J. CHOI, AND H. M. SRIVASTAVA, AND V. S. ADAMCHIK We begin by recalling the Barnes G-function (=G = being the so-called double Gamma function) which h

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1 MULTIPLE GAMMA AND RELATED FUNCTIONS Junesang Choi, H. M. Srivastava, and V. S. Adamchik Abstract. The authors give several new (and potentially useful) relationships between the multiple Gamma functions and other mathematical functions and constants. As by-products 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.. Introduction and Preliminaries The multiple Gamma functions were defined and studied by Barnes (cf. [7] and []) and others in about 9. 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 [, p. 6] and recorded also by Gradshteyn and Ryzhik [, p. 66, Entry 6.() p. 97, Entry.]. Recently, these functions were revived in the study of the determinants of the Laplacians on the n-dimensional unit sphere S n (see [], [7], [], [], [9], and []), and in evaluations of specific classes of definite integrals and infinite series involving, for example, the Riemann and Hurwitz Zeta functions (see [], [5], [6], [], and [9]). The subject of some of these developments can be traced back to an over two-century old theorem of Christian Goldbach (6976), as noted in the work of Srivastava [, p. ] who investigated this subject in a systematic and unified manner (see also [5], [6, Chapter ], and [7]). More recently, Adamchik ([] and []) 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 [], Choi and Seo [], and Vardi [9] 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, [6]). 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. Mathematics Subject Classification. Primary B99 Secondary M6, M99. Key words and phrases. Multiple Gamma functions, Riemann's -function, Hurwitz Zeta function, determinants of Laplacians, Clausen integrals, series involving the Zeta function. Typeset by AMS-TEX

2 J. CHOI, AND H. M. SRIVASTAVA, AND V. S. ADAMCHIK We begin by recalling the Barnes G-function (=G = being the so-called double Gamma function) which has several equivalent forms including (for example) the Weierstrass canonical product (.) f (z +)g = G(z +) =(ß) z e [(+fl)z +z] Y ρ + z k ff z+ z e k k where fl denotes the Euler-Mascheroni constant given by (.) fl = lim n! ψ! k log n ο= For sufficiently large real x and a C we have the Stirling formula for the G-function (.) log G(x + a +)= x + a x + log(ß) log A + x ax + a + ax log x + O(=x) (x!) where A is the Glaisher-Kinkelin constant given by (cf. [9] and []) (.) log A = () Based on the Binet integral representation, Adamchik [] derived the complete asymptotic expansion of the Barnes G-function. The G-function satisfies the following fundamental functional relationships (.5) G()= and G(z +)=(z) G(z) (z C ) where denotes the familiar Gamma function. Barnes [7] generalized the functional relationships in (.5) to the case of the multiple Gamma functions which are denoted usually by G n (z) or n (z). Throughout this paper, wechoose to follow the notations used (for example) by Vignéras [] (see also Srivastava and Choi [6]). Thus we have (.6) n (z) =fg n (z)g ()n (n N) so that (.7) G (z) = (z) =(z) G (z) = (z) = G(z) G (z) = (z) and so on. In terms of the multiple Gamma functions G n (z) of order n (n N), the aforementioned functional relationships of Barnes [7] can be rewritten as follows (cf. [, p. 9] see also [6, p., Theorem.])

