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

Authors: Junesang Choi
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 1. Introduction and Preliminaries

The double Gamma functionΓ2=1/G and the multiple Gamma functions Γn 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 Sn (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 {Aq|q∈ℕ:={1,2,3,…}} given in (1.11) involves naturally in the theory of the multiple Gamma functions Γn (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)]): (1.1)logG(x+a+1)=x+a2log(2π)-logA+112-3x24-ax+(x22-112+a22+ax)logx+O(x-1)(x→∞),
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 Aq  (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 functionf(x) in (1.2).Lemma 1.1.
Differentiating the function(1.2)f(x):=xqlogx(q∈ℕ;x>0)ℓ times, we obtain
(1.3)f(ℓ)(x)=xq-ℓ{∏j=1ℓ‍(q-j+1)logx+Pℓ(q)}(ℓ∈ℕ;1≦ℓ≦q),
where Pℓ(q) is a polynomial of degree ℓ-1 in q satisfying the following recurrence relation:
(1.4)Pℓ(q)={(q-ℓ+1)Pℓ-1(q)+∏j=1ℓ-1‍(q-j+1)(ℓ∈ℕ∖{1};2≦ℓ≦q),1(ℓ=1).
In fact, by mathematical induction on ℓ∈ℕ, we can give an explicit expression for Pℓ(q) as follows:
(1.5)Pℓ(q)=∏j=1ℓ‍(q-j+1)·∑j=1ℓ‍1q-j+1(ℓ∈ℕ;1≦ℓ≦q).
Setting ℓ=q in (1.3) and (1.5), respectively, we get
(1.6)f(q)(x)=q!(logx+Hq)(q∈ℕ),
where Hn are the harmonic numbers defined by
(1.7)Hn:=∑j=1n‍1j(n∈ℕ).
Differentiating f(q)(x) in (1.6) r times, we obtain
(1.8)f(q+r)(x)=(-1)r+1q!(r-1)!1xr(r∈ℕ).
Integrating the function f(x) in (1.2) from 1 to n, we get
(1.9)∫1n‍f(x)dx=nq+1q+1logn+1-nq+1(q+1)2(q,n∈ℕ).For eachq∈ℕ, define a sequence {Aq(n)}n=1∞ by
(1.10)logAq(n):=∑k=1n‍kqlogk-(nq+1q+1+nq2+∑r=1[(q+1)/2]‍B2r(2r)!·∏j=12r-1‍(q-j+1)·nq+1-2r)logn+nq+1(q+1)2-∑r=1[(q+1)/2]+((-1)q-1)/2‍B2r(2r)!P2r-1(q)nq+1-2r(n,q∈ℕ),
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
(1.11)logAq:=limn→∞logAq(n)(q∈ℕ).
The Bernoulli numbers Br are defined by the generating function (see [21, Section 1.6]; see also, [26, Section 1.7]):
(1.12)zez-1=∑r=0∞Brzrr!(|z|<2π).
We introduce a well-known formula (see [21, Section 2.3]):
(1.13)B2p=(-1)p+12(2p)!(2π)2pζ(2p)(p∈ℕ0:=ℕ∪{0}),
