Applied Mathematics and Computation 134 (2003) 515–533 www.elsevier.com/locate/amc
Multiple Gamma and related functions Junesang Choi a, H.M. Srivastava a
b,*
, V.S. Adamchik
c
Department of Mathematics, College of Natural Sciences, Dongguk University, Kyongju 780-714, South Korea b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada c Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA
Abstract The authors give several new (and potentially useful) relationships between the multiple Gamma functions and other mathematical functions and constants. As byproducts of some of these relationships, a classical definite integral due to Euler and other definite integrals are also considered together with closed-form evaluations of some series involving the Riemann and Hurwitz Zeta functions. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Multiple Gamma functions; Riemann’s f-function; Hurwitz Zeta function; Determinants of Laplacians; Clausen integrals; Series involving the Zeta function
1. Introduction and preliminaries The multiple Gamma functions were defined and studied by Barnes (cf. [7,8]) and others in about 1900. Although these functions did not appear in the tables of the most well-known special functions, yet the double Gamma function was cited in the exercises by Whittaker and Watson [42, p. 264] and recorded also by Gradshteyn and Ryzhik [24, p. 661, Entry 6.441(4); p. 937, Entry 8.333]. Recently, these functions were revived in the study of the
*
Corresponding author. E-mail addresses:
[email protected] (J. (H.M. Srivastava),
[email protected] (V.S. Adamchik).
Choi),
[email protected]
0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 3 0 1 - 0
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determinants of the Laplacians on the n-dimensional unit sphere Sn (see [11,17,18,30,39,41]), and in the evaluations of specific classes of definite integrals and infinite series involving, for example, the Riemann and Hurwitz Zeta functions (see [4,15,16,18,19,27]). The subject of some of these developments can be traced back to an over two-century old theorem of Christian Goldbach (1690–1764), as noted in the work of Srivastava [32, p. 1] who investigated this subject in a systematic and unified manner (see also [35], [36, Chapter 3], and [37]). More recently, Adamchik ([2,3]) presented new integral representations for the G-function, its asymptotic expansions, its various relations with other special functions, as well as a procedure for its efficient numerical computation (see also the works of Choi [12], Choi and Seo [14], and Vardi [39] for several closely related results). The theory of the double Gamma function has indeed found interesting applications in many other recent investigations (see, for details, [36]). In this paper we aim at presenting several new (and potentially useful) relationships between other mathematical functions and the multiple Gamma functions. As by-products of some of these relationships, we shall derive a classical definite integral due to Euler and evaluate some other related integrals and series involving the Riemann and Hurwitz Zeta functions. We begin by recalling the Barnes G-function (1=G ¼ C2 being the so-called double Gamma function) which has several equivalent forms including (for example) the Weierstrass canonical product: 1 z=2 fC2 ðz þ 1Þg ¼ Gðz þ 1Þ ¼ ð2pÞ exp 12½ð1 þ cÞz2 þ z 1 Y z k z2 1þ exp z þ ; ð1:1Þ
k 2k k¼1 where c denotes the Euler–Mascheroni constant given by ! n X 1 log n ffi 0:577215664901532860606512 : c ¼ lim n!1 k k¼1
ð1:2Þ
For sufficiently large real x and a 2 C, we have the Stirling formula for the G-function: log Gðx þ a þ 1Þ ¼
xþa 1 3x2 logð2pÞ log A þ ax 2 12 4 x 2 1 a2 þ þ þ ax log x þ Oð1=xÞ ðx ! 1Þ; 2 12 2
ð1:3Þ
where A is the Glaisher–Kinkelin constant given by (cf. [39,41]) log A ¼
1 f0 ð1Þ: 12
ð1:4Þ
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Based on the Binet integral representation, Adamchik [2] derived the complete asymptotic expansion of the Barnes G-function. The G-function satisfies the following fundamental functional relationships: Gð1Þ ¼ 1
and
Gðz þ 1Þ ¼ CðzÞGðzÞ ðz 2 CÞ;
ð1:5Þ
where C denotes the familiar Gamma function. Barnes [7] generalized the functional relationships in (1.5) to the case of the multiple Gamma functions which are denoted usually by Gn ðzÞ or Cn ðzÞ. Throughout this paper, we choose to follow the notations used (for example) by Vigneras [40] (see also [36]). Thus we have Cn ðzÞ :¼ fGn ðzÞgð1Þ
n1
ðn 2 NÞ;
ð1:6Þ
so that G1 ðzÞ ¼ C1 ðzÞ ¼ CðzÞ;
G2 ðzÞ ¼
1 ¼ GðzÞ; C2 ðzÞ
G3 ðzÞ ¼ C3 ðzÞ;
ð1:7Þ
and so on. In terms of the multiple Gamma functions Gn ðzÞ of order n ðn 2 NÞ, the aforementioned functional relationships of Barnes [7] can be rewritten as follows (cf. [40, p. 239]; see also [36, p. 14, Theorem 1.4]): Gn ð1Þ ¼ 1