3 MULTIPLE GAMMA AND RELATED FUNCTIONS (.) G n () = and G n+ (z +)=G n (z)g n+ (z) or, equivalently, (z C n N) (.9) n () = and n+ (z +)= n+(z) n (z) (z C n N) which may be used to define the multiple Gamma functions G n (z) and n (z) (n N). From (.) and (.) one can easily derive the Weierstrass canonical product form of the triple Gamma function (see []) (.) ( + z) =G ( + z) =» fl z + ß exp Y 6 ρ + z k fl + log(ß)+ z + k(k+) exp» (k +)z + k log(ß) log A + k z + 6k z ff Another form of the triple Gamma function appeared in the work of Choi [] who expressed, in terms of the double and triple Gamma functions, the analogous Weierstrass canonical product of the shifted sequence of the eigenvalues of the Laplacian on the unit sphere S with standard metric which was indispensable to evaluate the determinant of the Laplacian on S there. The Riemann Zeta function (s) is defined by (.) (s) = >< > k s = s s (k ) s (R(s) > ) () k+ k s (R(s) > s 6= ) which can, except for a simple pole only at s = with its residue, be continued analytically to the whole complex s-plane by means of the contour integral representation (cf. Whittaker and Watson [, p. 66]) or many other integral representations (cf. Erdélyi et al. [, p. ]). It satisfies the functional equation (see [, p. ]) (.) (s) = s ß s ( s)( s) sin ßs The generalized (or Hurwitz) Zeta function (s a) is defined by (.) (s a) = k= (k + a) s (R(s) > a C n Z Z = Z [fg) z

4 J. CHOI, AND H. M. SRIVASTAVA, AND V. S. ADAMCHIK which can, just as (s), be continued analytically to the whole complex s-plane except for a simple pole only at s =. Indeed, from the definitions (.) and (.), it is easily observed that (.) (s ) = (s) =( s ) (s ) and (s ) = (s) (.5) (s ma) = m X m s j= s a + j m (m N) and (.6) (s) = m s m X j= s j m (m N nfg) Many elementary identities such as those derivable especially from (.5) and (.6) will be required in our present investigation.. A Family of Generalized Glaisher-Kinkelin Constants Following the works of Glaisher [] and Alexeiewsky [5] on the asymptotic behavior of the product p p p n np when n!, Bendersky [9] considered the limit (.) log A k = lim m n!ψ k log m p(n k) m=! where (.) p(n k) = nk + k! nk+ log n + k + kx j= log n n kj B j+ (j + )!(k j)! ψ k + log n +( ffi jk ) kx l=! k l + and ffi jk is the Kronecker delta. For n =,(.) yields (.) log A = lim log m n!" m= + n log n + n This limit is fairly well-known. The finite sum on the right-hand side of (.) is equal to log n!, and thus, using the Stirling formula, we immediately obtain # (.) log A = log( ß)

5 MULTIPLE GAMMA AND RELATED FUNCTIONS 5 For n =, the limit (.) defines the Glaisher-Kinkelin constant A [cf. Eq. (.) above] n (.5) log A = log A = lim m log m n!" m= + n + # log n + n Using the functional relation (.5), we express the finite sum in (.5) in terms of the Barnes G-function (.6) m= m log m = log ψ ny m= m m! = n log n! log G(n +) By combining (.5), (.6), and the Stirling formula, we can establish the relationship (.7) log A = lim n!» n log G(n +)+ log n n + n log(ß)+ In a similar way, we can demonstrate that the higher-order Glaisher-Kinkelin constant A n is related to the multiple Barnes function. Upon setting n = in (.) and observing that (.) m= m log m = n log n! (n ) log G(n +) log G (n +) we find that» n log A = lim log G (n +) n! n + log n (.9) n + n n 6 + n n log(ß)+(n ) log A The generalized Glaisher-Kinkelin constants A n appear naturally in the asymptotic expansion of the multiple Barnes function. With a view to circumventing the problem of evaluation of the limits in (.7) and (.9), Adamchik [] found the following closed-form representation for the generalized Glaisher-Kinkelin constant A n (.) log A n = B n+ H n n + (n) (n N ) where B k and H k are the Bernoulli and harmonic numbers, respectively. The proof is based on the following analytical property of the Hurwitz Zeta function (s a) (.) m= m k log m = (k n +) (k) and its known asymptotic expansion at infinity. In fact, in light of the following consequence of the functional equation (.) [, p. 7, Eq. (.5)] (.) (n +)=() n (ß)n (n) (n N) (n)!