where ζ(s) is the Riemann Zeta function defined by
(1.14)ζ(s):={∑n=1∞‍1ns=11-2-s∑n=1∞‍1(2n-1)s(ℜ(s)>1)11-21-s∑n=1∞‍(-1)n-1ns(ℜ(s)>0;s≠1).
It is easy to observe from (1.13) that
(1.15)B4p<0,B4p+2>0(p∈ℕ0).Remark 1.2.
We find that the constantsA1, A2 and A3 correspond with the Glaisher-Kinkelin constant A, the constants B and C introduced by Choi and Srivastava, respectively:
(1.16)logA1=limn→∞{∑k=1nklogk-(n22+n2+112)logn+n24}=logA,
where A denotes the Glaisher-Kinkelin constant whose numerical value is
(1.17)A≅1.282427130⋯,logA2=limn→∞{∑k=1n‍k2logk-(n33+n22+n6)logn+n39-n12}=logB,logA3=limn→∞{∑k=1n‍k3logk-(n44+n32+n24-1120)logn+n416-n212}=logC.
Here B and C are constants whose approximate numerical values are given by
(1.18)B≅1.03091675⋯,C≅0.97955746⋯.
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)):
(2.1)∑k=1n‍f(k)~C0+∫anf(x)dx+12f(n)+∑r=1∞‍B2r(2r)!f(2r-1)(n),
where C0 is an arbitrary constant to be determined in each special case and Br are the Bernoulli numbers given in (1.12). For another useful summation formula, see Edwards [31, page 117].Letf be a function of class C2p+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
(2.2)∑k=0m‍f(a+kh)=1h∫ab‍f(x)dx+f(a)+f(b)2+∑k=1p‍h2k-1(2k)!B2k(f(2k-1)(b)-f(2k-1)(a))+h2p+2(2p+2)!B2p+2∑k=0m-1‍f(2p+2)(a+kh+θh),
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]):
(2.3)∑k=1m‍f(k)=∫1nf(x)dx+f(1)+f(n)2+∑k=1p‍B2k(2k)!(f(2k-1)(n)-f(2k-1)(1))+Rn(f,p),
where, for convenience, the remainder term Rn(f,p) is given by
(2.4)Rn(f,p):=B2p+2(2p+2)!∑k=1n-1‍f(2p+2)(k+θ)
which is seen to be bounded by
(2.5)|Rn(f,p)|≦2(2π)2p∫1n|f(2p+1)(x)|dx.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.
Letℓ∈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(2.6)an:=∑k=ℓn‍f(k)-∫ℓnf(x)dx-12f(n)-∑k=1p-1‍B2k(2k)!f(2k-1)(n)(n≧ℓ)
converges. Let a:=limn→∞an.(ii)
Forf(2p)(x)>0 and f(2p+2)(x)>0, we have
(2.7)0<(-1)p-1(a-an)<(-1)pB2p(2p)!f(2p-1)(n)(n≧ℓ).
For f(2p)(x)<0 and f(2p+2)(x)<0, we have
(2.8)0>(-1)p-1(a-an)>(-1)pB2p(2p)!f(2p-1)(n)(n≧ℓ).(iii)
There existsθ∈(0,1) such that
(2.9)∑k=ℓn‍f(k)=a+∫ℓnf(x)dx+12f(n)+∑k=1p-1‍B2k(2k)!f(2k-1)(n)+θ·B2p(2p)!f(2p-1)(n).