and
Gnþ1 ðz þ 1Þ ¼ Gn ðzÞGnþ1 ðzÞ ðz 2 C; n 2 NÞ
and
Cnþ1 ðz þ 1Þ ¼
ð1:8Þ
or, equivalently, Cn ð1Þ ¼ 1
Cnþ1 ðzÞ ðz 2 C; n 2 NÞ; Cn ðzÞ
ð1:9Þ
which may be used to define the multiple Gamma functions Gn ðzÞ and Cn ðzÞ ðn 2 NÞ. From (1.8) and (1.2) one can easily derive the Weierstrass canonical product form of the triple Gamma function C3 (see [18]): C3 ð1 þ zÞ ¼ G3 ð1 þ zÞ
z3 p2 3 1 1 2 c þ logð2pÞ þ cþ þ ¼ exp þ z 4 2 6 6 2 3 logð2pÞ log A z þ 8 4
1 Y z 12kðkþ1Þ 1 1 1 2
ðk þ 1Þz 1þ 1þ exp z k 2 4 k k¼1 1 1 3 1þ þ z : 6k k
ð1:10Þ
Another form of the triple Gamma function C3 appeared in the work of Choi [11] who expressed, in terms of the double and triple Gamma functions,
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the analogous Weierstrass canonical product of the shifted sequence of the eigenvalues of the Laplacian on the unit sphere S3 with standard metric which was indispensable to evaluate the determinant of the Laplacian on S3 there. The Riemann Zeta function fðsÞ is defined by 8X 1 1 X 1 1 1 > > ¼ ðRðsÞ > 1Þ; > < s s k 1 2 ð2k 1Þs k¼1 k¼1 ð1:11Þ fðsÞ :¼ 1 X > ð1Þkþ1 > 1s 1 > Þ ðRðsÞ > 0; s ¼ 6 1Þ; ð1 2 : ks k¼1 which can, except for a simple pole only at s ¼ 1 with its residue 1, be continued analytically to the whole complex s-plane by means of the contour integral representation (cf. [42, p. 266]) or many other integral representations (cf. [21, p. 33]). It satisfies the functional equation (see [38, p. 13]) ps fðsÞ ¼ 2s ps1 Cð1 sÞfð1 sÞ sin : ð1:12Þ 2 The generalized (or Hurwitz) Zeta function fðs; aÞ is defined by fðs; aÞ :¼
1 X k¼0
1 ðk þ aÞs
ðRðsÞ > 1; a 2 C n Z 0 ; Z0 :¼ Z [ f0gÞ;
ð1:13Þ which can, just as fðsÞ, be continued analytically to the whole complex s-plane except for a simple pole only at s ¼ 1. Indeed, from the definitions (1.11) and (1.13), it is easily observed that fðs; 1Þ ¼ fðsÞ ¼ ð2s 1Þ
fðs; maÞ ¼
1
1 f s; 2
m1 1 X j f s; a þ ms j¼0 m
and
fðs; 2Þ ¼ fðsÞ 1;
ðm 2 NÞ;
ð1:14Þ
ð1:15Þ
and m1 1 X j f s; fðsÞ ¼ s m 1 j¼1 m
ðm 2 N n f1gÞ:
ð1:16Þ
Many elementary identities such as those derivable especially from (1.15) and (1.16) will be required in our present investigation.
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2. A family of generalized Glaisher–Kinkelin constants Following the works of Glaisher [22] and Alexeiewsky [5] on the asymptotic behavior of the product: p
p
p
p
11 22 33 nn
when n ! 1, Bendersky [9] considered the limit formula: ! n X k log Ak ¼ lim m log m pðn; kÞ ; n!1
ð2:1Þ
m¼1
where pðn; kÞ ¼
nk nkþ1 1 log n þ log n kþ1 2 kþ1 ! j k X X nkj Bjþ1 1 þ k! log n þ ð1 dj;k Þ klþ1 ðj þ 1Þ!ðk jÞ! l¼1 j¼1 ð2:2Þ
and dj;k is the Kronecker delta. For n ¼ 0, (2.1) yields " # n X 1 log A0 ¼ lim log m þ n log n þ n : n!1 2 m¼1
ð2:3Þ
This limit formula is fairly well-known. The finite sum on the right-hand side of (2.3) is equal to log n!, and thus, using the Stirling formula, we immediately obtain log A0 ¼
logð2pÞ : 2
ð2:4Þ
For n ¼ 1, the limit (2.1) defines the Glaisher–Kinkelin constant A [cf. Eq. (1.4) above]: " # 2 n X n n 1 n2 log A1 ¼ log A ¼ lim m log m log n þ þ þ : ð2:5Þ n!1 2 2 12 4 m¼1 Using the functional relation (1.5), we express the finite sum in (2.5) in terms of the Barnes G-function: ! n n Y X m log m ¼ log mm ¼ n log n! log Gðn þ 1Þ: ð2:6Þ m¼1
m¼1
By combining (2.5) and (2.6), and the Stirling formula, we can establish the relationship:
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log A ¼ lim
n!1
log Gðn þ 1Þ þ
n2 1 2 12
3n2 n 1 þ logð2pÞ þ log n : 2 12 4 ð2:7Þ
In a similar way, we can demonstrate that the higher-order Glaisher–Kinkelin constant An is related to the multiple Barnes function. Upon setting n ¼ 3 in (2.1) and observing that n X
m2 log m ¼ n2 log n! ð2n 1Þ log Gðn þ 1Þ 2 log G3 ðn þ 1Þ;
ð2:8Þ
m¼1
we find that
3 n n2 1 logA2 ¼ lim 2logG3 ðn þ 1Þ logn þ n!1 3 2 12 2 11n3 3n2 n 1 n n þ logð2pÞ þ ð2n 1ÞlogA : þ 18 4 6 12 2 2 ð2:9Þ The generalized Glaisher–Kinkelin constants An appear naturally in the asymptotic expansion of the multiple Barnes function. With a view to circumventing the problem of evaluation of the limits in (2.7) and (2.9), Adamchik [1] found the following closed-form representation for the generalized Glaisher–Kinkelin constant An : log An ¼
Bnþ1 Hn f0 ðnÞ ðn 2 N0 Þ; nþ1
ð2:10Þ
where Bk and Hk are the Bernoulli and harmonic numbers, respectively. The proof is based on the following analytical property of the Hurwitz Zeta function fðs; aÞ: n X
mk log m ¼ f0 ðk; n þ 1Þ f0 ðkÞ
ð2:11Þ
m¼1
and its known asymptotic expansion at infinity. In fact, in light of the following consequence of the functional equation (1.12) [33, p. 387, Eq. (1.15)]: 2n
fð2n þ 1Þ ¼ ð1Þn
2 ð2pÞ 0 f ð2nÞ ð2nÞ!