6 6 J. CHOI, AND H. M. SRIVASTAVA, AND V. S. ADAMCHIK it is easily seen from (.) that (n)! (.) log A n =() n+ (n +) (n N) (ß) n The constants A, A, and A (denoted commonly by A, B, and C, respectively) were also considered by Voros [], Vardi [9], Choi and Srivastava (cf. [6] and [] see also [6, Chapter ]).. Special cases of the multiple Barnes function The Barnes G-function, given by the functional equation in (.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 A n, as well as other known constants. Barnes [7, p. ] evaluated the following integral (see also [6, p. 7, Eq..()]) (.) Z z log (t + a)dt = z [log(ß)+ a] z +(z + a ) log (z + a) log G(z + a)+( a) log (a) + log G(a) which, in the special case when a =, reduces immediately to Alexeiewsky's theorem (cf. [5] and [6, p., Eq.. ()]) (.) Z z or, equivalently, log (t +)dt = [log(ß) ] z z + z log (z +) log G(z +) (.) since (.) Z z log (t) dt = z( z) Z z + z log(ß) ( z) log (z) log G(z) log tdt= z log z z The integral formula (.), evaluated recently by Gosper [] and Adamchik [], happens to be a source for many closed-form representations of the Barnes G- function. The simplest non-trivial case z = is due to Barnes [7, p., Section 7] (.5) G = ß e A By recalling the following special value of the Gamma function (see Spiegel [, p. ]) (.6) ο=

7 MULTIPLE GAMMA AND RELATED FUNCTIONS 7 and making use of a duplication formula for the G-function (cf. Barnes [7, p. 9] for the general case see also Vardi [9]), Choi and Srivastava [6] showed that (.7) G = e G ß A Φ 9 Ψ ο = 9756 or, equivalently, that = ß e + G ß A 9 Φ Ψ ο = 7 (.) G where G is the Catalan constant defined by (.9) G = Z K(k)dk = m= () m (m +) ο = and K is the complete elliptic integral of the first kind, given by (.) K(k) = = dt p k sin t The Catalan constant G becomes a special case of many other functions (one of which will be considered in the next section) and has, among other things, been used to evaluate integrals, such as (see [, p. 56, Entry.]) (.) = log(sin t) dt = ß log G and to give closed-form evlauations of a certain class of series involving the Zeta function (cf. Choi and Srivastava [6]). Other particular cases of the Barnes function G(z) when (.) z = 6 and 5 6 were given by Adamchik (see [] and []). The identity (.5) was generalized by Vardi [9] to the multiple Barnes function. The formula is much too complicated to be reproduced here. Here is one particular case when n =and z = (see also [, p. 95, Eq. (.)]) = ß 6 e A A 7 = (.) G where A k denotes the generalized Glaisher-Kinkelin constants given by (.) with, of course, A = A and A = B Other (known or new) closed-form representations for the triple Barnes function should emerge from the following integral (see [] and Eq. (.) below with a = ) (.) Z z log G(t) dt = log G (z +)+z log G(z)+ z(z ) log(ß) z log A z(z 6z +) The integral (.) has an interesting application to the constants A and A (or, alternatively, A and B) Z (.5) 7 log(a A )= 6 log( ß 9 ) log G(t) dt

8 J. CHOI, AND H. M. SRIVASTAVA, AND V. S. ADAMCHIK. Relationships between multiple Gamma and other functions The Clausen function (or the Clausen integral) Cl (t) (see Lewin [, p. ] see also Chen and Srivastava [, p. ]) defined by (.) Cl (t) = Z t log sin u Λ du = n= sin(nt) n stems actually from the imaginary part of the Dilogarithm function Li e it defined by (.) Li (z) = Z z log( u) du = u n= z n n (jzj») Now, by employing integration by parts in the following well-known integral formula (see Barnes [7, p. 79] see also Choi and Srivastava [6, p. 9, Eq. (.)]) due originally to Kinkelin (.) Z t ßucot(ßu) du = t log(ß) + log we have (cf. Choi and Srivastava [6, p. 95, Eq. (.)]) Z t sin (ßt) (.) log [sin (ßu)] du = t log + log ß G( t) G( + t) G( + t) G( t) If we combine (.) and (.), we obtain the following relationship between the Clausen function and the G-function sin(ßnt) sin (ßt) (.5) Cl (ßt) = = ßt log ß log n ß n= which, for t =, readily yields the Catalan constant (.6) Cl ß = G G( + t) G( t) by just observing, for the third member of (.5), the following identity derivable from Eqs. (.7), (.), and (.5) (.7) G 5 G = ß exp G ß Next we recall a relationship (cf. Cvijović and Klinowski [, p. ]) (.) S (ff) =Cl (ff) Cl (ff) where S ν (ff) denotes the classical trigonometric series defined by (cf. [] and []) (.9) S ν (ff) = k= sin(k +)ff (k +) ν

