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Asymptotic formulas for the triple Gamma function Γ3 by means of its integral representation
Applied Mathematics and Computation 218 (2011) 2631–2640
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Asymptotic formulas for the triple Gamma function C3 by means of its integral representation Junesang Choi a, H.M. Srivastava b,⇑ a b
Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
a r t i c l e
i n f o
a b s t r a c t There is an abundant literature on inequalities for the Gamma function C and its various related functions as well as their approximations. Only very recently, several authors began to investigate various inequalities for the double Gamma function C2 and its approximation. Here, in this sequel to some of these recent works, we aim at presenting an integral representation of the triple Gamma function C3, which is then used to derive an asymptotic formula for C3. As a by-product of the results presented here, integral representations and asymptotic formulas for the Gamma function C and the double Gamma function C2 are also given. The methods and techniques used in this paper can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple Gamma functions Cn (n = 4). Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Gamma, double, triple and multiple Gamma functions Riemann Zeta function Hurwitz (or generalized) Zeta function Psi (or Digamma) function Bernoulli numbers Bohr-Mollerup theorem Euler-Mascheroni and Glaisher-Kinkelin constant Determinants of the Laplacians
1. Introduction, definitions and preliminaries The multiple Gamma functions Cn were defined and studied systematically by Barnes [10–13] and by others (cf., e.g., [5,38–40]) in about 1900 (see also [37, p. 649, Entry 6.441(4); p. 887, Entry 8.333] and [60, p. 264]). Three decades ago, these functions were revived in the study of the determinants of the Laplacians on the n-dimensional unit sphere Sn (cf. [21,23,26,27,43,57,59]) and have been investigated in various other ways (cf. [56, p. 24]; see also [3,4,16– 18,24,25,28,35,42,44,45,47,48,53,54]). There is a remarkably abundant literature on inequalities and asymptotic formulas (see, for details, [46]) for the classical Gamma function C and its such related functions as, for example, the Psi (or Digamma) function defined by
wðzÞ :¼
d C0 ðzÞ log CðzÞ ¼ dz CðzÞ
or
log CðzÞ ¼
Z
z
wðsÞds
1
(see, for example, [6,7,51,52]; see also the references cited in these earlier works). Among several equivalent useful expressions of the Gamma function, its Weierstrass canonical product form is recalled here as follows:
CðzÞ ¼
1 z ecz Y z 1 ðz 2 C n Z0 Þ; 1þ exp k k z k¼1
where C is the set of complex numbers,
Z0 :¼ f0; 1; 2; . . .g ⇑ Corresponding author. E-mail addresses: [email protected] (J. Choi), [email protected] (H.M. Srivastava). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.002
ð1:1Þ
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and c denotes the Euler–Mascheroni constant defined by (see also a recent work [30])
! n X 1 log n ffi 0:57721 56649 01532 86060 65120 90082 40243 1042 : k k¼1
c :¼ n!1 lim
ð1:2Þ
On the other hand, there are only a few recent papers on inequalities (see, for example, Batir [14], Batir and Cancan [15], Chen [19], and Chen and Srivastava [20]) and asymptotic formulas (see, for example, Ferreira [35] and Koumandos [41]) for the double Gamma function C2 :¼ 1/G. Barnes [10] gave several explicit Weierstrass canonical product forms of the double Gamma function C2 :¼ 1/G, one of which is recalled here as follows:
Y 1 1 1 z k z2 ; fC2 ðz þ 1Þg1 ¼ Gðz þ 1Þð2pÞz=2 exp z ðc þ 1Þz2 1þ exp z þ 2 2 k 2k k¼1
ð1:3Þ
where c denotes the Euler–Mascheroni constant given by (1.2). Analogous to the familiar relations:
Cð1Þ ¼ 1 and Cðz þ 1Þ ¼ zCðzÞ;
ð1:4Þ
the double Gamma function C2 :¼ 1/G satisfies the following fundamental relations:
Gð1Þ ¼ 1 and Gðz þ 1Þ ¼ CðzÞGðzÞ:
ð1:5Þ
Just as the Euler–Mascheroni constant c in (1.2), there is a set of constants which are naturally involved in the theory of multiple Gamma functions (see [2,16,36], and [56, p. 128]). Some of these constants are recalled here. First of all, A denotes the Glaisher–Kinkelin constant defined by (see [10])
log A ¼ lim
" n X
n!1
k log k
k¼1
# n2 n 1 n2 ffi 1:282427130 : log n þ þ þ 2 2 12 4
ð1:6Þ
The constants B and C are analogous to the Glaisher–Kinkelin constant A and are defined by (see [27])
"
# 3 n n2 n n3 7 log B ¼ lim k log k log n þ þ þ n!1 3 2 6 9 12 k¼1 n X
2
" n X
3
ð1:7Þ
and
# 4 n n3 n2 1 n4 n2 log n þ ; log C ¼ lim k log k þ þ n!1 4 2 4 120 16 12 k¼1
ð1:8Þ
respectively. The approximate numerical values of the constant B and C are given by
B ffi 1:03091675
and C ffi 0:97955746 :
The constants A, B and C are also known to be expressible as follows:
log A ¼
1 f0 ð1Þ; 12
log B ¼ f0 ð2Þ and
log C ¼
11 f0 ð3Þ 720
ð1:9Þ
in terms of special values of the derivative of the Riemann zeta function f(s) defined by
fðsÞ :¼
8 1 1 X 1 X 1 1 > > > ¼ s > s < n 1 2 n¼1 ð2n 1Þs n¼1 > > > > :
1 1 21s
1 X ð1Þn1 ns n¼1
RðsÞ > 1 RðsÞ > 0; s – 1 :
ð1:10Þ
The Riemann zeta function f(s) is a special case of the Hurwitz (or generalized) Zeta function f(s, a) defined by
fðs; aÞ :¼
1 X k¼0
1 ðk þ aÞs
RðsÞ > 1; a 2 C n Z0 ;
ð1:11Þ
each of which can be continued meromorphically to the whole complex s-plane except for a simple pole at s = 1 with its residue 1 (see, for details, [56, pp. 88–103]). From (1.10) and (1.11) it is easy to observe the following properties:
1 ¼ 1 þ fðs; 2Þ: fðsÞ ¼ fðs; 1Þ ¼ ð2s 1Þ1 f s; 2
ð1:12Þ
It is well-known that fð2nÞðn 2 NÞ can be expressed in terms of p and the Bernoulli numbers Bn (see, for example, [56, p. 98]):
fð2nÞ ¼ ð1Þnþ1
ð2pÞ2n B2n 2 ð2nÞ!