ðn 2 NÞ;
ð2:12Þ
it is easily seen from (2.10) that log A2n ¼ ð1Þnþ1
ð2nÞ! 2 ð2pÞ2n
fð2n þ 1Þ ðn 2 NÞ:
ð2:13Þ
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The constants A1 , A2 , and A3 (denoted commonly by A, B, and C, respectively) were also considered by Voros [41], Vardi [39], Choi and Srivastava (cf. [16,18]; see also [36, Chapter 1]).
3. Special cases of the multiple Barnes function The Barnes G-function, given by the functional equation in (1.5), is a generalization of the Euler Gamma function. Thus it is not surprising to anticipate that the G-function would have closed-form representations for particular values of its argument. As a matter of fact, the G-function can be expressed in finite terms by means of the generalized Glaisher–Kinkelin constants An , as well as other known constants. Barnes [7, p. 283] evaluated the following integral (see also [36, p. 207, Eq. 3.4(444)]): Z z 1 z2 log Cðt þ aÞ dt ¼ ½logð2pÞ þ 1 2a z þ ðz þ a 1Þ 2 2 0
log Cðz þ aÞ log Gðz þ aÞ þ ð1 aÞ
log CðaÞ þ log GðaÞ; ð3:1Þ which, in the special case when a ¼ 1, reduces immediately to Alexeiewsky’s theorem (cf. [5] and [36, p. 32, Eq. 1.3 (42)]): Z z 1 z2 log Cðt þ 1Þ dt ¼ ½logð2pÞ 1 z þ z log Cðz þ 1Þ log Gðz þ 1Þ 2 2 0 ð3:2Þ or, equivalently, Z z zð1 zÞ z þ logð2pÞ ð1 zÞ log CðzÞ log GðzÞ; log CðtÞ dt ¼ 2 2 0 ð3:3Þ since Z
z
log t dt ¼ z log z z:
ð3:4Þ
0
The integral formula (3.3), evaluated recently by Gosper [23] and Adamchik [1], happens to be a source for many closed-form representations of the Barnes G-function. The simplest non-trivial case z ¼ 12 is due to Barnes [7, p. 288, Section 7]: ð3:5Þ G 12 ¼ 21=24 p1=4 e1=8 A3=2 :
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By recalling the following special value of the Gamma function (see [31, p. 1]): C 14 ffi 3:625609908221908
ð3:6Þ and making use of a duplication formula for the G-function (cf. [7, p. 291] for the general case; see also [39]), Choi and Srivastava [16] showed that 3=4 ffi 0:293756
ð3:7Þ G 14 ¼ eð3=32ÞðG=4pÞ A9=8 C 14 or, equivalently, that 1=4 G 34 ¼ 21=8 p1=4 eð3=32ÞþðG=4pÞ A9=8 C 14 ffi 0:848718 ; where G is the Catalan constant defined by Z 1 X 1 1 ð1Þm G :¼ KðkÞ dk ¼ ffi 0:915965594177219015 ; 2 2 0 m¼0 ð2m þ 1Þ and K is the complete elliptic integral of the first kind, given by Z p=2 dt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : KðkÞ :¼ 0 1 k 2 sin2 t
ð3:8Þ
ð3:9Þ
ð3:10Þ
The Catalan constant G becomes a special case of many other functions (one of which will be considered in the following section) and has, among other things, been used to evaluate integrals, such as (see [24, p. 526, Entry 4.224(2)]): Z p=4 p 1 ð3:11Þ logðsin tÞ dt ¼ log 2 G; 4 2 0 and to give closed-form evaluations of a certain class of series involving the Zeta function (cf. [16]). Other particular cases of the Barnes function GðzÞ when 1 1 2 5 z ¼ ; ; ; and 6 3 3 6
ð3:12Þ
were recorded by (for example) Srivastava and Choi [36, p. 269, Problem 1], who also provided references to relevant earlier works dealing with these evaluations (see also [2,3]). The identity (3.5) was generalized by Vardi [39] to the multiple Barnes function. The formula is much too complicated to be reproduced here. Here is one particular case when n ¼ 3 and z ¼ 12 (see also [18, p. 95, Eq. (4.8)]): 7=8 G3 12 ¼ 21=24 p3=16 e1=8 A3=2 ¼ C3 12 ; ð3:13Þ 1 A2 where Ak denotes the generalized Glaisher–Kinkelin constants given by (2.1) with, of course,
J. Choi et al. / Appl. Math. Comput. 134 (2003) 515–533
A1 ¼ A