9 MULTIPLE GAMMA AND RELATED FUNCTIONS 9 A combination of (.5) and (.) gives the following relationship between S (ff) and the G-function (.) S (ßt) =ßt log[ cot(ßt)] + ß log G( + t) G( t) ß log G( + t) G( t) Integrating both sides of the second equality of(.5) from to t, we find an interesting relationship among a series involving a cosine function, a definite integral, and integrals of G-function (.) ß n= cos(nßt) = n + Z t Z t log G( + u) du + u log[sin(ßu)] du t log ß Z t log G( + u) du Now we recall an integral formula related to the multiple Gamma functions (see Choi and Srivastava [, p. 96, Eq. (5.7)]) in the form Z t» a log G(u + a)du = (a ) log(ß) log A + a t (.) + [log(ß)+ a] t 6 t +(t + a ) log G(t + a) log (t + a) +( a) log G(a) + log (a) which, for a =, yields [cf. Eq. (.) above] (.) Z t log G(u +)du = log A t + t log(ß) t 6 +(t ) log G(t +) log (t +) by virtue of Eqs. (.5) and (.) with n =. If we apply (.) in (.), we obtain an interesting identity involving a series and an integral through multiple Gamma functions (.) ß n= cos(nßt) n = ß t u log(sin u) du + t log +(t ) log G(t +) ( + t) log G( t) log [ ( + t) ( t)] the left-hand summation part of which can, in terms of the higher-order Clausen function Cl n (t) defined, for all n N nfg, by (cf. Lewin [, p. 9]) (.5) Cl n (t) = >< > sin(kt) k n cos(kt) k n if n is even if n is odd

10 J. CHOI, AND H. M. SRIVASTAVA, AND V. S. ADAMCHIK be expressed as follows (.6) n= cos(nßt) n =Cl (ßt) () It is known that (cf., e.g., Hansen [6, p. 56, Entry (5.5.)] see also Chen and Srivastava [, p., Eq. (.)]) (.7) ßt Cl (ßt) = log[ sin(ßt)] (k) k + tk A combination of (.5) and (.7) gives another proof of a special case of the following known identity for series involving the Riemann Zeta function (see Choi and Srivastava [6, p. 7, Eq. (.)]) (.) (k) k + tk+ = [ log(ß)] t + log G( + t) G( t) (jtj < ) The special case of (.) when t =, with the aid of (.5), (.6) with n =, (.), (.) with n =,(.), and (.), yields a known integral formula (.9) u log(sin u) du = 7 ß () log 6 which is due to Euler (77) who proved it through a striking and elaborate scheme as noted by Ayoub [6, p. ]. We now give here another short proof of (.9). Indeed, if we multiply the well-known formula (.) u k u cot u = (k) ß (juj <ß) k= by u and integrate the resulting equation with respect to u from to ß, we find that (.) u cot udu= u log(sin u) du = ß k= (k) (k +) k which, when combined with a known result (cf., e.g., [, p. 9, Eq. (.9)]), proves (.9). From (.7) and a result of Grosjean [5, p., Eq. ()] we readily obtain a closed-form evaluation of a class of series involving the Riemann Zeta function p pß (.) (k) k + q q X pqß k =» log» sin k q q k q (p q N» p» q (p q) =) pkß sin q