ðn 2 N0 Þ:
ð1:13Þ
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 218 (2011) 2631–2640
2633
It is known that (cf., e.g., Srivastava [55, p. 387, Eq. (1.15)])
f0 ð2nÞ ¼ lim
fð2n þ Þ
!0
¼ ð1Þn
ð2nÞ! 2ð2pÞ2n
fð2n þ 1Þ ðn 2 NÞ
ð1:14Þ
and (see [56, p. 92, Eq. (58)])
d 1 fðs; aÞ ¼ log CðaÞ logð2pÞ; ds 2 s¼0
ð1:15Þ
which is equivalent to the identity:
CðaÞ ¼
pffiffiffiffiffiffiffi 2p exp½f0 ð0; aÞ:
ð1:16Þ
The n-ple (or, simply, multiple) Hurwitz Zeta function fn(s, a) is defined by
fn ðs; aÞ :¼
1 X
ða þ k1 þ þ kn Þs
RðsÞ > n; a 2 C n Z0 ;
ð1:17Þ
k1 ;...;kn ¼0
which can be continued meromorphically to the complex s-plane except for simple poles at s = k (1 5 k 5 n) (see, for details, [56, Chapter 2]; see also [31,32]). Here, in this paper, we aim at presenting an integral representation of the triple Gamma function C3 which we then use to derive an asymptotic formula for C3. As a by-product of the results presented in this paper, integral representations and asymptotic formulas for the Gamma function C and the double Gamma function C2 are also given. The methods and techniques used here can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple Gamma functions Cn (n = 4). 2. Multiple Gamma functions There are two known ways to define the n-ple Gamma functions Cn. First of all, Barnes [13] (see also Vardi [57]) defined Cn by using the n-ple Hurwitz Zeta functions (see, e.g., [24], [56, Chapter 2]). Secondly, a recurrence relation of the Weierstrass canonical product forms of the n-ple Gamma functions Cn was given by Vignéras [58] who used the theorem of Dufresnoy and Pisot [34] which provides the existence, uniqueness, and expansion of the series of Weierstrass satisfying a certain functional equation. By making use of the aforementioned Dufresnoy-Pisot theorem and starting with
f1 ðxÞ ¼ cx þ
1 h X x n¼1
x i log 1 þ ; n n
ð2:1Þ
Vignéras [58] obtained a recurrence relation of Cn ðn 2 NÞ which is stated here as Theorem 1 below. Theorem 1. The n-ple Gamma functions Cn are defined by
Cn ðzÞ ¼ ½Gn ðzÞð1Þ
n1
ðn 2 NÞ;
ð2:2Þ
where
Gn ðz þ 1Þ ¼ exp½fn ðzÞ
ð2:3Þ
and the functions fn(z) are given by
fn ðzÞ ¼ zAn ð1Þ þ
n1 i X pk ðzÞ h ðkÞ fn1 ð0Þ AðkÞ n ð1Þ þ An ðzÞ; k! k¼1
ð2:4Þ
with
An ðzÞ ¼
X m2Nn1 N 0
" n n1 # 1 z 1 z z z þ ð1Þn log 1 þ ; þ þ ð1Þn1 n LðmÞ n 1 LðmÞ LðmÞ LðmÞ
where
LðmÞ ¼ m1 þ m2 þ m3 þ þ mn if
m ¼ ðm1 ; m2 ; m3 ; . . . ; mn Þ 2 Nn1 N 0 and the polynomials pn(z) given by
ð2:5Þ
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J. Choi, H.M. Srivastava / Applied Mathematics and Computation 218 (2011) 2631–2640
( pn ðzÞ :¼
1n þ 2n þ 3n þ þ ðN 1Þn
ðz ¼ N; N 2 N n f1gÞ
Bnþ1 ðzÞBnþ1 nþ1
ðz 2 CÞ
ð2:6Þ
satisfy the following relations:
p0n ðzÞ ¼
B0nþ1 ðzÞ ¼ Bn ðzÞ and pn ð0Þ ¼ 0; nþ1
Bn(z) being the nth Bernoulli polynomials in z. By analogy with the Bohr–Mollerup theorem (see [9, p. 14]; see also [56, p. 13]), which guarantees the uniqueness of the Gamma function C, one can give, for the double Gamma function and (more generally) for the multiple Gamma functions of order nðn 2 NÞ, a definition of Artin [9] by means of the following theorem (see Vignéras [58, p. 239]). Theorem 2. For all n 2 N; there exists a unique meromorphic function Gn(z) satisfying each of the following properties: (a) Gn ðz þ 1Þ ¼ Gn1 ðzÞGn ðzÞ ðz 2 CÞ; (b) Gn(1) = 1; (c) For x = 1, Gn(x) are infinitely differentiable and nþ1
d
nþ1
dx
flog Gn ðxÞg = 0;
(d) G0(x) = x. It is not difficult to verify (see, e.g., [56, pp. 40–41]) that {Cn(z)}1 is an entire function with zeros at z ¼ k ðk 2 N0 Þ with multiplicities
nþk1 n1
ðn 2 N; k 2 N0 Þ:
ð2:7Þ
In our earlier investigations, we gave explicit forms of the multiple Gamma functions Cn (n = 3, 4, 5) (see, e.g., [23,33]). The Weierstrass canonical product form of the triple Gamma function C3 is recalled here as follows:
8 9 kþ1 > > > > < = 2 3 Y
kþ1 1 z z z z 2 exp 2þ 3 C3 ð1 þ zÞ ¼ exp c1 z þ c2 z2 þ c3 z3 k¼1 1 þ ; > > k k 2k 2 3k > > : ;
ð2:8Þ
where
c1 ¼
3 1 logð2pÞ log A; 8 4
c2 ¼
1 1 c 1 p2 c þ logð2pÞ þ and c3 ¼ : 8 4 4 36 6 4
The logarithm of C3(1 + z) is expressed as two series associated with the Riemann Zeta function (see [32]):
log C3 ð1 þ zÞ ¼ d1 z þ d2 z2 þ d3 z3 þ
1 1 1X ð1Þkþ1 1X ð1Þk fðkÞzkþ1 þ fðkÞzkþ2 ; 2 k¼2 k þ 1 2 k¼2 k þ 2
ð2:9Þ
where
d1 ¼
3 1 logð2pÞ log A; 8 4
d2 ¼
1 1 c 1 c þ logð2pÞ þ and d3 ¼ : 8 4 4 6 4
On the other hand, Vardi [57, p. 498] gave another expression for the multiple Gamma functions Cn(a) whose general form was also studied by Barnes [13]. Theorem 3. The multiple Gamma functions Cn(a) can be expressed in terms of f0n ð0; aÞ as follows:
91 8 a > > m > > ð1Þ n < =C BY m1 CGn ðaÞ ðn 2 NÞ; Cn ðaÞ ¼ B R nmþ1 @ A > > > m¼1 > : ; 0
where
Gn ðaÞ :¼ exp½f0n ð0; aÞ with f0n ðs; aÞ ¼
@ f ðs; aÞ @s n
ð2:10Þ
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 218 (2011) 2631–2640
2635
and
Rm :¼ exp
m X
! f0k ð0; 1Þ
with R0 ¼ 1:
ð2:11Þ
k¼1
It is noted that the special case of (2.10) when n = 1 is given in (1.16). Choi [22] (see also [31,32]) found that fn(s, a) is expressible as the following finite linear combination of the generalized zeta functions f(s, a) with polynomial coefficients in a:
fn ðs; aÞ ¼
n1 X
pn;j ðaÞfðs j; aÞ;
ð2:12Þ
j¼0
where
pn;j ðaÞ ¼
n1 X ‘ 1 sðn; ‘ þ 1Þa‘j ; ð1Þnþ1j ðn 1Þ! ‘¼j j
ð2:13Þ
and s(n,‘) denotes the Stirling numbers of the first kind (see [56, Section 1.5]). The series for fn(s, a) can be evaluated explicitly for the first few values of n:
f2 ðs; aÞ ¼ ð1 aÞfðs; aÞ þ fðs 1; aÞ;
f3 ðs; aÞ ¼ 12 ða2 3a þ 2Þfðs; aÞ þ 32 a fðs 1; aÞ þ 12 fðs 2; aÞ; f4 ðs; aÞ ¼ 16 fða3 þ 6a2 11a þ 6Þfðs; aÞ þ ð3a2 12a þ 11Þfðs 1; aÞ
ð2:14Þ
ð3a 6Þfðs 2; aÞ þ fðs 3; aÞg: 3. Integral representation for C3 and its asymptotic formula We begin by recalling a known integral representation of the generalized Zeta function f(s, a) as Lemma 1 below (see, e.g., [56, Section 2.2]). Lemma 1. The following integral representation holds true: n asþ1 as X Cðk þ s 1Þ Bk ksþ1 þ þ a s1 2 CðsÞ k! k¼2 ! Z 1 n X 1 1 Bk k1 at s1 e t dt þ t t e 1 k¼0 k! CðsÞ 0 RðsÞ > ð2n 1Þ; RðaÞ > 0; n 2 N ;
fðs; aÞ ¼
ð3:1Þ
where Bk ðk 2 N0 Þ are the Bernoulli numbers. The Bernoulli numbers Bk ðk 2 N0 Þ involved in the assertion (3.1) of Lemma 1 are usually defined by means of the following generating function (see [56, Section 1.6]):