and
523
A2 ¼ B:
Other (known or new) closed-form representations for the triple Barnes function should emerge from the following integral (see [1] and Eq. (4.12) below with a ¼ 1): Z z zðz 2Þ logð2pÞ log GðtÞ dt ¼ 2 log G3 ðz þ 1Þ þ z log GðzÞ þ 4 0 2z log A1
zð2z2 6z þ 3Þ : 12
ð3:14Þ
The integral (3.14) has an interesting application to the constants A1 and A2 (or, alternatively, A and B): Z 1=2 1 logð28 p9 Þ 4 7 logðA1 A2 Þ ¼ log GðtÞ dt: ð3:15Þ 6 12 0
4. Relationships between multiple Gamma and other functions The Clausen function (or the Clausen integral) Cl2 ðtÞ (see Lewin [28, p. 101]; see also Chen and Srivastava [10, p. 184]) defined by Z t 1 X sinðntÞ Cl2 ðtÞ :¼ log 2 sin 12u du ¼ ð4:1Þ n2 0 n¼1 stems actually from the imaginary part of the Dilogarithm function Li2 ðeit Þ defined by Z z 1 X logð1 uÞ zn du ¼ Li2 ðzÞ :¼ ðjzj 6 1Þ: ð4:2Þ u n2 0 n¼1 Now, by employing integration by parts in the following well-known integral formula (see [7, p. 279]; see also [16, p. 94, Eq. (2.1)]) due originally to Kinkelin: Z t Gð1 tÞ pu cotðpuÞ du ¼ t logð2pÞ þ log ; ð4:3Þ Gð1 þ tÞ 0 we have (cf. [16, p. 95, Eq. (2.2)]): Z t sinðp tÞ Gð1 þ tÞ log½sinðpuÞ du ¼ t log þ log : 2p Gð1 tÞ 0
ð4:4Þ
If we combine (4.1) and (4.4), we obtain the following relationship between the Clausen function and the G-function:
524
J. Choi et al. / Appl. Math. Comput. 134 (2003) 515–533 1 X sinð2pntÞ n2 n¼1 sinðptÞ Gð1 þ tÞ ¼ 2pt log 2p log ; p Gð1 tÞ
Cl2 ð2ptÞ ¼
which, for t ¼ 14, readily yields the Catalan constant: p ¼G Cl2 2
ð4:5Þ
ð4:6Þ
by just observing, for the third member of (4.5), the following identity derivable from Eqs. (3.7), (3.8), and (1.5): G 54 G 1=8 1=4 ¼ 2 p exp : ð4:7Þ 2p G 34 Next we recall a relationship (cf. [20, p. 210]): S2 ðaÞ ¼ Cl2 ðaÞ 14 Cl2 ð2aÞ;
ð4:8Þ
where Sm ðaÞ denotes the classical trigonometric series defined by (cf. [20,34]) Sm ðaÞ :¼
1 X sinð2k þ 1Þa m : ð2k þ 1Þ k¼0
ð4:9Þ
A combination of (4.5) and (4.8) gives the following relationship between S2 ðaÞ and the G-function: p Gð1 þ 2tÞ Gð1 þ tÞ S2 ð2ptÞ ¼ pt log½2 cotðptÞ þ log 2p log : 2 Gð1 2tÞ Gð1 tÞ ð4:10Þ Integrating both sides of the second equality of (4.5) from 0 to t, we find an interesting relationship among a series involving a cosine function, a definite integral, and integrals of the G-function: 1 1 X cosð2nptÞ 1 ¼ 2 4p n¼1 n3
Z
t
t2 log p 2 0 Z t Z t þ log Gð1 þ uÞ du þ log Gð1 þ uÞ du: ð4:11Þ ulog½sinðpuÞ du
0
0
Now we recall an integral formula related to the multiple Gamma functions (see [18, p. 96, Eq. (5.7)]) in the form:
J. Choi et al. / Appl. Math. Comput. 134 (2003) 515–533
Z
525
t
log Gðu þ aÞ du ¼
0
1 a2 1 ða 1Þ logð2pÞ 2 log A þ a t 2 4 2 1 1 þ ½logð2pÞ þ 2 2a t2 t3 4 6 þ ðt þ a 2Þ log Gðt þ aÞ 2 log C3 ðt þ aÞ þ ð2 aÞ log GðaÞ þ 2 log C3 ðaÞ;
ð4:12Þ
which, for a ¼ 1, yields [cf. Eq. (3.14) above] Z t 1 t2 t3 log Gðu þ 1Þ du ¼ 2 log A t þ logð2pÞ 4 4 6 0 þ ðt 1Þ log Gðt þ 1Þ 2 log C3 ðt þ 1Þ;
ð4:13Þ
by virtue of Eqs. (1.5) and (1.8) with n ¼ 2. If we apply (4.13) in (4.11), we obtain an interesting identity involving a series and an integral through multiple Gamma functions: Z pt 1 1 X cosð2nptÞ 1 1 t2 ¼ u logðsin uÞ du þ log 2 2 3 2 4p n¼1 n p 0 2 þ ðt 1Þ log Gðt þ 1Þ ð1 þ tÞ log Gð1 tÞ 2 log½C3 ð1 þ tÞC3 ð1 tÞ ;
ð4:14Þ
the left-hand summation part of which can, in terms of the higher-order Clausen function Cln ðtÞ be defined, for all n 2 N n f1g, by (cf. [28, p. 191]) 8X 1 sinðktÞ > > if n is even; > < kn k¼1 ð4:15Þ Cln ðtÞ :¼ X 1 > cosðktÞ > > if n is odd; : kn k¼1 be expressed as follows: 1 X cosð2nptÞ 1 ¼ Cl3 ð2ptÞ fð3Þ: n3 n¼1