11 MULTIPLE GAMMA AND RELATED FUNCTIONS which, in the special cases when (respectively) p = q =, p = q =, and p = q =,yields (.) (k) (k +) k = log which is equivalent to a known result [, p. 9, Eq. (.)] (.) (k) (k +) k = log G ß which is precisely the same as the known result [6, p., Eq. (5.9)] (k) (.5) (k +) 6 k = p + ß p ß which corresponds to the special case of the known result (.7) above when t = 6 (see also [6, Problem (Chapter ) and Problem (Chapter )]). In fact, closed-form evaluations of series involving the Zeta function has an over two-century old history as commented in Section and has attracted many mathematicians ever since then (cf. Srivastava [] see also Srivastava and Choi [6, Chapter ]). Here we also give a general formula for this subject by recalling (see Choi and Nash [, p., Eq. (.)]) (.6) k= () k a k+fi (k ff) = k + fi Z a t fi [ψ(t + ff) ψ(ff)] dt (fi ) where the Digamma function (or the ψ-function) is the logarithmic derivative of the Gamma function, and a closed-form solution to the integral (see Adamchik [] and []) (.7) Z a t n ψ(t) dt =() n (n)+ ()n n + B n+ H n + k= () k n k a nk (k a) B k+(a) H k k + where B n and B n (a) are the Bernoulli numbers and polynomials (see [, pp. 5-6]), respectively, and H n are the harmonic numbers defined by [cf. Eq. (.) above] (.) H n = k (n N) Combining (.6) and (.), we obtain the desired result (.9) k= () k a k+n k + n (k) = an n + flan+ n + +()n (n)+ ()n n + B n+ H n + k= () k n k a nk (<(a) > jaj < n N) where fl is the Euler-Mascheroni constant given by (.). (k a) B k+(a) H k k +

12 J. CHOI, AND H. M. SRIVASTAVA, AND V. S. ADAMCHIK 5. Evaluations of some definite integrals We begin by combining the formulas (.) and (.6) in the form (5.) ß [Cl (ßt) ()] = ß I(t)+ t log +(t ) log G(t +) ( + t) log G( t) log A(t) where, for convenience, I(t) and A(t) are defined by (5.) I(t) = and t u log(sin u) du (5.) A(t) = ( + t) ( t) so that, obviously, Euler's integral is precisely I. Now we evaluate the integral I. First of all, it follows from (.7) that Λ fl + log(ß)+ t (5.) log A(t) = + ψ X (k) k + tk+ +! (k +) t k+ k + Furthermore, the special case of a known result [6, p. 7, Eq. (.)] when a =gives (5.5) (k +) t k+ = (fl +)t log [G( + t)g( t)] (jtj < ) k + which is a companion formula of (5.) for the G-function. We thus find from (.5), (.7), and (.) that (5.6) (k +) h (k +) k = fl + log ß A 6 Φ Ψ i The identity of Chen and Srivastava [, p. 9, Eq. (.)] is rewritten here in the following form (5.7) (k) (k +) k = log G ß + 5 ß () Next, the special case of another known result [6, p., Eq. (.)] when a =yields (5.)» (k) k + zk+ =[ log(ß)] z G( + z) + z log G( z) Z z log G(t +)dt Z z log G(t + )dt (jtj < )

13 MULTIPLE GAMMA AND RELATED FUNCTIONS which, upon setting z = and using (5.7), gives the following interesting identity Z Z (5.9) log G(t+) dt+ log G(t+) dt = log(ß)+ G () ß 5 ß In order to illustrate the usefulness of (5.9), we integrate the special case of [6, p. 7, Eq. (.)] when a =from to, and we readily find from (5.9) that (5.) () = ß 5 ψ + G ß! (k) (k + )(k +) k which is an interesting addition to the results recorded by Chen and Srivastava [] who treated the subject of series representations for () in a rather systematic manner and presented new results and some relevant connections with other functions. From the definition (.5) and (.), it is easy to see that (cf. Lewin [, p. ]) ß (5.) Cl n+ = n (n +) (n N) n+ Finally, upon setting t = in (5.), and making use of (5.), (5.6), (5.7), and (5.) (with n = ), we obtain the desired integral evaluation (5.) 5 u log(sin u) du = I = () ßG ß log which, upon employing integration by parts, yields (5.) u cot udu= 5 6 () + ßG + ß log We evaluate some more definite integrals and series involving the Zeta function. By applying the known result [, p., Eq. (c)] (5.) S ß = h p p+ ß 6 p G in (.) with t =, and using the relationship (.7), we obtain (5.5) log If we set t = (5.6) ψ G 9 G 7! = h 6 log + p i ß + p G ß p+ + 6ß h ß p i in (.), (.), and (.), and make use of (5.5), we find that u cot udu= ß h 6 log p i ß + p G p+ + 6 h p i ß i

