦𝑞ℓ=1(1.4) In fact, by mathematical induction on ℓ∈ℕ, we can give an explicit expression for 𝑃_ℓ(𝑞) as follows: 𝑃_ℓ𝑞=^ℓ_𝑗=1𝑞−𝑗+1⋅^ℓ_𝑗=11.𝑞−𝑗+1ℓ∈ℕ;1≦ℓ≦𝑞(1.5) Setting ℓ=𝑞 in (1.3) and (1.5), respectively, we get 𝑓^(𝑞)(𝑥)=𝑞!log𝑥+𝐻_𝑞𝑞∈ℕ,(1.6) where 𝐻_𝑛 are the harmonic numbers defined by 𝐻_𝑛∶=^𝑛_𝑗=11𝑗(𝑛∈ℕ).(1.7) Differentiating 𝑓^(𝑞)(𝑥) in (1.6) 𝑟 times, we obtain 𝑓^(𝑞+𝑟)(𝑥)=(−1)^𝑟+1()!1𝑞!𝑟−1𝑥^𝑟(𝑟∈ℕ).(1.8) Integrating the function 𝑓(𝑥) in (1.2) from 1 to 𝑛, we get ^𝑛₁𝑓(𝑥)𝑛𝑑𝑥=^𝑞+1𝑞+1log𝑛+1−𝑛^𝑞+1𝑞+1²𝑞,𝑛∈ℕ.(1.9)

For each 𝑞∈ℕ, define a sequence {𝐴_𝑞(𝑛)}^∞_𝑛=1 by log𝐴_𝑞(𝑛)∶=^𝑛_𝑘=1𝑘^𝑞−𝑛log𝑘_𝑞+1+𝑛𝑞+12^𝑞+^[(𝑞+1)/2]_𝑟=1𝐵_2𝑟()!⋅2𝑟^2𝑟−1_𝑗=1𝑞−𝑗+1⋅𝑛^{𝑞+1−2𝑟}+𝑛log𝑛_𝑞+1𝑞+1²−^{[(𝑞+1)/2]+((−1)𝑞−1)/2}_𝑟=1𝐵_2𝑟()!𝑃2𝑟^2𝑟−1𝑞𝑛^{𝑞+1−2𝑟}𝑛,𝑞∈ℕ,(1.10) 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 log𝐴_𝑞∶=lim_𝑛→∞log𝐴_𝑞(𝑛)𝑞∈ℕ.(1.11) The Bernoulli numbers 𝐵_𝑟 are defined by the generating function (see [21, Section 1.6]; see also, [26, Section 1.7]): 𝑧𝑒^𝑧=−1^∞_𝑟=0𝐵_𝑟𝑧^𝑟𝑧|).𝑟!(|<2𝜋(1.12) We introduce a well-known formula (see [21, Section 2.3]): 𝐵_2𝑝=(−1)^𝑝+122𝑝!(2𝜋)^2𝑝𝜁2𝑝𝑝∈ℕ₀∶=ℕ∪{0},(1.13) where 𝜁(𝑠) is the Riemann Zeta function defined by 𝜁(𝑠)∶=^∞_𝑛∑₌₁1𝑛^𝑠=11−2^∞−𝑠_𝑛∑₌₁1(2𝑛−1)^𝑠(ℜ(𝑠))1>11−2^∞1−𝑠_𝑛∑₌₁(−1)^𝑛−1𝑛^𝑠(ℜ(𝑠)>0;𝑠≠1).(1.14) It is easy to observe from (1.13) that 𝐵_4𝑝<0,𝐵_4𝑝+2>0𝑝∈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=1𝑛𝑘log𝑘−²2+𝑛2+1𝑛12log𝑛+²4=log𝐴,(1.16) where 𝐴 denotes the Glaisher-Kinkelin constant whose numerical value is 𝐴≅1.282427130⋯,log𝐴₂=lim_𝑛→∞^𝑛_𝑘=1𝑘²𝑛log𝑘−³3+𝑛²2+𝑛6𝑛log𝑛+³9−𝑛12=log𝐵,log𝐴₃=lim_𝑛→∞^𝑛_𝑘=1𝑘³𝑛log𝑘−⁴4+𝑛³2+𝑛²4−1𝑛120log𝑛+⁴−𝑛1612²=log𝐶.(1.17) Here 𝐵 and 𝐶 are constants whose approximate numerical values are given by 𝐵≅1.03091675⋯,𝐶≅0.97955746⋯.(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)): ^𝑛_𝑘=1𝑓(𝑘)∼𝐶₀+^𝑛_𝑎𝑓(𝑥)1𝑑𝑥+2𝑓(𝑛)+^∞_𝑟=1𝐵_2𝑟()!𝑓2𝑟^(2𝑟−1)(𝑛),(2.1) where 𝐶₀ 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].