ð4:16Þ
It is known that (cf., e.g., [26, p. 356, Entry (54.5.4)]; see also [10, p. 184, Eq. (2.23)]) 1 X 1 1 1 fð2kÞ 2k Cl2 ð2ptÞ ¼ log½2 sinðptÞ t : 4pt 2 2 2k þ1 k¼1
ð4:17Þ
A combination of (4.5) and (4.17) gives another proof of a special case of the following known identity for series involving the Riemann Zeta function (see [16, p. 107, Eq. (4.10)]):
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1 X fð2kÞ 2kþ1 1 1 Gð1 þ tÞ t ¼ ½1 logð2pÞ t þ log 2k þ 1 2 2 Gð1 tÞ k¼1
ðjtj < 1Þ: ð4:18Þ
The special case of (4.14) when t ¼ 12, with the aid of (1.5), (1.6) with n ¼ 2, (1.8), (2.13) with n ¼ 1, (3.2) and (3.10), yields a known integral formula: Z p=2 7 p2 ð4:19Þ u logðsin uÞ du ¼ fð3Þ log 2; 16 8 0 which is due to Euler (1772) who proved it through a striking and elaborate scheme as noted by Ayoub [6, p. 1084]. We now give here another short proof of (4.19). Indeed, if we multiply the well-known formula: 1 u 2k X u cot u ¼ 2 fð2kÞ ðjuj < pÞ ð4:20Þ p k¼0 by u and integrate the resulting equation with respect to u from 0 to p=2, we find that (cf. [37, p. 832, Eq. (2.4) with x ¼ s ) 1 ¼ 2]) Z p=2 Z p=2 1 p2 X fð2kÞ u2 cot u du ¼ 2 u logðsin uÞ du ¼ ; ðk þ 1Þ 22k 4 0 0 k¼0 ð4:21Þ which, when combined with a known result (cf., e.g., [10, p. 191, Eq. (3.19)]), proves (4.19). From (4.17) and a result of Grosjean [25, p. 334, Eq. (12)] we readily obtain a closed-form evaluation of a class of series involving the Riemann Zeta function: 2k
1 X fð2kÞ p 1 1 pp ¼ log 2 sin 2k þ 1 2q 2 2 2q k¼1
q1 1 X k k pkp f 2; f 2;1 sin 8pqp k¼1 2q 2q q ðp;q 2 N; 1 6 p 6 2q 1; ðp;qÞ ¼ 1Þ;
ð4:22Þ
which, in the special cases when (respectively) p ¼ q ¼ 1, p ¼ q 1 ¼ 1, and p ¼ q 2 ¼ 1, yields 1 X k¼1
fð2kÞ 1 1 ¼ log 2; ð2k þ 1Þ 22k 2 2
which is equivalent to a known result [11, p. 109, Eq. (4.22)];
ð4:23Þ
J. Choi et al. / Appl. Math. Comput. 134 (2003) 515–533 1 X k¼1
fð2kÞ 1 1 G ¼ log 2 ; 2k ð2k þ 1Þ 4 2 4 p
527
ð4:24Þ
which is precisely the same as the known result [16, p. 114, Eq. (5.9)]; pffiffiffi 1 X 3 fð2kÞ 1 p 1 p ffiffi ffi f 2; þ ¼ ; ð4:25Þ ð2k þ 1Þ 62k 2 2 3 4p 3 k¼1 which corresponds to the special case of the known result (4.17) above when t ¼ 16 (see also [36, Problem 2 (Chapter 3) and Problem 13 (Chapter 4)]). In fact, closed-form evaluations of series involving the Zeta function have an over two-century old history as commented in Section 1 and have attracted many mathematicians ever since then (cf. [32]; see also [36, Chapter 3]). Here we also give a general formula for this subject by recalling (see [13, p. 224, Eq. (1.11)]) Z a 1 X ð1Þk akþb fðk; aÞ ¼ tb ½wðt þ aÞ wðaÞ dt ðb P 0Þ; ð4:26Þ k þ b 0 k¼2 where the Digamma function (or the w-function) is the logarithmic derivative of the Gamma function C, and a closed-form solution to the integral (see [1] and [2]): Z a ð1Þn n1 Bnþ1 Hn tn wðtÞ dt ¼ ð1Þ f0 ðnÞ þ nþ1 0 n X Bkþ1 ðaÞHk k n 0 nk a þ ð1Þ f ð k; aÞ ; ð4:27Þ k kþ1 k¼0 where Bn and Bn ðaÞ are the Bernoulli numbers and Bernoulli polynomials (see [21, pp. 35–36]), respectively, and Hn are the harmonic numbers defined by [cf. Eq. (2.10) above] Hn :¼
n X 1 k k¼1
ðn 2 NÞ:
ð4:28Þ
Combining (4.26) and (4.28), we obtain the desired result: k n 1 X ð1Þ akþn an canþ1 ð1Þ n1 fðkÞ ¼ þ þ ð1Þ f0 ðnÞ þ Bnþ1 Hn kþn n nþ1 nþ1 k¼2 n X Bkþ1 ðaÞHk k n 0 nk þ ð1Þ f ð k; aÞ a kþ1 k k¼0
ðRðaÞ > 0; jaj < 1; n 2 NÞ; where c is the Euler–Mascheroni constant given by (1.2).