et
1 X t tk Bk ¼ 1 n¼0 k!
ðjtj < 2pÞ:
ð3:2Þ
There are various known and useful properties and formulas for the Bernoulli numbers Bk ðk 2 N0 Þ, some of which are being recalled here for later use in this investigation (see [8, p. 267, Theorem 12.18]). Lemma 2. The Bernoulli numbers B2k alternate in sign. That is,
ð1Þkþ1 B2k > 0: Moreover,
jB2k j ! 1
ðk ! 1Þ
and
ð1Þkþ1 B2k
2ð2kÞ! ð2pÞ2k
ðk ! 1Þ:
ð3:3Þ
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Differentiating f(s, a) in (3.1) with respect to s and setting s ¼ p ðp 2 N0 Þ in the resulting identity, we obtain Lemma 3 below (see [29]). Lemma 3. The following integral representation holds true:
apþ1 log a apþ1 ap log a þ Aðn; a; pÞ 2 pþ1 2 ðp þ 1Þ ! Z 1 n X 1 Bk k1 eat p dt t þ ð1Þ p! et 1 k¼0 k! t pþ1 0 RðaÞ > 0; n 2 N; p 2 N0 ; 0 5 p < 2n 1 ;
f0 ðp; aÞ ¼
where, for convenience,
ð3:4Þ
13 6 C7 B 6 n k2 k2 B Y k2 C7 X Y X 6 C7 B k Bk kþpþ1 6 : Aðn; a; pÞ :¼ ð1Þ ðlog aÞ ðp jÞ þ ðp jÞ a C7 B 6 C7 B k! 7 6 ‘¼0 @ j ¼ 0 j¼0 k¼2 A 5 4 ðj–‘Þ 2
0
ð3:5Þ
Proof. Differentiating both sides of Eq. (3.1) with respect to s under the sign of integration (see, for validity of this process, [60, p. 74, Corollary]), we obtain
! n k1 X a1s log a a1s 1 s Cðs þ k 1Þ Bk ksþ1 X 1 a log a log a þ a s1 sþj1 CðsÞ k! ðs 1Þ2 2 j¼1 k¼2 ! ! Z 1 Z n n 1 X X wðsÞ 1 Bk k1 at s1 1 1 Bk k1 at s1 t dt þ t t e e t log tdt et 1 k¼0 k! et 1 k¼0 k! CðsÞ 0 CðsÞ 0 RðsÞ > ð2n 1Þ; RðaÞ > 0; n 2 N :
f0 ðs; aÞ ¼
ð3:6Þ
Moreover, it is observed that
‘ X wðsÞ 1 ¼ CðsÞ s¼k Cð‘ k þ 1Þ j¼0
‘ Y
ðp kÞ ¼ ð1Þkþ1 k! ðk 2 N0 Þ
p¼0
ð3:7Þ
ðp–jÞ and
1 ¼ 0 ðk 2 N0 Þ: CðsÞs¼k
By setting s = p ðp 2 N0 Þ in Eq. (3.6), and with the aid of Eqs. (3.7) and (3.8), we readily arrive at Eq. (3.4). h Some properties of Aðn; a; pÞ are observed as in the following lemma. Lemma 4. The following result holds true:
Að1; a; pÞ ¼ 0: Furthermore, for n 2 N n f1; 2g;
Aðn; a; 0Þ ¼
n X 1 ðk 2Þ! þ Bk akþ1 ; 12a k¼4 k!
Aðn; a; 1Þ ¼
n X 1 ðk 3Þ! ð1 þ log aÞ Bk akþ2 12 k! k¼4
Aðn; a; 2Þ ¼
n X a ðk 4Þ! ð1 þ 2 log aÞ þ 2 Bk akþ3 : 12 k! k¼4
and
We now give an asymptotic formula for the integral in Eq. (3.4) as asserted by Lemma 5 below.
ð3:8Þ
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Lemma 5. The following asymptotic formula holds true:
! n X 1 Bk k1 eat 1 dt ¼ O t pþ1 et 1 k¼0 k! a‘þ1 t 0 a ! 1; RðaÞ > 0; p; ‘ 2 N0 ; n ¼ p þ 1 þ ‘ : Z
1
ð3:9Þ
Proof. We prove the assertion (3.9) of Lemma 5 only for the case when n = p + 1. Then it is easy to see that the same argument will prove Lemma 5 for the cases when n > p + 1. We begin by separating the integral into two parts:
Z 0
1
! pþ1 X 1 Bk k1 eat dt ¼ I1 ðaÞ þ I2 ðaÞ; t et 1 k¼0 k! t pþ1
where
I1 ðaÞ :¼
! pþ1 X 1 Bk k1 eat dt t et 1 k¼0 k! t pþ1
Z p 0
and
I2 ðaÞ :¼
Z
! pþ1 X 1 Bk k1 eat dt: t et 1 k¼0 k! tpþ1
1
p
Since
jeat j ¼ eRðaÞt and we need RðaÞ > 0, we assume that a is any positive real number. In view of (3.2), I1(a) can be written as follows:
I1 ðaÞ :¼
Z p 0
! 1 X Bk kp2 at e dt: t k! k¼pþ2
The involved series is convergent when jtj < 2p; it is bounded for jtj < 2p. That is, for some L1 > 0, we have
Z p X Z p Z 1 1 Bk kp2 at L1 jI1 ðaÞj 5 eat dt 5 L1 eat dt ¼ : t e dt 5 L1 a 0 k¼pþ2 k! 0 0
It is also easy to observe that, in view of (3.3), for the second integral I2(a),
et
pþ1 X Bk k1 ¼ Oðt pþ1 Þ ðt ! 1Þ: t k! k¼0
1 ¼ Oð1=et Þ and 1
Thus there exist a large T > 0, L2 > 0 and L3 > 0 such that
pþ1 X Bk k1 and t 5 L3 t pþ1 k¼0 k!
1 L2 5 t et 1 e
for all t = T. Now we find that
jI2 ðaÞj 5
Z
T
jfp ðtÞjeat dt þ L2
p
Z T
1
1 eat dt þ L3 et tpþ1
Z
1
T
t pþ1
eat dt; t pþ1
where, for convenience,
fp ðtÞ :¼
! pþ1 X 1 Bk k1 1 : t et 1 k¼0 k! tpþ1
Since fp(t) is continuous on the closed bounded interval [p, T], for some L4 > 0, we have jfp(t)j 5 L4 for all t 2 [p, T]. We, therefore, get
jI2 ðaÞj 5 ðL2 þ L3 þ L4 Þ
Z 0
1
1 eat dt ¼ ðL2 þ L3 þ L4 Þ : a
This completes the proof of Lemma 5. h
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By mainly using Eq. (2.10) with the aid of the formulas and properties presented and recalled here, we can give an integral representation of the triple Gamma function C3 and its asymptotic formula as in Theorem 4 below. Here, as well as elsewhere in this paper, O() denotes the usual big O-notation (see, e.g., [49, pp. 4–8]). Theorem 4. Each of the following formulas holds true:
11 3 log C3 ðaÞ ¼ 18 þ 12 logð2pÞ þ 32 log A þ 2fð3Þ p2 12 þ 4 logð2pÞ þ log A a 3 a3 þ a6 34 a2 þ a 12 log a þ 98 þ 14 logð2pÞ a2 11 36
þ 12 ða2 3a þ 2ÞAðn; a; 0Þ þ 32 a Aðn; a; 1Þ þ 12 Aðn; a; 2Þ n
R1 P Bk k1 eat t dt þ 0 12 ða2 3a þ 2Þt 2 þ a 32 t þ 1 et 11 k! t3
ð3:10Þ
k¼0
ðRðaÞ > 0; n 2 N n f1; 2gÞ and
11 3 log C3 ðaÞ ¼ 18 þ 12 logð2pÞ þ 32 log A þ 2fð3Þ p2 12 þ 4 logð2pÞ þ log A a
3 1 þ 98 þ 14 logð2pÞ a2 11 a3 þ a6 34 a2 þ a 38 log a þ 12a 36 þ
‘þ3 P k¼4
ðk4Þ! Bk akþ1 12 ða2 k!