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 ^𝑚_𝑘=0𝑓()=1𝑎+𝑘ℎℎ^𝑏_𝑎𝑓(𝑥)𝑓(𝑎)(𝑏)𝑑𝑥++𝑓2+^𝑝_𝑘=1ℎ^2𝑘−1()!𝐵2𝑘^2𝑘𝑓^(2𝑘−1)(𝑏)−𝑓^(2𝑘−1)(𝑎)+ℎ^2𝑝+2!𝐵2𝑝+2^{2𝑝+2𝑚−1}_𝑘=0𝑓^(2𝑝+2)(𝑎+𝑘ℎ+𝜃ℎ),(2.2) 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]): ^𝑚_𝑘=1𝑓(𝑘)=^𝑛₁𝑓(𝑥)𝑓(1)(𝑛)𝑑𝑥++𝑓2+^𝑝_𝑘=1𝐵_2𝑘()!𝑓2𝑘^(2𝑘−1)(𝑛)−𝑓^(2𝑘−1)(1)+𝑅_𝑛𝑓,𝑝,(2.3) where, for convenience, the remainder term 𝑅_𝑛(𝑓,𝑝) is given by 𝑅_𝑛𝐵𝑓,𝑝∶=^2𝑝+22𝑝+2!^𝑛−1_𝑘=1𝑓^(2𝑝+2)(𝑘+𝜃)(2.4) which is seen to be bounded by 𝑅_𝑛≦2𝑓,𝑝(2𝜋)^2𝑝𝑛1𝑓^(2𝑝+1)(𝑥)𝑑𝑥.(2.5)

Zhu and Yang [33] established some useful formulas originated from the Euler-Maclaurin summation formula (2.1) (see also [25]) asserted by the following lemma.

Lemma 2.1. Let ℓ∈ℕ and let 𝑓 have its first 2𝑝+2 derivatives on an interval [ℓ,∞) such that 𝑓^(2𝑝)(𝑥)>0 and 𝑓^(2𝑝+2)(𝑥)>0 (or 𝑓^(2𝑝)(𝑥)<0 and 𝑓^(2𝑝+2)(𝑥)<0) and 𝑓^(2𝑝−1)(∞)=0. Then the following results hold true: (i)The sequence 𝐚_𝐧∶=^𝑛_𝑘=ℓ𝑓(𝑘)−^𝑛_ℓ𝑓(𝑥)1𝑑𝑥−2𝑓(𝑛)−^𝑝−1_𝑘=1𝐵_2𝑘()!𝑓2𝑘^(2𝑘−1)(𝑛)(𝑛≧ℓ)(2.6) converges. Let 𝐚∶=lim_𝑛→∞𝐚_𝐧. (ii)For 𝑓^(2𝑝)(𝑥)>0 and 𝑓^(2𝑝+2)(𝑥)>0, we have 0<(−1)^𝑝−1𝐚−𝐚_𝐧<(−1)^𝑝𝐵_2𝑝!𝑓2𝑝^(2𝑝−1)(𝑛)(𝑛≧ℓ).(2.7) For 𝑓^(2𝑝)(𝑥)<0 and 𝑓^(2𝑝+2)(𝑥)<0, we have 0>(−1)^𝑝−1𝐚−𝐚_𝐧>(−1)^𝑝𝐵_2𝑝!𝑓2𝑝^(2𝑝−1)(𝑛)(𝑛≧ℓ).(2.8)(iii)There exists 𝜃∈(0,1) such that ^𝑛_𝑘=ℓ𝑓(𝑘)=𝐚+^𝑛_ℓ𝑓(𝑥)1𝑑𝑥+2𝑓(𝑛)+^𝑝−1_𝑘=1𝐵_2𝑘()!𝑓2𝑘^(2𝑘−1)(𝑛)𝐵+𝜃⋅^2𝑝!𝑓2𝑝^(2𝑝−1)(𝑛).(2.9)