## 3. Asymptotic Formulas and Inequalities forAq

Applying the functionf(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 Aq(n) as in the following theorem.Theorem 3.1.
The following asymptotic formulas for the constantsAq(n) and Aq hold true:
(3.1)logAq(n)~Cq+1(q+1)2+1-(-1)q2Bq+1Hqq+1+(-1)qq!∑r=[(q+1)/2]+1∞‍B2r(2r)!(2r-q-2)!n2r-q-1,
where Cq's are constants dependent on each q and an empty sum is understood (as usual) to be nil. And
(3.2)logAq=limn→∞logAq(n)=Cq+1(q+1)2+1-(-1)q2Bq+1Hqq+1.Proof.
We only note that(i)
1≦r≦[(q+1)/2](3.3)f(2r-1)(n)=nq+1-2r·∏j=12r-1‍(q-j+1)·logn+nq+1-2rP2r-1(q).(ii)
r≧[(q+1)/2]+1(3.4)f(2r-1)(n)=(-1)qq!(2r-q-2)!n2r-q-1.Applying the functionf(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 logAq(n) and logAq asserted by Theorem 3.2.Theorem 3.2.
The following inequalities hold true:(3.5)q!∑r=[(q+1)/2]+12p‍(2r-q-2)!(2r)!B2rn2r-q-1<(-1)q(logAq(n)-logAq)<q!∑r=[(q+1)/2]+12p+1‍(2r-q-2)!(2r)!B2rn2r-q-1(p,q,n∈ℕ).Proof.
Setting the functionf(x) in (1.2) in the formula (2.9) with ℓ=1, and using the results presented in Lemma 1.1, we get
(3.6)logAq(n)=logAq+(-1)qq!∑r=[(q+1)/2]+1p-1‍(2r-q-2)!(2r)!B2rn2r-q-1+(-1)qq!(2p-q-2)!(2p)!B2pn2p-q-1θ,
for some θ∈(0,1).
Replacingp by 2p+1 and 2p+2 in (3.6), respectively, we obtain
(3.7)logAq(n)-logAq=(-1)qq!∑r=[(q+1)/2]+12p‍(2r-q-2)!(2r)!B2rn2r-q-1+(-1)qq!(4p-q)!(4p+2)!B4p+2n4p+1-qθ.logAq(n)-logAq=(-1)qq!∑r=[(q+1)/2]+12p+1‍(2r-q-2)!(2r)!B2rn2r-q-1+(-1)qq!(4p+2-q)!(4p+4)!B4p+4n4p+3-qθ.
In view of (1.15), we find the following inequalities:
(3.8)q!∑r=[(q+1)/2]+12p‍(2r-q-2)!(2r)!B2rn2r-q-1<logAq(n)-logAq<q!∑r=[(q+1)/2]+12p+1‍(2r-q-2)!(2r)!B2rn2r-q-1(qiseven),q!∑r=[(q+1)/2]+12p‍(2r-q-2)!(2r)!B2rn2r-q-1<logAq-logAq(n)<q!∑r=[(q+1)/2]+12p+1‍(2r-q-2)!(2r)!B2rn2r-q-1(qisodd).
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 functionf(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 logAq asserted by Theorem 3.4.Theorem 3.4.
The following inequalities hold true:(3.9)αq+q!∑k=[(q+1)/2]+12p‍(2k-q-2)!(2k)!B2k<logAq<αq+q!∑k=[(q+1)/2]+12p+1‍(2k-q-2)!(2k)!B2k(qisodd),(3.10)αq-q!∑k=[(q+1)/2]+12p+1‍(2k-q-2)!(2k)!B2k<logAq<αq-q!∑k=[(q+1)/2]+12p‍(2k-q-2)!(2k)!B2k(qiseven),
where, for convenience,
(3.11)αq:=1(q+1)2+1-(-1)q2Bq+1Hqq+1-∑k=1[(q+1)/2]‍B2k(2k)!P2k-1(q)(q∈ℕ).Proof.
Setting the functionf(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)logAq(n)=αq+(-1)qq!∑k=[(q+1)/2]+1p‍(2k-2-q)!(2k)!B2k(1n2k-1-q-1)+(-1)q+1q!(2p+1-q)!(2p+2)!B2p+2∑k=1n-1‍1(k+θ)2p+2-q.
Replacing p by 2p and 2p+1 in (3.12), respectively, we obtain
(3.13)logAq(n)=αq+(-1)qq!∑k=[(q+1)/2]+12p‍(2k-2-q)!(2k)!B2k(1n2k-1-q-1)+(-1)q+1q!(4p+1-q)!(4p+2)!B4p+2∑k=1n-1‍1(k+θ)4p+2-q,logAq(n)=αq+(-1)qq!∑k=[(q+1)/2]+12p+1‍(2k-2-q)!(2k)!B2k(1n2k-1-q-1)+(-1)q+1q!(4p+3-q)!(4p+4)!B4p+4∑k=1n-1‍1(k+θ)4p+4-q.
In view of (1.15), we find from  that
(3.14)αq-q!∑k=[(q+1)/2]+12p‍(2k-2-q)!(2k)!B2k(1n2k-1-q-1)<logAq(n)<αq-q!∑k=[(q+1)/2]+12p+1‍(2k-2-q)!(2k)!B2k(1n2k-1-q-1)(qisodd),αq+q!∑k=[(q+1)/2]+12p+1‍(2k-2-q)!(2k)!B2k(1n2k-1-q-1)<logAq(n)<αq+q!∑k=[(q+1)/2]+12p‍(2k-2-q)!(2k)!B2k(1n2k-1-q-1)(qiseven).
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.



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