ð4:29Þ
528
J. Choi et al. / Appl. Math. Comput. 134 (2003) 515–533
5. Evaluations of some definite integrals We begin by combining formulas (4.14) and (4.16) in the form: 1 1 t2 ½Cl3 ð2ptÞ fð3Þ ¼ 2 IðtÞ þ log 2 þ ðt 1Þ log Gðt þ 1Þ 2 4p p 2 ð1 þ tÞ log Gð1 tÞ 2 log AðtÞ; where, for convenience, IðtÞ and AðtÞ are defined by Z pt IðtÞ :¼ u logðsin uÞ du
ð5:1Þ
ð5:2Þ
0
and AðtÞ :¼ C3 ð1 þ tÞC3 ð1 tÞ;
ð5:3Þ
so that, obviously, Euler’s integral (4.19) is precisely Ið12Þ. Now we evaluate the integral Ið14Þ. First of all, it follows from (1.7) that
1 1 log AðtÞ ¼ c þ logð2pÞ þ t2 2 2 ! 1 1 X 1 fð2kÞ 2kþ2 X fð2k þ 1Þ 2kþ2 t t þ þ : ð5:4Þ 2 k¼1 k þ 1 kþ1 k¼1 Furthermore, the special case of a known result [16, p. 107, Eq. (4.11)] when a ¼ 1 gives 1 X fð2k þ 1Þ 2kþ2 t ¼ ðc þ 1Þt2 log½Gð1 þ tÞGð1 tÞ kþ1 k¼1
ðjtj < 1Þ; ð5:5Þ
which is a companion formula of (5.4) for the G-function. We thus find from (1.5), (3.7), and (3.8) that 1 h X 18 i fð2k þ 1Þ 4 36 ¼ 4 c þ log 4
p
A
C 4 : ðk þ 1Þ 42k k¼1
ð5:6Þ
The identity of Chen and Srivastava [10, p. 191, Eq. (3.20)] is rewritten here in the following form: 1 X k¼1
fð2kÞ 1 1 4G 35 þ 2 fð3Þ: ¼ log 2 ðk þ 1Þ 42k 2 2 p 4p
ð5:7Þ
Next, the special case of another known result [16, p. 108, Eq. (4.14)] when a ¼ 1 yields
J. Choi et al. / Appl. Math. Comput. 134 (2003) 515–533
1 X fð2kÞ k¼1
z2 Gð1 þ zÞ z2kþ2 ¼ ½1 logð2pÞ þ zlog k þ1 Gð1 zÞ 2 Z z Z z logGðt þ 1Þ dt logGðt þ 1Þ dt ðjtj < 1Þ; 0
529
ð5:8Þ
0
which, upon setting z ¼ 14 and using (5.7), gives the following interesting identity: Z 1=4 Z 1=4 log Gðt þ 1Þ dt þ log Gðt þ 1Þ dt 0 0 1 4G 35 logð2pÞ þ 2 fð3Þ : ¼ ð5:9Þ 32 p 2p In order to illustrate the usefulness of (5.9), we integrate the special case of [16, p. 107, Eq. (4.10)] when a ¼ 1 from 0 to 14, and we readily find from (5.9) that ! 1 4p2 1 2G X fð2kÞ fð3Þ ¼ þ ; ð5:10Þ p ðk þ 1Þð2k þ 1Þ 42k 35 2 k¼1 which is an interesting addition to the results recorded by Chen and Srivastava [10] who treated the subject of series representations for fð3Þ in a rather systematic manner and presented new results and some relevant connections with other functions. From the definitions (4.15) and (1.11), it is easy to see that (cf. [28, p. 300]) p 1 2n Cl2nþ1 ð5:11Þ ¼ 4nþ1 fð2n þ 1Þ ðn 2 NÞ: 2 2 Finally, upon setting t ¼ 14 in (5.1), and making use of (5.4), (5.6), (5.7), and (5.11) (with n ¼ 1), we obtain the desired integral evaluation (cf., e.g., [29, p. 539, Entry 2.6.34.2]): Z p=4 35 pG p2 fð3Þ log 2; ð5:12Þ u logðsin uÞ du ¼ I 14 ¼ 128 8 32 0 which, upon employing integration by parts, yields Z p=4 35 pG p2 þ log 2: u2 cot u du ¼ fð3Þ þ 64 4 32 0
ð5:13Þ
We evaluate some more definite integrals and series involving the Zeta function. By applying the known result [20, p. 210, Eq. (13c)]:
pffiffiffi pffiffiffi 1 pffiffiffi 1 S2 p4 ¼ 2f 2; ð5:14Þ 2 2 þ 1 p2 16 2G 32 8
530
J. Choi et al. / Appl. Math. Comput. 134 (2003) 515–533