3. 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: log𝐴_𝑞(𝑛)∼𝐶_𝑞+1𝑞+1²+1−(−1)^𝑞2𝐵_𝑞+1𝐻_𝑞+()𝑞+1−1^𝑞𝑞!^∞_{𝑟=𝑞+1/2+1}𝐵_2𝑟()!!2𝑟2𝑟−𝑞−2𝑛^{2𝑟−𝑞−1},(3.1) where 𝐶_𝑞's are constants dependent on each 𝑞 and an empty sum is understood (as usual) to be nil. And log𝐴_𝑞=lim_𝑛→∞log𝐴_𝑞(𝑛)=𝐶_𝑞+1𝑞+1²+1−(−1)^𝑞2𝐵_𝑞+1𝐻_𝑞.𝑞+1(3.2)

Proof. We only note that(i)1≦𝑟≦[(𝑞+1)/2]𝑓^(2𝑟−1)(𝑛)=𝑛^{𝑞+1−2𝑟}⋅^2𝑟−1_𝑗=1𝑞−𝑗+1⋅log𝑛+𝑛^{𝑞+1−2𝑟}𝑃_2𝑟−1𝑞.(3.3)(ii)𝑟≧[(𝑞+1)/2]+1𝑓^(2𝑟−1)(𝑛)=(−1)^𝑞!𝑞!2𝑟−𝑞−2𝑛^{2𝑟−𝑞−1}.(3.4)

Applying the function 𝑓(𝑥) in (1.2) to the formula (2.9) with ℓ=1, and using the results presented in Lemma 1.1, we get two sided inequalities for the difference of log𝐴_𝑞(𝑛) and log𝐴_𝑞 asserted by Theorem 3.2.

Theorem 3.2. The following inequalities hold true: 𝑞!^2𝑝_{𝑟=𝑞+1/2+1}2𝑟−𝑞−2!()!𝐵2𝑟𝑛^{2𝑟2𝑟−𝑞−1}<(−1)^𝑞log𝐴_𝑞(𝑛)−log𝐴_𝑞<𝑞!^2𝑝+1_{𝑟=𝑞+1/2+1}2𝑟−𝑞−2!()!𝐵2𝑟𝑛^{2𝑟2𝑟−𝑞−1}𝑝,𝑞,𝑛∈ℕ.(3.5)

Proof. Setting the function 𝑓(𝑥) in (1.2) in the formula (2.9) with ℓ=1, and using the results presented in Lemma 1.1, we get log𝐴_𝑞(𝑛)=log𝐴_𝑞+(−1)^𝑞𝑞!^𝑝−1_{𝑟=𝑞+1/2+1}2𝑟−𝑞−2!()!𝐵2𝑟𝑛^{2𝑟2𝑟−𝑞−1}+(−1)^𝑞!𝑞!2𝑝−𝑞−2!𝐵2𝑝𝑛^{2𝑝2𝑝−𝑞−1}𝜃,(3.6) for some 𝜃∈(0,1).
Replacing 𝑝 by 2𝑝+1 and 2𝑝+2 in (3.6), respectively, we obtain log𝐴_𝑞(𝑛)−log𝐴_𝑞=(−1)^𝑞𝑞!^2𝑝_{𝑟=𝑞+1/2+1}2𝑟−𝑞−2!()!𝐵2𝑟𝑛^{2𝑟2𝑟−𝑞−1}+(−1)^𝑞!𝑞!4𝑝−𝑞!𝐵4𝑝+2𝑛^{4𝑝+24𝑝+1−𝑞}𝜃.log𝐴_𝑞(𝑛)−log𝐴_𝑞=(−1)^𝑞𝑞!^2𝑝+1_{𝑟=𝑞+1/2+1}2𝑟−𝑞−2!()!𝐵2𝑟𝑛^{2𝑟2𝑟−𝑞−1}+(−1)^𝑞!𝑞!4𝑝+2−𝑞!𝐵4𝑝+4𝑛^{4𝑝+44𝑝+3−𝑞}𝜃.(3.7) In view of (1.15), we find the following inequalities: 𝑞!^2𝑝_{𝑟=𝑞+1/2+1}2𝑟−𝑞−2!()!𝐵2𝑟𝑛^{2𝑟2𝑟−𝑞−1}<log𝐴_𝑞(𝑛)−log𝐴_𝑞<𝑞!^2𝑝+1_{𝑟=𝑞+1/2+1}2𝑟−𝑞−2!()!𝐵2𝑟𝑛^{2𝑟2𝑟−𝑞−1}𝑞iseven,𝑞!^2𝑝_{𝑟=𝑞+1/2+1}2𝑟−𝑞−2!()!𝐵2𝑟𝑛^{2𝑟2𝑟−𝑞−1}<log𝐴_𝑞−log𝐴_𝑞(𝑛)<𝑞!^2𝑝+1_{𝑟=𝑞+1/2+1}2𝑟−𝑞−2!()!𝐵2𝑟𝑛^{2𝑟2𝑟−𝑞−1}𝑞isodd.(3.8) Finally it is easily seen that the two-sided inequalities 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 log𝐴_𝑞 asserted by Theorem 3.4.