3a þ 2Þðk 2Þðk 3Þ þ a a 32 ðk 3Þ þ a2
ð3:11Þ
1 þO a‘þ1 ða ! 1; RðaÞ > 0; ‘ 2 N0 Þ; where Aðn; a; pÞðp ¼ 0; 1; 2Þ are given as in Lemma 4. 4. Concluding remarks and observations From Eq. (1.16) or the special case of Eq. (2.10) when n = 1, together with Eq. (3.4), we obtain an integral representation of log C(a):
log CðaÞ ¼ 12 logð2pÞ þ a 12 log a a þ Aðn; a; 0Þ n R1 P Bk k1 eat þ 0 et 11 t dt ðRðaÞ > 0; n 2 NÞ; k! t
ð4:1Þ
k¼0
where Aðn; a; 0Þ is given as in Lemma 4. By using Eq. (4.1) with the aid of Eq. (3.9), we get the following familiar asymptotic formula for log C(a) (see, e.g., [1, p. 257, Entry 6.1.40]): ‘þ1
P 1 log CðaÞ ¼ 12 logð2pÞ þ a 12 log a a þ 12a þ k¼4
1 þO a‘þ1 ðRðaÞ > 0; a ! 1; ‘ 2 N0 Þ:
ðk2Þ! Bk akþ1 k!
ð4:2Þ
From the special case of Eq. (2.10) when n = 2, together with some of the above previously recorded results, we obtain an integral representation and an asymptotic formula for logC2(a) (see, e.g., [56, p. 96, Eq. (40)]):
1 1 1 þ logð2pÞ þ log A 1 þ logð2pÞ a 12 2 2 1 3 2 2 ða 1Þ log a þ a þ ð1 aÞAðn; a; 0Þ þ Aðn; a; 1Þ 2 4 ! Z 1 n X 1 Bk k1 eat ½ð1 aÞt 1 t dt t þ e 1 k¼0 k! t2 0
log C2 ðaÞ ¼
ð4:3Þ
ðRðaÞ > 0; n 2 N n f1gÞ and
1 log C2 ðaÞ ¼ 12 þ 12 logð2pÞ þ log A 1 þ 12 logð2pÞ a
1 12 a2 2a þ 56 log a þ 34 a2 þ 12a þ
‘þ2 P k¼4
ðk3Þ! Bk akþ1 ½ð1 k!
aÞk þ a 2
1 ða ! 1; RðaÞ > 0; ‘ 2 N0 Þ: þO a‘þ1
ð4:4Þ
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 218 (2011) 2631–2640
2639
We choose to conclude our present investigation by remarking further that the methods and techniques used above can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple Gamma functions Cn (n = q4). Acknowledgements This research was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology of the Republic of Korea (2011-0005224). The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. References [1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Tenth Printing, National Bureau of Standards, Applied Mathematics Series 55, National Bureau of Standards, Washington, DC, 1972; Reprinted by Dover Publications, New York, 1965 (see also [50]). [2] V.S. Adamchik, Polygamma functions of negative order, J. Comput. Appl. Math. 100 (1998) 191–199. [3] V.S. Adamchik, On the Barnes function, in: B. Mourrain (Ed.), Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation (London, Ontario; July 22–25, 2001), ACM Press, New York, 2001, pp. 15–20. [4] V.S. Adamchik, The multiple gamma function and its application to computation of series, Ramanujan J. 9 (2005) 271–288. [5] W.P. Alexeiewsky, Über eine Classe von Funktionen, die der Gammafunktion analog sind, Leipzig Weidmannsche Buchhandlung 46 (1894) 268–275. [6] H. Alzer, Inequalities involving C(x) and C(1/x), J. Comput. Appl. Math. 192 (2006) 460–480. [7] H. Alzer, Sub-and super-additive property of Euler’s Gamma function, Proc. Amer. Math. Soc. 135 (2007) 3641–3648. [8] T.M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, Heidelberg and Berlin, 1976. [9] E. Artin, The Gamma Function, Holt, Rinehart and Winston, New York, Chicago, San Francisco, Toronto and London, 1964. [10] E.W. Barnes, The theory of the G-function, Quart. J. Math. 31 (1899) 264–314. [11] E.W. Barnes, Genesis of the double Gamma function, Proc. London Math. Soc. (Ser. 1) 31 (1900) 358–381. [12] E.W. Barnes, The theory of the double Gamma function, Philos. Trans. Roy. Soc. London Ser. A 196 (1901) 265–388. [13] E.W. Barnes, On the theory of the multiple Gamma functions, Trans. Cambridge Philos. Soc. 19 (1904) 374–439. [14] N. Batir, Inequalities for the double Gamma function, J. Math. Anal. Appl. 351 (2009) 182–185. [15] N. Batir, M. Cancan, A double inequality for the double Gamma function, Internat. J. Math. Anal. 2 (2008) 329–335. [16] L. Bendersky, Sur la fonction Gamma généralisée, Acta Math. 61 (1933) 263–322. [17] J. Billingham, A.C. King, Uniform asymptotic expansions for the Barnes double Gamma function, Proc. Roy. Soc. London Ser. A 453 (1997) 1817– 1829. [18] P. Cassou-Noguès, Analogues p-adiques des fonctions C-multiples, in: Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978), Astérisque, vol. 61, Soc. Math. France, Paris, 1979, pp. 43–55. [19] C.-P. Chen, Inequalities associated with Barnes G-function, Exposition. Math. 29 (2011) 119–125. [20] C.-P. Chen, H.M. Srivastava, Some inequalities and monotonicity properties associated with the Gamma and Psi functions and the Barnes G-function, Integral Transforms Spec. Funct. 22 (2011) 1–15. [21] J. Choi, Determinant of Laplacian on S3, Math. Japon. 40 (1994) 155–166. [22] J. Choi, Explicit formulas for the Bernolli polynomials of order n, Indian J. Pure Appl. Math. 27 (1996) 667–674. [23] J. Choi, Y.J. Cho, H.M. Srivastava, Series involving the Zeta function and multiple Gamma functions, Appl. Math. Comput. 159 (2004) 509–537. [24] J. Choi, J.R. Quine, E.W. Barnes’ approach of the multiple Gamma functions, J. Korean Math. Soc. 29 (1992) 127–140. [25] J. Choi, T.Y. Seo, The double Gamma function, East Asian Math. J. 13 (1997) 159–174. [26] J. Choi, H.M. Srivastava, An application of the theory of the double Gamma function, Kyushu J. Math. 53 (1999) 209–222. [27] J. Choi, H.M. Srivastava, Certain classes of series associated with the Zeta function and multiple Gamma functions, J. Comput. Appl. Math. 118 (2000) 87–109. [28] J. Choi, H.M. Srivastava, A note on a multiplication formula for the multiple Gamma function Cn, Italian J. Pure Appl. Math. 23 (2008) 179–188. [29] J. Choi, H.M. Srivastava, Integral representations for the Gamma function, the Beta function, and the double Gamma function, Integral Transforms Spec. Funct. 20 (2009) 859–869. [30] J. Choi, H.M. Srivastava, Integral representations for the Euler–Mascheroni constant c, Integral Transforms Spec. Funct. 