in (4.10) with t ¼ 18, and using the relationship (4.7), we obtain ! h pffiffiffi i 1 pffiffiffi G 98 1 7 ¼ log 2 2 þ 2 p þ 2 21 G log 16 8p G 8 h p ffiffi ffi p ffiffiffi i 1 þ 2 2 þ 1 p2 2f 2; 18 : 64p
ð5:15Þ
If we set t ¼ 18 in (4.3), (4.4), and (4.18), and make use of (5.15), we find that Z p=8 h pffiffiffi i 1 pffiffiffi p log 2 2 p þ 1 2 2 G u cot u du ¼ 16 8 0 h i p ffiffi ffi p ffiffi ffi 1 2f 2; 18 2 2 þ 1 p2 ; þ ð5:16Þ 64 Z p=8 p 1 pffiffiffi logð4pÞ þ 2 2 1 G logðsin uÞ du ¼ 16 8 0 pffiffiffi i 1 h pffiffiffi 2 2 þ 1 p2 2f 2; 18 ; ð5:17Þ þ 64 and 1 X k¼1
h pffiffiffi i 1 pffiffiffi fð2kÞ 1 1 log 2 2 p þ 2 2 1 G ¼ ð2k þ 1Þ 82k 2 4 2p pffiffiffi i 1 h pffiffiffi þ 2 2 þ 1 p2 2f 2; 18 : 16p
By setting t ¼ 16 in (4.18) and applying (4.25), we obtain ! pffiffiffi pffiffiffi G 76 1 3 3 1 5 ¼ p þ logð2pÞ f 2; 3 : log 6 18 12p G 6
ð5:18Þ
ð5:19Þ
In view of (5.19) and the relationship (cf. [20, p. 210, Eq. (13e)]): S2
p 6
¼
2G ; 3
ð5:20Þ
(4.10) with t ¼ 121 yields ! pffiffiffi h pffiffiffi i G 13 3 p 1 1 G 1 12 11 ¼ þ log 4 2 þ 3 p þ log f 2; 3 : 24 3 2p 3p 24 G 12 ð5:21Þ If we set t ¼ 121 in (4.3), (4.4), and (4.18), and apply (5.21), we see that pffiffiffi ! pffiffiffi Z p=12 3 1 1 p2 G p 2þ 3 log f 2; 3 u cot u du ¼ þ ; ð5:22Þ 24 2 3 24 p 3 0
J. Choi et al. / Appl. Math. Comput. 134 (2003) 515–533
Z
p=12
logðsin uÞ du ¼
0
pffiffiffi 2 3 p G p 1 logð4pÞ þ f 2; 13 ; 24 3 2 3 24
531
ð5:23Þ
and 1 X k¼1
fð2kÞ 1 2G 1 þ log ¼ 2k ð2k þ 1Þ 12 2 p 4
pffiffiffi ! pffiffiffi 2þ 3 3 p 1 1 þ f 2; 3 : p 4 3 2p ð5:24Þ
Just as in getting (4.23), (4.24), and (4.25), the special cases of (4.22) when p ¼q3¼1
and
p ¼q5¼1
can also be seen to yield (5.18) and (5.24), respectively. In view of the r^ oles played by (5.15) and (5.21) in closed-form evaluations of definite integrals, we choose to record here the following general identity (cf. [36, p. 352, Problem 9]): 1 0
G 1 þ 2qp p 1 pp A ¼ log sin log @ 2q p q G 1 p 2q
q1 1 X k k pkp þ 2 f 2; 1 f 2; sin 8q p k¼1 2q 2q q ðp; q 2 N; 1 6 p 6 2q 1; ðp; qÞ ¼ 1Þ;
ð5:25Þ
which follows immediately from (4.18) (with t ¼ p=ð2qÞ) and (4.22). We conclude this paper by remarking that (easily computable) closed-form evaluations of Ið1=2n Þ ðn 2 N n f1; 2gÞ do not seem to have been carried out in the mathematical literature so far. The following integral formula, which is complementary to the identity (5.25), is worthy of note in this connection: Z pp=q p I u logðsin uÞ du :¼ q 0 q1 p2 p2 1 1 pp X 2pkp k 1 3 fð3Þ 3 sin ¼ 2 log 2 þ f 2; 4 q 2q k¼1 q q 2q q1 1 X 2pkp k 3 cos f 3; ðp; q 2 N; 1 6 p 6 qÞ; ð5:26Þ 4q k¼1 q q which generalizes (4.19) as well as (5.12), since
1 1 3 f 2; f 2; : G¼ 16 4 4
ð5:27Þ
532
J. Choi et al. / Appl. Math. Comput. 134 (2003) 515–533
Acknowledgements For the first-named author, this work was supported by Grant No. 2001-110200-004-2 from the Basic Research Program of the Korea Science and Engineering Foundation. The second-named author was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.
