International Journal of Mathematics and M Computer Science, 9(2014), no. 1, 11–30 CS Contributions to the Theory of the Barnes Function Victor S. Adamchik Computer Science Department Carnegie Mellon University 5000 Forbes Ave Pittsburgh, PA 15213, USA email: [email protected] (Received November 13, 2013, Accepted May 10, 2014) Abstract This paper presents a family of new integral representations and asymptotic series of the multiple gamma function. The numerical schemes for high-precision computation of the Barnes gamma function and Glaisher’s constant are also discussed. 1 Introduction In a sequence of papers published between 1899-1904, Barnes introduced and studied (see [9, 10, 11, 12]) a generalization of the classical Euler gamma function, called the multiple gamma function Γn(z). The function Γn(z) satisfies the following recurrence-functional equation [31, 32]: Γn+1(z) Γn+1(z +1)= ,z∈ C,n∈ N, Γn(z) (1.1) Γ1(z)=Γ(z), Γn(1) = 1, Key words and phrases: Barnes function, gamma function, Riemann zeta function, Hurwitz zeta function, Stirling numbers, harmonic numbers, Glaisher’s constant. AMS (MOS) Subject Classifications: Primary 33E20, 11M35 Secondary 11Y. This work was supported by grant CCR-0204003 from the National Science Foundation. ISSN 1814-0432, 2014, http://ijmcs.future-in-tech.net 12 Victor S. Adamchik where Γ(z) is the Euler gamma function. The system (1.1) has a unique solution if we add the condition of convexity [32] dn+1 − n+1 z ≥ ,z>. ( 1) dzn+1 log Γn( ) 0 0 In this paper we aim at the most interesting case G(z)=1/Γ2(z), being the so-called double gamma function or the Barnes G-function: G(z +1)=Γ(z) G(z),z∈ C, G(1) = G(2) = G(3) = 1, d3 log G(z) ≥ 0,z>0. dz3 The G-function has several equivalent forms including the Weierstrass canonical product 2 ∞ 2 z z + z (1 + γ) z k z G z π 2 − − z ( +1)=(2 ) exp 1+k exp k 2 k=1 2 and the functional relationship with the Hurwitz zeta function ζ(t, z), due to Gosper [20] and Vardi [31] log G(z +1)− z log Γ(z)=ζ(−1) − ζ(−1,z), (z) > 0, (1.3) where γ denotes the Euler-Mascheroni constant and the derivative ζ(t, z)= d ζ t, z dt ( ) is defined as an analytic continuation provided by the Hermite inte- gral (3.19): z2 z2 z ζ(−1,z)= log z − − log z + 2 4 2 (1.4) ∞ arctan( x ) ∞ x x2 z2 z z dx log( + ) dx. z > , 2 2πx + 2πx ( ) 0 0 e − 1 0 e − 1 In particular, ∞ x x ζ − ζ − , log dx. ( 1) = ( 1 0) = 2 2πx (1.5) 0 e − 1 Contributions to the Theory of the Barnes Function 13 For z →∞,theG-function has the Stirling asymptotic expansion z 3z2 z2 1 1 log G(z +1)= log(2π) − log A − + − log z + O , 2 4 2 12 z where A is the Glaisher-Kinkelin constant given by 1 log A = − ζ(−1). (1.6) 12 Combining this definition with the representation (1.5), we obtain a new integral for log A: π ∞ x x A 1+log(2 ) − 1 log dx. log = 2 x (1.7) 12 2π 0 e − 1 The constant A originally appeared in papers by Kinkelin [23] and Glaisher [17, 18, 19] on the asymptotic expansion (when n →∞) of the following product p p p 11 22 ...nn ,p∈ N. Similar to the Euler gamma function, the G-function satisfies the multipli- cation formula [31]: 2 2 2 G(nz)=eζ (−1)(1−n )nn z /2−nz+5/12 n−1 n−1 n−1 (1−nz) i + j (1.8) π 2 G z ,n∈ N (2 ) + n i=0 j=0 The Barnes function is not listed in the tables of the most well-known spe- cial functions. However, it is cited in the exercises by Whittaker and Watson [33], and in entries 6.441, 8.333 by Gradshteyn and Ryzhik [21]. Recently, Vardi [31], Quine and Choi [27], and Kumagai [24] computed the functional determinants of Laplacians of the n-sphere in terms of the Γn-function. Choi, Srivastava [13, 14] and Adamchik [3, 16] presented relationships (including integrals and series) between other mathematical functions and the multiple gamma function. The Γn function has several interesting applications in pure and applied mathematics and theoretical physics. The most intriguing one is the applica- tion of G-function to the Riemann