14 J. CHOI, AND H. M. SRIVASTAVA, AND V. S. ADAMCHIK (5.7) log sin udu= ß 6 log(ß)+ h + 6 p+ p G ß p i and (5.) (k) (k +) k = h log p i ß + p G ß p+ + h 6ß ß p i By setting t = 6 (5.9) log in (.) and applying (.5), we obtain ψ! 7 G 6 G 5 6 = p ß + 6 log(ß) p ß In view of (5.9) and the relationship (cf. [, p., Eq. (e)]) (5.) S ß 6 (.) with t = yields (5.) log ψ If we set t = (5.) G G! = G = G ß + h log + p i p» ß ß + ß in (.), (.), and (.), and apply (5.), we see that ψ u cot udu= G ß log + p! p» + ß ß (5.) log sin udu= G ß log(ß)+ p» ß and (5.) ψ (k) (k +) k = G ß + log + p ß! + p» ß ß Just as in getting (.), (.), and (.5), the special cases of (.) when p = q = and p = q 5=

15 MULTIPLE GAMMA AND RELATED FUNCTIONS 5 can also be seen to yield (5.) and (5.), respectively. In view of the r^oles played by (5.5) and (5.) in closed-form evaluations of definite integrals, we choose to record here the following general identity (5.5) + log@ G p q G p q + q X q ß»» p A pß = q log ß sin q k q k q (p q N» p» q (p q) =) pkß sin q which follows immediately from (.) (with t = p q ) and (.). We conclude this paper by remarking that closed-form evaluation of I p q p» q p q N) can be done in terms of the Hurwitz function (5.6) Z pß q x log sin(x)dx= p ß q pß q () log + q () q X n= sin( npß ) ( n q q ) q q X n= cos( npß ) ( n q q ) (» ACKNOWLEDGEMENTS For the first-named author, this work was supported by grant No 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 OGP75. References. V. S. Adamchik, Polygamma functions of negative order, J. Comput. Appl. Math. (99), V. S. Adamchik, Integral representations for the Barnes function (Preprint ).. V. S. Adamchik, On the Barnes function, in Proceedings of the International Symposium on Symbolic and Algebraic Computation (London, Ontario July -5, ), ACM Press, New York,.. V. S. Adamchik and H.M. Srivastava, Some series of the Zeta and related functions, Analysis (99),. 5. W. P. Alexeiewsky, Über eine Classe von Funktionen, die der Gammafunktion analog sind, Leipzig Weidmannsche Buchhandlung 6 (9), R. Ayoub, Euler and the Zeta function, Amer. Math. Monthly (97), E. W. Barnes, The theory of the G-function, Quart. J. Math. (99), 6.. E. W. Barnes, On the theory of the multiple Gamma function, Trans. Cambridge Philos. Soc. 9 (9), L. Bendersky, Sur la fonction Gamma généralisée, Acta Math. 6 (9), 6.. M.-P. Chen and H. M. Srivastava, Some families of series representations for the Riemann (), Resultate Math. (99), J. Choi, Determinant of Laplacian on S, Math. Japon. (99), 5566.