Theorem 3.4. The following inequalities hold true: 𝛼_𝑞+𝑞!^2𝑝_{𝑘=𝑞+1/2+1}2𝑘−𝑞−2!()!𝐵2𝑘^2𝑘<log𝐴_𝑞<𝛼_𝑞+𝑞!^2𝑝+1_{𝑘=𝑞+1/2+1}2𝑘−𝑞−2!()!𝐵2𝑘^2𝑘,𝛼𝑞isodd(3.9)_𝑞−𝑞!^2𝑝+1_{𝑘=𝑞+1/2+1}2𝑘−𝑞−2!()!𝐵2𝑘^2𝑘<log𝐴_𝑞<𝛼_𝑞−𝑞!^2𝑝_{𝑘=𝑞+1/2+1}2𝑘−𝑞−2!()!𝐵2𝑘^2𝑘𝑞iseven,(3.10) where, for convenience, 𝛼_𝑞1∶=𝑞+1²+1−(−1)^𝑞2𝐵_𝑞+1𝐻_𝑞−𝑞+1^[(𝑞+1)/2]_𝑘=1𝐵_2𝑘()!𝑃2𝑘^2𝑘−1𝑞𝑞∈ℕ.(3.11)

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), log𝐴_𝑞(𝑛)=𝛼_𝑞+(−1)^𝑞𝑞!^𝑝_{𝑘=𝑞+1/2+1}2𝑘−2−𝑞!()!𝐵2𝑘^2𝑘1𝑛^{2𝑘−1−𝑞}+()−1−1^𝑞+1!𝑞!2𝑝+1−𝑞!𝐵2𝑝+2^{2𝑝+2𝑛−1}_𝑘=11(𝑘+𝜃)^{2𝑝+2−𝑞}.(3.12) Replacing 𝑝 by 2𝑝 and 2𝑝+1 in (3.12), respectively, we obtain log𝐴_𝑞(𝑛)=𝛼_𝑞+(−1)^𝑞𝑞!^2𝑝_{𝑘=𝑞+1/2+1}2𝑘−2−𝑞!()!𝐵2𝑘^2𝑘1𝑛^{2𝑘−1−𝑞}+()−1−1^𝑞+1!𝑞!4𝑝+1−𝑞!𝐵4𝑝+2^{4𝑝+2𝑛−1}_𝑘=11(𝑘+𝜃)^{4𝑝+2−𝑞},log𝐴_𝑞(𝑛)=𝛼_𝑞+(−1)^𝑞𝑞!^2𝑝+1_{𝑘=𝑞+1/2+1}2𝑘−2−𝑞!()!𝐵2𝑘^2𝑘1𝑛^{2𝑘−1−𝑞}+()−1−1^𝑞+1!𝑞!4𝑝+3−𝑞!𝐵4𝑝+4^{4𝑝+4𝑛−1}_𝑘=11(𝑘+𝜃)^{4𝑝+4−𝑞}.(3.13) In view of (1.15), we find from that 𝛼_𝑞−𝑞!^2𝑝_{𝑘=𝑞+1/2+1}2𝑘−2−𝑞!()!𝐵2𝑘^2𝑘1𝑛^{2𝑘−1−𝑞}−1<log𝐴_𝑞(𝑛)<𝛼_𝑞−𝑞!^2𝑝+1_{𝑘=𝑞+1/2+1}2𝑘−2−𝑞!()!𝐵2𝑘^2𝑘1𝑛^{2𝑘−1−𝑞},𝛼−1𝑞isodd_𝑞+𝑞!^2𝑝+1_{𝑘=𝑞+1/2+1}2𝑘−2−𝑞!()!𝐵2𝑘^2𝑘1𝑛^{2𝑘−1−𝑞}−1<log𝐴_𝑞(𝑛)<𝛼_𝑞+𝑞!^2𝑝_{𝑘=𝑞+1/2+1}2𝑘−2−𝑞!()!𝐵2𝑘^2𝑘1𝑛^{2𝑘−1−𝑞}−1𝑞iseven.(3.14) Now, taking the limit on each side of the inequalities in 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.

Acknowledgments

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).