21 (2010) 675–690. [31] J. Choi, H.M. Srivastava, The multiple Hurwitz Zeta function and the multiple Hurwitz–Euler Eta function, Taiwanese J. Math. 15 (2011) 501–522. [32] J. Choi, H.M. Srivastava, Inequalities for the multiple Gamma functions, Preprint 2011. [33] J. Choi, H.M. Srivastava, J.R. Quine, Some series involving the Zeta function, Bull. Austral. Math. Soc. 51 (1995) 383–393. [34] J. Dufresnoy, Ch. Pisot, Sur la relation fonctionnelle f(x + 1) f(x) = u(x), Bull. Soc. Math. Belg. 15 (1963) 259–270. [35] C. Ferreira, An asymptotic expansion of the double Gamma function, J. Approx. Theory 111 (2001) 298–314. [36] J.W.L. Glaisher, On the product 11 22 33 nn, Messenger Math. 7 (1877) 43–47. [37] I.S. Gradshteyn, I.M. Ryzhik (Eds.), Sixth ed., Tables of Integrals, Series, and Products (Corrected and Enlarged edition prepared by A. Jeffrey), Academic Press, New York and London, 2000. [38] O. Hölder, Über eine Transcendente Funktion, Göttingen, Dieterichsche Verlags-Buchhandlung 1886 (1886) 514–522. [39] O. Hölder, Über eine von Abel untersuchte Transzendente und eine merkwürdige Funktionalbeziehung, Berichte über die Verhandlungen der Saechsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse 80 (1928) 312–325. [40] V.H. Kinkelin, Über eine mit der Gamma Funktion verwandte Transcendente und deren Anwendung auf die Integralrechnung, J. Reine Angew. Math. 57 (1860) 122–158. [41] S. Koumandos, On Ruijsenaars’ asymptotic expansion of the logarithm of the double gamma function, J. Math. Anal. Appl. 341 (2008) 1125–1132. [42] S. Koumandos, H.L. Pedersen, Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function, J. Math. Anal. Appl. 355 (2009) 33–40. [43] H. Kumagai, The determinant of the Laplacian on the n-sphere, Acta Arith. 91 (1999) 199–208. kyu roku 958 (1996) 162–165. [44] K. Matsumoto, Asymptotic series for double Zeta and double Gamma functions of Barnes, RIMS Ko [45] K. Matsumoto, Asymptotic series for double Zeta, double Gamma, and Hecke L-functions, Math. Proc. Cambridge Philos. Soc. 123 (1998) 385–405. Corrigendum and Addendum, ibid. 132 (2002), 377–384.. [46] C. Mortici, New improvements of the Stirling formula, Appl. Math. Comput. 217 (2010) 699–704. [47] M. Nishizawa, On a q-analogue of the multiple Gamma functions, Lett. Math. Phys. 37 (1996) 201–209. [48] M. Nishizawa, Multiple Gamma function, its q-and elliptic analogue, Rocky Mountain J. Math. 32 (2002) 793–811. [49] F.W.J. Olver, Asymptotics and Special Functions, A.K. Peters, Wellesley, Massachusetts, 1997.
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[50] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (Eds.), NIST Handbook of Mathematical Functions [With 1 CD-ROM (Windows, Macintosh and UNIX)], US Department of Commerce, National Institute of Standards and Technology, Washington, DC, 2010; Cambridge University Press, Cambridge, London and New York, 2010 (see also [1]). [51] F. Qi, S.-X. Chen, W.-S. Cheung, Logarithmically completely monotonic functions concerning gamma and digamma functions, Integral Transforms Spec. Funct. 18 (2007) 435–443. [52] F. Qi, R.-Q. Cui, C.-P. Chen, B.-N. Guo, Some completely monotonic functions involving Polygamma functions and an application, J. Math. Anal. Appl. 310 (2005) 303–308. [53] S. Ruijsenaars, On Barnes’ multiple zeta and gamma functions, Adv. Math. 156 (2000) 107–132. [54] T. Shintani, A proof of the classical Kronecker limit formula, Tokyo J. Math. 3 (1980) 191–199. [55] H.M. Srivastava, Some rapidly converging series for f(2n + 1), Proc. Amer. Math. Soc. 127 (1999) 385–396. [56] H.M. Srivastava, J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001. [57] I. Vardi, Determinants of Laplacians and multiple Gamma functions, SIAM J. Math. Anal. 19 (1988) 493–507. [58] M.-F. Vignéras, L’équation fonctionnelle de la fonction zêta de Selberg du groupe moudulaire PSL(2,Z), in: Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978), Astérisque, vol. 61, Soc. Math. France, Paris, 1979, pp. 235–249. [59] A. Voros, Special functions, spectral functions and the Selberg Zeta function, Commun. Math. Phys. 110 (1987) 439–465. [60] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, Fourth ed., Cambridge University Press, Cambridge, London and New York, 1963.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Asymptotic formulas for the triple Gamma function C3 by means of its integral representation Junesang Choi a, H.M. Srivastava b,⇑ a b
Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
a r t i c l e
i n f o
a b s t r a c t There is an abundant literature on inequalities for the Gamma function C and its various related functions as well as their approximations. Only very recently, several authors began to investigate various inequalities for the double Gamma function C2 and its approximation. Here, in this sequel to some of these recent works, we aim at presenting an integral representation of the triple Gamma function C3, which is then used to derive an asymptotic formula for C3. As a by-product of the results presented here, integral representations and asymptotic formulas for the Gamma function C and the double Gamma function C2 are also given. The methods and techniques used in this paper can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple Gamma functions Cn (n = 4). Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Gamma, double, triple and multiple Gamma functions Riemann Zeta function Hurwitz (or generalized) Zeta function Psi (or Digamma) function Bernoulli numbers Bohr-Mollerup theorem Euler-Mascheroni and Glaisher-Kinkelin constant Determinants of the Laplacians
1. Introduction, definitions and preliminaries The multiple Gamma functions Cn were defined and studied systematically by Barnes [10–13] and by others (cf., e.g., [5,38–40]) in about 1900 (see also [37, p. 649, Entry 6.441(4); p. 887, Entry 8.333] and [60, p. 264]). Three decades ago, these functions were revived in the study of the determinants of the Laplacians on the n-dimensional unit sphere Sn (cf. [21,23,26,27,43,57,59]) and have been investigated in various other ways (cf. [56, p. 24]; see also [3,4,16– 18,24,25,28,35,42,44,45,47,48,53,54]). There is a remarkably abundant literature on inequalities and asymptotic formulas (see, for details, [46]) for the classical Gamma function C and its such related functions as, for example, the Psi (or Digamma) function defined by