Hypothesis. Montgomery [26] and Sarnak [29] (see [22] for additional references) have conjectured that the non-trivial 14 Victor S. Adamchik zeros of the Riemann zeta function are pairwise distributed like eigenval- ues of matrices in the Gaussian Unitary Ensemble (GUE) of random matrix theory. It has been shown in works by Mehta, Sarnak, Conrey, Keating, and Snaith that a closed representation for statistical averages over GUE of N × N unitary matrices, when N →∞can be expressed in terms of the Barnes functions. Another interesting appearance of the G-function is in its relation to the determinants of the Hankel matrices. Consider a matrix M of the Bell numbers: ⎛ ⎞ B1 B2 ... Bn ⎜ B B ... B ⎟ M ⎜ 2 3 n+1 ⎟ n = ⎝ ... ⎠ Bn Bn+1 ... B2n−1 where Bn are the Bell numbers, defined by n n B n = k k=0 n and k are the Stirling subset numbers [28] n n − 1 n − 1 n 1,n=0, = k + , = (1.9) k k k − 1 0 0,n=0 It was proved in [5], that det Mn = G(n +1),n∈ N. 2 Special cases of the Barnes Function For some particular arguments, the Barnes function can be expressed in terms of the known special functions and constants. The simplest (but not-trivial) special case is due to Barnes [9]: 1 1 log 2 log π 3 log G = + − − log A, (2.10) 2 8 24 4 2 where A is the Glaisher-Kinkelin constant (1.6). The next two special cases are due to Srivastava and Choi [14]: Contributions to the Theory of the Barnes Function 15 1 3 G 3 1 9 log G = − − log Γ − log A, (2.11) 4 32 4π 4 4 8 3 1 G log 2 log π 1 log G =logG + − − +logΓ , (2.12) 4 4 2π 8 4 4 where G denotes Catalan’s constant: ∞ (−1)k G . = k 2 k=0 (2 +1) Identities (2.11) and (2.12) are proven by using the duplication formula (1.8) n z 1 , 1 , 2 5 (setting = 2) together with (2.10). More special cases for = 6 3 3 and 6 can be derived as well. Here is one of such formulas (see [3] for the others): Proposition 2.1. 1 1 log 3 π log G = + + √ − 3 9 72 18 3 (2.13) 2 1 4 ψ(1) 1 log Γ − log A − √ 3 , 3 3 3 12 3π ψ(1) z ∂2 z where ( )= ∂z2 log Γ( ) is the polygamma function. Proof. We recall the Lerch functional equation for the Hurwitz zeta function (see [2, 8]) n πi ζ (−n, z)+(−1) ζ (−n, 1 − z)= Bn+1(z)+ n +1 −πin/2 n! 2πiz e Lin+1 e , 0 <z<1,n∈ N, (2π)n ζ t, z d ζ t, z B z z where ( )= dt ( ), n( ) are the Bernoulli polynomials and Lin( )is the polylogarithm. Setting n = 1 and using (1.3) yields the reflexion formula for the G-function (see also [14, 16]): G(1 + z) sin(πz) 1 log = −z log − Cl2(2 πz), 0 <z<1, (2.14) G(1 − z) π 2π 16 Victor S. Adamchik where Cl2(z) is the Clausen function defined by −iz Cl2(z)=− (Li2(e )). (2.15) z 1 Upon setting = 3 to (2.14), and making use of ψ(1) 1 2 2π 3 2π Cl2 = √ − √ , 3 3 3 9 3 we obtain √ G 1 3 2π 2π2 − 3ψ(1) 1 3 = √ exp √ 3 . (2.16) G 2 6 1 π 3 3Γ 3 18 3 z 1 On the other hand, using the multiplication formula (1.8) with = 3 and n =3,wefind 1 2 37/12e2/3 G G = 3 . (2.17) πA8 1 3 3 2 Γ 3 Now, combining (2.16) and (2.17), we conclude the proof. It remains to be seen if the Barnes function (or the multiple gamma function) can be expressed as a finite combination of elementary functions and constants for other rational arguments. The most general result available z z 1 for the multiple gamma function Γn( ) is a closed form at = 2 : Proposition 2.2. For n ∈ N n 1 (2n − 3)!! log π (−1) (n − 1)! log Γn( )=− + 2 2n n 1 n k (2.18) Pk,n( ) Bk+1 − 2 2 1P 1 ζ −k , log 2 k k + k k,n( ) ( ) k=1 ( +1)2 k=1 2 2 P 1 xk k ≤ n where k,n( 2 ) are coefficients by , in expansion of n−1 1 (x + j − ) j=1 2 z 1 Proof. The proof follows directly from [4], formula (17), by setting = 2 . Contributions to the Theory of the Barnes Function 17 Here are a few particular cases: the gamma function (n =1): 1 √ Γ( )= π 2 the Barnes function (n =2): 1 21/24 e1/8 G( )= 2 A3/2 π1/4 the triple gamma function (n =3): 1 A3/2 π3/16 7 ζ(3) 1 Γ3( )= exp − 2 21/24 32 π2 8 3 Hermite integrals In this section we derive a few integral representations for the Barnes func- tion.