References [1] V.S. Adamchik, Polygamma functions of negative order, J. Comput. Appl. Math. 100 (1998) 191–199. [2] V.S. Adamchik, Integral representations for the Barnes function, 2001 (Preprint). [3] V.S. Adamchik, On the Barnes function, in: B. Mourrain (Ed.), Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, London, July 22–25, 2001, ACM Press, New York, 2001, pp. 15–20. [4] V.S. Adamchik, H.M. Srivastava, Some series of the Zeta and related functions, Analysis 18 (1998) 131–144. € ber eine Classe von Funktionen, die der Gammafunktion analog sind, [5] W.P. Alexeiewsky, U Leipzig Weidmannsche Buchhandlung 46 (1894) 268–275. [6] R. Ayoub, Euler and the Zeta function, Amer. Math. Monthly 81 (1974) 1067–1086. [7] E.W. Barnes, The theory of the G-function, Quart. J. Math. 31 (1899) 264–314. [8] E.W. Barnes, On the theory of the multiple Gamma function, Trans. Cambridge Philos. Soc. 19 (1904) 374–439. [9] L. Bendersky, Sur la fonction Gamma generalisee, Acta Math. 61 (1933) 263–322. [10] M.-P. Chen, H.M. Srivastava, Some families of series representations for the Riemann fð3Þ, Resultate Math. 33 (1998) 179–197. [11] J. Choi, Determinant of Laplacian on S3 , Math. Japon. 40 (1994) 155–166. [12] J. Choi, Explicit formulas for Bernoulli polynomials of order n, Indian J. Pure Appl. Math. 27 (1996) 667–674. [13] J. Choi, C. Nash, Integral representations of the Kinkelin’s constant A, Math. Japon. 45 (1997) 223–230. [14] J. Choi, T.Y. Seo, The double Gamma function, East Asian Math. J. 13 (1997) 159–174. [15] J. Choi, H.M. Srivastava, Sums associated with the Zeta function, J. Math. Anal. Appl. 206 (1997) 103–120. [16] J. Choi, H.M. Srivastava, Certain classes of series involving the Zeta function, J. Math. Anal. Appl. 231 (1999) 91–117. [17] J. Choi, H.M. Srivastava, An application of the theory of the double Gamma function, Kyushu J. Math. 53 (1999) 209–222. [18] J. Choi, H.M. Srivastava, Certain classes of series associated the Zeta function and multiple Gamma functions, J. Comput. Appl. Math. 118 (2000) 87–109. [19] J. Choi, H.M. Srivastava, J.R. Quine, Some series involving the Zeta function, Bull. Austral. Math. Soc. 51 (1995) 383–393. [20] D. Cvijovic, J. Klinowski, Closed-form summation of some trigonometric series, Math. Comput. 64 (1995) 205–210. [21] A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953. [22] J.W.L. Glaisher, On the product 11 22 33 nn , Messenger Math. 7 (1877) 43–47.
J. Choi et al. / Appl. Math. Comput. 134 (2003) 515–533 R m=6
533
[23] R.W. Gosper Jr., n=4 ln CðzÞ dz, Fields Inst. Comm. 14 (1997) 71–76. [24] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980 (Corrected and Enlarged Ed. prepared by A. Jeffrey). [25] C.C. Grosjean, Formulae concerning the computation of the Clausen integral Cl2 ðhÞ, J. Comput. Appl. Math. 11 (1984) 331–342. [26] E.R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975. [27] S. Kanemitsu, H. Kumagai, M. Yoshimoto, Sums involving the Hurwitz Zeta function, Ramanujan J. 5 (2001) 5–19. [28] L. Lewin, Polylogarithms and Associated Functions, Elsevier, Amsterdam, 1981. [29] A.P. Prudnikov, Yu.A. Bryckov, O.I. Maricev, Integrals and Series (Elementary Functions), Nauka, Moscow, 1981 (in Russian); see also Integrals and Series, Vol. I: Elementary Functions (Translated from the Russian by N.M. Queen), Gordon and Breach, New York, 1986. [30] J.R. Quine, J. Choi, Zeta regularized products and functional determinants on spheres, Rocky Mountain J. Math. 26 (1996) 719–729. [31] M.R. Spiegel, Mathematical Handbook, McGraw-Hill, New York, 1968. [32] H.M. Srivastava, A unified presentation of certain classes of series of the Riemann Zeta function, Riv. Mat. Univ. Parma (4) 14 (1988) 1–23. [33] H.M. Srivastava, Some rapidly converging series for fð2n þ 1Þ, Proc. Amer. Math. Soc. 127 (1999) 385–396. [34] H.M. Srivastava, A note on the closed-form summation of some trigonometric series, Kobe J. Math. 16 (1999) 177–182. [35] H.M. Srivastava, Some simple algorithms for the evaluations and representations of the Riemann Zeta function at positive integer arguments, J. Math. Anal. Appl. 246 (2000) 331– 351. [36] H.M. Srivastava, J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, 2001. [37] H.M. Srivastava, M.L. Glasser, V.S. Adamchik, Some definite integrals associated with the Riemann Zeta function, Z. Anal. Anwendungen 19 (2000) 831–846. [38] E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, Second Ed., Clarendon Press, Oxford, 1951 (Revised by D.R. Heath-Brown, 1986). [39] I. Vardi, Determinants of Laplacians and multiple Gamma functions, SIAM J. Math. Anal. 19 (1988) 493–507. [40] M.-F. Vigneras, L’equation fonctionnelle de la fonction Z^eta de Selberg du groupe moudulaire PSLð2; ZÞ, in: Journees Arithmetiques de Luminy, Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978, Asterisque, 61, Soc. Math. France, Paris, 1979, pp. 235–249. [41] A. Voros, Special functions, spectral functions and the Selberg Zeta function, Comm. Math. Phys. 110 (1987) 439–465. [42] 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, 1963.