16 6 J. CHOI, AND H. M. SRIVASTAVA, AND V. S. ADAMCHIK. J. Choi, Explicit formulas for Bernoulli polynomials of order n, Indian J. Pure Appl. Math. 7 (996), J. Choi and C. Nash, Integral representations of the Kikelin's constant A, Math. Japon. 5 (997),.. J. Choi and T. Y. Seo, The double Gamma function, East Asian Math. J. (997), J. Choi and H. M. Srivastava, Sums associated with the Zeta function, J. Math. Anal. Appl. 6 (997),. 6. J. Choi and H. M. Srivastava, Certain classes of series involving the Zeta function, J. Math. Anal. Appl. (999), J. Choi and H. M. Srivastava, An application of the theory of the double Gamma function, Kyushu J. Math. 5 (999), 9.. J. Choi and H. M. Srivastava, Certain classes of series associated the Zeta function and multiple Gamma functions, J. Comput. Appl. Math. (), J. Choi, H. M. Srivastava, and J. R. Quine, Some series involving the Zeta function, Bull. Austral. Math. Soc. 5 (995), 9.. D. Cvijović and J. Klinowski, Closed-form summation of some trigonometric series, Math. Comput. 6 (995), 5.. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, Toronto, and London, 95.. J. W. L. Glaisher, On the product n n, Messenger Math. 7 (77), 7.. R. W. Gosper, Jr., Z m=6 n= ln (z) dz, Fields Inst. Comm. (997), I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Corrected and Enlarged Ed. prepared by A. Jeffrey), Academic Press, New York, London, and Toronto, C. C. Grosjean, Formulae concerning the computation of the Clausen integral Cl (), J. Comput. Appl. Math. (9),. 6. E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, New Jersey, G. H. Hardy, Divergent Series, Clarendon (Oxford University) Press, Oxford, London, and New York, 99.. L. Lewin, Polylogarithms and Associated Functions, Elsevier (North-Holland), New York, London, and Amsterdam, J. Miller and V. S. Adamchik, Derivatives of the Hurwitz Zeta function for rational arguments, J. Comput. Appl. Math. (99), 6.. J. R. Quine and J. Choi, Zeta regularized products and functional determinants on spheres, Rocky Mountain J. Math. 6 (996), M. R. Spiegel, Mathematical Handbook, McGraw-Hill Book Company, New York, 96.. H. M. Srivastava, A unified presentation of certain classes of series of the Riemann Zeta function, Riv. Mat. Univ. Parma () (9),.. H. M. Srivastava, Some rapidly converging series for (n + ), Proc. Amer. Math. Soc. 7 (999), H. M. Srivastava, A note on the closed-form summation of some trigonometric series, Kobe J. Math. 6 (999), H. M. Srivastava, Some simple algorithms for the evaluations and representations of the Riemann Zeta function at positive integer arguments, J. Math. Anal. Appl. 6 (), H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Series on Mathematics and Its Applications, Vol. 5, Kluwer Academic Publishers, Dordrecht, Boston, and London,. 7. H. M. Srivastava, M. L. Glasser, and V. S. Adamchik, Some definite integrals associated with the Riemann Zeta function, Z. Anal. Anwendungen 9 (), 6.. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Clarendon (Oxford University) Press, Oxford, London, and New York, 95 Second Ed. (Revised by D. R. Heath-Brown), I. Vardi, Determinants of Laplacians and multiple Gamma functions, SIAM J. Math. Anal. 9 (9), 957.

17 MULTIPLE GAMMA AND RELATED FUNCTIONS 7. M.-F. Vignéras, L'équation fonctionnelle de la fonction Z^eta de Selberg du groupe moudulaire PSL(Z), in Journées Arithmétiques de Luminy" (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 97), pp. 59, Astérisque, 6, Soc. Math. France, Paris, A. Voros, Special functions, spectral functions and the Selberg Zeta function, Comm. Math. Phys. (97), E. T. Whittaker and 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, 96. Junesang Choi Department of Mathematics College of Natural Sciences Dongguk University Kyongju 7-7 Korea junesang@mail.dongguk.ac.kr H. M. Srivastava Department of Mathematics and Statistics University of Victoria Victoria, British Columbia VW P Canada hmsri@uvvm.uvic.ca V. S. Adamchik Department of Computer Science Carnegie Mellon University Pittsburgh, Pennsylvania 5-9 U.S.A. adamchik@cs.cmu.edu