wðzÞ :¼
d C0 ðzÞ log CðzÞ ¼ dz CðzÞ
or
log CðzÞ ¼
Z
z
wðsÞds
1
(see, for example, [6,7,51,52]; see also the references cited in these earlier works). Among several equivalent useful expressions of the Gamma function, its Weierstrass canonical product form is recalled here as follows:
CðzÞ ¼
1 z ecz Y z 1 ðz 2 C n Z0 Þ; 1þ exp k k z k¼1
where C is the set of complex numbers,
Z0 :¼ f0; 1; 2; . . .g ⇑ Corresponding author. E-mail addresses: [email protected] (J. Choi), [email protected] (H.M. Srivastava). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.002
ð1:1Þ
2632
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 218 (2011) 2631–2640
and c denotes the Euler–Mascheroni constant defined by (see also a recent work [30])
! n X 1 log n ffi 0:57721 56649 01532 86060 65120 90082 40243 1042 : k k¼1
c :¼ n!1 lim
ð1:2Þ
On the other hand, there are only a few recent papers on inequalities (see, for example, Batir [14], Batir and Cancan [15], Chen [19], and Chen and Srivastava [20]) and asymptotic formulas (see, for example, Ferreira [35] and Koumandos [41]) for the double Gamma function C2 :¼ 1/G. Barnes [10] gave several explicit Weierstrass canonical product forms of the double Gamma function C2 :¼ 1/G, one of which is recalled here as follows:
Y 1 1 1 z k z2 ; fC2 ðz þ 1Þg1 ¼ Gðz þ 1Þð2pÞz=2 exp z ðc þ 1Þz2 1þ exp z þ 2 2 k 2k k¼1
ð1:3Þ
where c denotes the Euler–Mascheroni constant given by (1.2). Analogous to the familiar relations:
Cð1Þ ¼ 1 and Cðz þ 1Þ ¼ zCðzÞ;
ð1:4Þ
the double Gamma function C2 :¼ 1/G satisfies the following fundamental relations:
Gð1Þ ¼ 1 and Gðz þ 1Þ ¼ CðzÞGðzÞ:
ð1:5Þ
Just as the Euler–Mascheroni constant c in (1.2), there is a set of constants which are naturally involved in the theory of multiple Gamma functions (see [2,16,36], and [56, p. 128]). Some of these constants are recalled here. First of all, A denotes the Glaisher–Kinkelin constant defined by (see [10])
log A ¼ lim
" n X
n!1
k log k
k¼1
# n2 n 1 n2 ffi 1:282427130 : log n þ þ þ 2 2 12 4
ð1:6Þ
The constants B and C are analogous to the Glaisher–Kinkelin constant A and are defined by (see [27])
"
# 3 n n2 n n3 7 log B ¼ lim k log k log n þ þ þ n!1 3 2 6 9 12 k¼1 n X
2
" n X
3
ð1:7Þ
and
# 4 n n3 n2 1 n4 n2 log n þ ; log C ¼ lim k log k þ þ n!1 4 2 4 120 16 12 k¼1
ð1:8Þ
respectively. The approximate numerical values of the constant B and C are given by
B ffi 1:03091675
and C ffi 0:97955746 :
The constants A, B and C are also known to be expressible as follows:
log A ¼
1 f0 ð1Þ; 12
log B ¼ f0 ð2Þ and
log C ¼
11 f0 ð3Þ 720
ð1:9Þ
in terms of special values of the derivative of the Riemann zeta function f(s) defined by
fðsÞ :¼
8 1 1 X 1 X 1 1 > > > ¼ s > s < n 1 2 n¼1 ð2n 1Þs n¼1 > > > > :
1 1 21s
1 X ð1Þn1 ns n¼1
RðsÞ > 1 RðsÞ > 0; s – 1 :
ð1:10Þ
The Riemann zeta function f(s) is a special case of the Hurwitz (or generalized) Zeta function f(s, a) defined by
fðs; aÞ :¼
1 X k¼0
1 ðk þ aÞs
RðsÞ > 1; a 2 C n Z0 ;
ð1:11Þ
each of which can be continued meromorphically to the whole complex s-plane except for a simple pole at s = 1 with its residue 1 (see, for details, [56, pp. 88–103]). From (1.10) and (1.11) it is easy to observe the following properties:
1 ¼ 1 þ fðs; 2Þ: fðsÞ ¼ fðs; 1Þ ¼ ð2s 1Þ1 f s; 2
ð1:12Þ
It is well-known that fð2nÞðn 2 NÞ can be expressed in terms of p and the Bernoulli numbers Bn (see, for example, [56, p. 98]):
fð2nÞ ¼ ð1Þnþ1
ð2pÞ2n B2n 2 ð2nÞ!
ðn 2 N0 Þ:
ð1:13Þ
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 218 (2011) 2631–2640
2633
It is known that (cf., e.g., Srivastava [55, p. 387, Eq. (1.15)])
f0 ð2nÞ ¼ lim
fð2n þ Þ
!0
¼ ð1Þn
ð2nÞ! 2ð2pÞ2n
fð2n þ 1Þ ðn 2 NÞ
ð1:14Þ
and (see [56, p. 92, Eq. (58)])
d 1 fðs; aÞ ¼ log CðaÞ logð2pÞ; ds 2 s¼0
ð1:15Þ
which is equivalent to the identity:
CðaÞ ¼
pffiffiffiffiffiffiffi 2p exp½f0 ð0; aÞ:
ð1:16Þ
The n-ple (or, simply, multiple) Hurwitz Zeta function fn(s, a) is defined by
fn ðs; aÞ :¼
1 X
ða þ k1 þ þ kn Þs
RðsÞ > n; a 2 C n Z0 ;
ð1:17Þ
k1 ;...;kn ¼0
which can be continued meromorphically to the complex s-plane except for simple poles at s = k (1 5 k 5 n) (see, for details, [56, Chapter 2]; see also [31,32]). Here, in this paper, we aim at presenting an integral representation of the triple Gamma function C3 which we then use to derive an asymptotic formula for C3. As a by-product of the results presented in this paper, integral representations and asymptotic formulas for the Gamma function C and the double Gamma function C2 are also given. The methods and techniques used here can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple Gamma functions Cn (n = 4). 2. Multiple Gamma functions There are two known ways to define the n-ple Gamma functions Cn. First of all, Barnes [13] (see also Vardi [57]) defined Cn by using the n-ple Hurwitz Zeta functions (see, e.g., [24], [56, Chapter 2]). Secondly, a recurrence relation of the Weierstrass canonical product forms of the n-ple Gamma functions Cn was given by Vignéras [58] who used the theorem of Dufresnoy and Pisot [34] which provides the existence, uniqueness, and expansion of the series of Weierstrass satisfying a certain functional equation. By making use of the aforementioned Dufresnoy-Pisot theorem and starting with
f1 ðxÞ ¼ cx þ
1 h X x n¼1
x i log 1 þ ; n n
ð2:1Þ
Vignéras [58] obtained a recurrence relation of Cn ðn 2 NÞ which is stated here as Theorem 1 below. Theorem 1. The n-ple Gamma functions Cn are defined by
Cn ðzÞ ¼ ½Gn ðzÞð1Þ
n1
ðn 2 NÞ;
ð2:2Þ
where
Gn ðz þ 1Þ ¼ exp½fn ðzÞ
ð2:3Þ
and the functions fn(z) are given by
fn ðzÞ ¼ zAn ð1Þ þ
n1 i X pk ðzÞ h ðkÞ fn1 ð0Þ AðkÞ n ð1Þ þ An ðzÞ; k! k¼1
ð2:4Þ
with
An ðzÞ ¼
X m2Nn1 N 0
" n n1 # 1 z 1 z z z þ ð1Þn log 1 þ ; þ þ ð1Þn1 n LðmÞ n 1 LðmÞ LðmÞ LðmÞ
where
LðmÞ ¼ m1 þ m2 þ m3 þ þ mn if
m ¼ ðm1 ; m2 ; m3 ; . . . ; mn Þ 2 Nn1 N 0 and the polynomials pn(z) given by
ð2:5Þ
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J. Choi, H.M. Srivastava / Applied Mathematics and Computation 218 (2011) 2631–2640
( pn ðzÞ :¼
1n þ 2n þ 3n þ þ ðN 1Þn
ðz ¼ N; N 2 N n f1gÞ
Bnþ1 ðzÞBnþ1 nþ1
ðz 2 CÞ
ð2:6Þ
satisfy the following relations:
p0n ðzÞ ¼
B0nþ1 ðzÞ ¼ Bn ðzÞ and pn ð0Þ ¼ 0; nþ1
Bn(z) being the nth Bernoulli polynomials in z. By analogy with the Bohr–Mollerup theorem (see [9, p. 14]; see also [56, p. 13]), which guarantees the uniqueness of the Gamma function C, one can give, for the double Gamma function and (more generally) for the multiple Gamma functions of order nðn 2 NÞ, a definition of Artin [9] by means of the following theorem (see Vignéras [58, p. 239]). Theorem 2. For all n 2 N; there exists a unique meromorphic function Gn(z) satisfying each of the following properties: (a) Gn ðz þ 1Þ ¼ Gn1 ðzÞGn ðzÞ ðz 2 CÞ; (b) Gn(1) = 1; (c) For x = 1, Gn(x) are infinitely differentiable and nþ1
d
nþ1
dx
flog Gn ðxÞg = 0;
(d) G0(x) = x. It is not difficult to verify (see, e.g., [56, pp. 40–41]) that {Cn(z)}1 is an entire function with zeros at z ¼ k ðk 2 N0 Þ with multiplicities
nþk1 n1
ðn 2 N; k 2 N0 Þ:
ð2:7Þ
In our earlier investigations, we gave explicit forms of the multiple Gamma functions Cn (n = 3, 4, 5) (see, e.g., [23,33]). The Weierstrass canonical product form of the triple Gamma function C3 is recalled here as follows:
8 9 kþ1 > > > > < = 2 3 Y
kþ1 1 z z z z 2 exp 2þ 3 C3 ð1 þ zÞ ¼ exp c1 z þ c2 z2 þ c3 z3 k¼1 1 þ ; > > k k 2k 2 3k > > : ;
ð2:8Þ
where
c1 ¼
3 1 logð2pÞ log A; 8 4
c2 ¼
1 1 c 1 p2 c þ logð2pÞ þ and c3 ¼ : 8 4 4 36 6 4
The logarithm of C3(1 + z) is expressed as two series associated with the Riemann Zeta function (see [32]):
log C3 ð1 þ zÞ ¼ d1 z þ d2 z2 þ d3 z3 þ
1 1 1X ð1Þkþ1 1X ð1Þk fðkÞzkþ1 þ fðkÞzkþ2 ; 2 k¼2 k þ 1 2 k¼2 k þ 2
ð2:9Þ
where
d1 ¼
3 1 logð2pÞ log A; 8 4
d2 ¼
1 1 c 1 c þ logð2pÞ þ and d3 ¼ : 8 4 4 6 4
On the other hand, Vardi [57, p. 498] gave another expression for the multiple Gamma functions Cn(a) whose general form was also studied by Barnes [13]. Theorem 3. The multiple Gamma functions Cn(a) can be expressed in terms of f0n ð0; aÞ as follows:
91 8 a > > m > > ð1Þ n < =C BY m1 CGn ðaÞ ðn 2 NÞ; Cn ðaÞ ¼ B R nmþ1 @ A > > > m¼1 > : ; 0
where
Gn ðaÞ :¼ exp½f0n ð0; aÞ with f0n ðs; aÞ ¼
@ f ðs; aÞ @s n
ð2:10Þ
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 218 (2011) 2631–2640
2635
and
Rm :¼ exp
m X
! f0k ð0; 1Þ
with R0 ¼ 1:
ð2:11Þ
k¼1
It is noted that the special case of (2.10) when n = 1 is given in (1.16). Choi [22] (see also [31,32]) found that fn(s, a) is expressible as the following finite linear combination of the generalized zeta functions f(s, a) with polynomial coefficients in a:
fn ðs; aÞ ¼
n1 X
pn;j ðaÞfðs j; aÞ;
ð2:12Þ
j¼0
where
pn;j ðaÞ ¼
n1 X ‘ 1 sðn; ‘ þ 1Þa‘j ; ð1Þnþ1j ðn 1Þ! ‘¼j j
ð2:13Þ
and s(n,‘) denotes the Stirling numbers of the first kind (see [56, Section 1.5]). The series for fn(s, a) can be evaluated explicitly for the first few values of n:
f2 ðs; aÞ ¼ ð1 aÞfðs; aÞ þ fðs 1; aÞ;
f3 ðs; aÞ ¼ 12 ða2 3a þ 2Þfðs; aÞ þ 32 a fðs 1; aÞ þ 12 fðs 2; aÞ; f4 ðs; aÞ ¼ 16 fða3 þ 6a2 11a þ 6Þfðs; aÞ þ ð3a2 12a þ 11Þfðs 1; aÞ
ð2:14Þ
ð3a 6Þfðs 2; aÞ þ fðs 3; aÞg: 3. Integral representation for C3 and its asymptotic formula We begin by recalling a known integral representation of the generalized Zeta function f(s, a) as Lemma 1 below (see, e.g., [56, Section 2.2]). Lemma 1. The following integral representation holds true: n asþ1 as X Cðk þ s 1Þ Bk ksþ1 þ þ a s1 2 CðsÞ k! k¼2 ! Z 1 n X 1 1 Bk k1 at s1 e t dt þ t t e 1 k¼0 k! CðsÞ 0 RðsÞ > ð2n 1Þ; RðaÞ > 0; n 2 N ;
fðs; aÞ ¼
ð3:1Þ
where Bk ðk 2 N0 Þ are the Bernoulli numbers. The Bernoulli numbers Bk ðk 2 N0 Þ involved in the assertion (3.1) of Lemma 1 are usually defined by means of the following generating function (see [56, Section 1.6]):
et
1 X t tk Bk ¼ 1 n¼0 k!
ðjtj < 2pÞ:
ð3:2Þ
There are various known and useful properties and formulas for the Bernoulli numbers Bk ðk 2 N0 Þ, some of which are being recalled here for later use in this investigation (see [8, p. 267, Theorem 12.18]). Lemma 2. The Bernoulli numbers B2k alternate in sign. That is,
ð1Þkþ1 B2k > 0: Moreover,
jB2k j ! 1
ðk ! 1Þ
and
ð1Þkþ1 B2k
2ð2kÞ! ð2pÞ2k
ðk ! 1Þ:
ð3:3Þ
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Differentiating f(s, a) in (3.1) with respect to s and setting s ¼ p ðp 2 N0 Þ in the resulting identity, we obtain Lemma 3 below (see [29]). Lemma 3. The following integral representation holds true:
apþ1 log a apþ1 ap log a þ Aðn; a; pÞ 2 pþ1 2 ðp þ 1Þ ! Z 1 n X 1 Bk k1 eat p dt t þ ð1Þ p! et 1 k¼0 k! t pþ1 0 RðaÞ > 0; n 2 N; p 2 N0 ; 0 5 p < 2n 1 ;
f0 ðp; aÞ ¼
where, for convenience,
ð3:4Þ
13 6 C7 B 6 n k2 k2 B Y k2 C7 X Y X 6 C7 B k Bk kþpþ1 6 : Aðn; a; pÞ :¼ ð1Þ ðlog aÞ ðp jÞ þ ðp jÞ a C7 B 6 C7 B k! 7 6 ‘¼0 @ j ¼ 0 j¼0 k¼2 A 5 4 ðj–‘Þ 2
0
ð3:5Þ
Proof. Differentiating both sides of Eq. (3.1) with respect to s under the sign of integration (see, for validity of this process, [60, p. 74, Corollary]), we obtain
! n k1 X a1s log a a1s 1 s Cðs þ k 1Þ Bk ksþ1 X 1 a log a log a þ a s1 sþj1 CðsÞ k! ðs 1Þ2 2 j¼1 k¼2 ! ! Z 1 Z n n 1 X X wðsÞ 1 Bk k1 at s1 1 1 Bk k1 at s1 t dt þ t t e e t log tdt et 1 k¼0 k! et 1 k¼0 k! CðsÞ 0 CðsÞ 0 RðsÞ > ð2n 1Þ; RðaÞ > 0; n 2 N :
f0 ðs; aÞ ¼
ð3:6Þ
Moreover, it is observed that
‘ X wðsÞ 1 ¼ CðsÞ s¼k Cð‘ k þ 1Þ j¼0
‘ Y
ðp kÞ ¼ ð1Þkþ1 k! ðk 2 N0 Þ
p¼0
ð3:7Þ
ðp–jÞ and
1 ¼ 0 ðk 2 N0 Þ: CðsÞs¼k
By setting s = p ðp 2 N0 Þ in Eq. (3.6), and with the aid of Eqs. (3.7) and (3.8), we readily arrive at Eq. (3.4). h Some properties of Aðn; a; pÞ are observed as in the following lemma. Lemma 4. The following result holds true:
Að1; a; pÞ ¼ 0: Furthermore, for n 2 N n f1; 2g;
Aðn; a; 0Þ ¼
n X 1 ðk 2Þ! þ Bk akþ1 ; 12a k¼4 k!
Aðn; a; 1Þ ¼
n X 1 ðk 3Þ! ð1 þ log aÞ Bk akþ2 12 k! k¼4
Aðn; a; 2Þ ¼
n X a ðk 4Þ! ð1 þ 2 log aÞ þ 2 Bk akþ3 : 12 k! k¼4
and
We now give an asymptotic formula for the integral in Eq. (3.4) as asserted by Lemma 5 below.
ð3:8Þ
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Lemma 5. The following asymptotic formula holds true:
! n X 1 Bk k1 eat 1 dt ¼ O t pþ1 et 1 k¼0 k! a‘þ1 t 0 a ! 1; RðaÞ > 0; p; ‘ 2 N0 ; n ¼ p þ 1 þ ‘ : Z
1
ð3:9Þ
Proof. We prove the assertion (3.9) of Lemma 5 only for the case when n = p + 1. Then it is easy to see that the same argument will prove Lemma 5 for the cases when n > p + 1. We begin by separating the integral into two parts:
Z 0
1
! pþ1 X 1 Bk k1 eat dt ¼ I1 ðaÞ þ I2 ðaÞ; t et 1 k¼0 k! t pþ1
where
I1 ðaÞ :¼
! pþ1 X 1 Bk k1 eat dt t et 1 k¼0 k! t pþ1
Z p 0
and
I2 ðaÞ :¼
Z
! pþ1 X 1 Bk k1 eat dt: t et 1 k¼0 k! tpþ1
1
p
Since
jeat j ¼ eRðaÞt and we need RðaÞ > 0, we assume that a is any positive real number. In view of (3.2), I1(a) can be written as follows:
I1 ðaÞ :¼
Z p 0
! 1 X Bk kp2 at e dt: t k! k¼pþ2
The involved series is convergent when jtj < 2p; it is bounded for jtj < 2p. That is, for some L1 > 0, we have
Z p X Z p Z 1 1 Bk kp2 at L1 jI1 ðaÞj 5 eat dt 5 L1 eat dt ¼ : t e dt 5 L1 a 0 k¼pþ2 k! 0 0
It is also easy to observe that, in view of (3.3), for the second integral I2(a),
et
pþ1 X Bk k1 ¼ Oðt pþ1 Þ ðt ! 1Þ: t k! k¼0
1 ¼ Oð1=et Þ and 1
Thus there exist a large T > 0, L2 > 0 and L3 > 0 such that
pþ1 X Bk k1 and t 5 L3 t pþ1 k¼0 k!
1 L2 5 t et 1 e
for all t = T. Now we find that
jI2 ðaÞj 5
Z
T
jfp ðtÞjeat dt þ L2
p
Z T
1
1 eat dt þ L3 et tpþ1
Z
1
T
t pþ1
eat dt; t pþ1
where, for convenience,
fp ðtÞ :¼
! pþ1 X 1 Bk k1 1 : t et 1 k¼0 k! tpþ1
Since fp(t) is continuous on the closed bounded interval [p, T], for some L4 > 0, we have jfp(t)j 5 L4 for all t 2 [p, T]. We, therefore, get
jI2 ðaÞj 5 ðL2 þ L3 þ L4 Þ
Z 0
1
1 eat dt ¼ ðL2 þ L3 þ L4 Þ : a
This completes the proof of Lemma 5. h
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By mainly using Eq. (2.10) with the aid of the formulas and properties presented and recalled here, we can give an integral representation of the triple Gamma function C3 and its asymptotic formula as in Theorem 4 below. Here, as well as elsewhere in this paper, O() denotes the usual big O-notation (see, e.g., [49, pp. 4–8]). Theorem 4. Each of the following formulas holds true:
11 3 log C3 ðaÞ ¼ 18 þ 12 logð2pÞ þ 32 log A þ 2fð3Þ p2 12 þ 4 logð2pÞ þ log A a 3 a3 þ a6 34 a2 þ a 12 log a þ 98 þ 14 logð2pÞ a2 11 36
þ 12 ða2 3a þ 2ÞAðn; a; 0Þ þ 32 a Aðn; a; 1Þ þ 12 Aðn; a; 2Þ n
R1 P Bk k1 eat t dt þ 0 12 ða2 3a þ 2Þt 2 þ a 32 t þ 1 et 11 k! t3
ð3:10Þ
k¼0
ðRðaÞ > 0; n 2 N n f1; 2gÞ and
11 3 log C3 ðaÞ ¼ 18 þ 12 logð2pÞ þ 32 log A þ 2fð3Þ p2 12 þ 4 logð2pÞ þ log A a
3 1 þ 98 þ 14 logð2pÞ a2 11 a3 þ a6 34 a2 þ a 38 log a þ 12a 36 þ
‘þ3 P k¼4
ðk4Þ! Bk akþ1 12 ða2 k!
3a þ 2Þðk 2Þðk 3Þ þ a a 32 ðk 3Þ þ a2
ð3:11Þ
1 þO a‘þ1 ða ! 1; RðaÞ > 0; ‘ 2 N0 Þ; where Aðn; a; pÞðp ¼ 0; 1; 2Þ are given as in Lemma 4. 4. Concluding remarks and observations From Eq. (1.16) or the special case of Eq. (2.10) when n = 1, together with Eq. (3.4), we obtain an integral representation of log C(a):
log CðaÞ ¼ 12 logð2pÞ þ a 12 log a a þ Aðn; a; 0Þ n R1 P Bk k1 eat þ 0 et 11 t dt ðRðaÞ > 0; n 2 NÞ; k! t
ð4:1Þ
k¼0
where Aðn; a; 0Þ is given as in Lemma 4. By using Eq. (4.1) with the aid of Eq. (3.9), we get the following familiar asymptotic formula for log C(a) (see, e.g., [1, p. 257, Entry 6.1.40]): ‘þ1
P 1 log CðaÞ ¼ 12 logð2pÞ þ a 12 log a a þ 12a þ k¼4
1 þO a‘þ1 ðRðaÞ > 0; a ! 1; ‘ 2 N0 Þ:
ðk2Þ! Bk akþ1 k!
ð4:2Þ
From the special case of Eq. (2.10) when n = 2, together with some of the above previously recorded results, we obtain an integral representation and an asymptotic formula for logC2(a) (see, e.g., [56, p. 96, Eq. (40)]):
1 1 1 þ logð2pÞ þ log A 1 þ logð2pÞ a 12 2 2 1 3 2 2 ða 1Þ log a þ a þ ð1 aÞAðn; a; 0Þ þ Aðn; a; 1Þ 2 4 ! Z 1 n X 1 Bk k1 eat ½ð1 aÞt 1 t dt t þ e 1 k¼0 k! t2 0
log C2 ðaÞ ¼
ð4:3Þ
ðRðaÞ > 0; n 2 N n f1gÞ and
1 log C2 ðaÞ ¼ 12 þ 12 logð2pÞ þ log A 1 þ 12 logð2pÞ a
1 12 a2 2a þ 56 log a þ 34 a2 þ 12a þ
‘þ2 P k¼4
ðk3Þ! Bk akþ1 ½ð1 k!
aÞk þ a 2
1 ða ! 1; RðaÞ > 0; ‘ 2 N0 Þ: þO a‘þ1
ð4:4Þ
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We choose to conclude our present investigation by remarking further that the methods and techniques used above can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple Gamma functions Cn (n = q4). Acknowledgements This research was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology of the Republic of Korea (2011-0005224). The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. References [1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Tenth Printing, National Bureau of Standards, Applied Mathematics Series 55, National Bureau of Standards, Washington, DC, 1972; Reprinted by Dover Publications, New York, 1965 (see also [50]). [2] V.S. 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