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 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

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. We start with recalling the Hermite integral (see [7, 25]) for the Hurwitz zeta function: z−s z1−s ∞ s x/z ζ s, z sin( arctan( )) dx, ( )= + +2 s/2 (3.19) 2 s − 1 0 (x2 + z2) (e2πx − 1) which provides analytic continuation of ζ(s, z) to the domain s ∈ C −{1}. Differentiating both sides of (3.19) with respect to s, letting s = −1(and s = −2), and using the second Binet formula for log Γ(z) (see [7]) π ∞ x/z z z − 1 z − z log(2 ) arctan( ) dx log Γ( )=( )log + +2 2πx (3.20) 2 2 0 e − 1 together with (1.3), we readily obtain:

2 z   log G(z +1)= log z − H2 − zζ(0) − ζ (−1) − 2 (3.21) ∞ x x2 z2 log( + ) dx,  z > , 2πx ( ) 0 0 e − 1 18 Victor S. Adamchik

p  where Hp = k=1 1/k are the harmonic numbers and ζ (z) is a derivative of the Riemann zeta function. In the same manner, repeating the above steps n times, we derive the general integral representations.

Proposition 3.1. For (z) > 0

(2n)! log Γ2n+1(z +1)= z2n+1 z − H 2n n (log 2n+1) − − k 2 ζ −k z2n−k − n ( 1) k ( ) 2 +1 k=0

2n−1 (3.22) n − kk 2 z ( 1) ! k log Γk+1( +1)+ k=1 ∞ x2n arctan x/z − n dx 2( 1) 2πx 0 e − 1 and

2n z (log z − H2n) (2n − 1)! log Γ2n(z +1)=− + 2n 2n−1 n − − k 2 1 ζ −k z2n−k−1 ( 1) k ( ) + k=0 (3.23) 2n−2 n − − kk 2 1 z − ( 1) ! k log Γk+1( +1) k=1 ∞ x2n−1 x2 z2 − n log( + ) dx ( 1) 2πx 0 e − 1 n where k are the Stirling subset numbers, defined in (1.9).

Upon substituting z = 1 into formulas (3.22) and (3.23), we obtain new integrals Contributions to the Theory of the Barnes Function 19

∞ 2n 2n x arctan(x) H2n+1 k 2n 2(−1)n dx = + (−1) ζ(−k) (3.24) e2πx − n k 0 1 2 +1 k=0

∞ x2n−1 x2 H 2n−1 n − − n log(1 + ) dx 2n − k 2 1 ζ −k ( 1) e2πx − = n + ( 1) k ( ) (3.25) 0 1 2 k=0

Performing simple transformations of the integrands in (3.24) and (3.25) will apparently lead to a large number of special integrals: ∞ x x log dx ζ − 2 2πx = ( 1) 0 e − 1 ∞ x x log dx ζ − 2πx =12 ( 1) + log 2 0 e +1 ∞ x x2 log(1 + ) dx 3 − 23 1ζ − 2πx = log 2 + ( 1) 0 e +1 4 24 2

∞ arctan x π z dx z z − z log(2 ) − z 1 ,  z > 2 2πx = log + log Γ + ( ) 0 0 e +1 2 2 ∞ x arctan( ) dx − 4 2πx = 2+3log2 0 e +1 ∞ xdx π ψ z 1 − z, z  2 2πx 2 2 = + log arg( ) = 0 (e +1)(x + z ) 2 2

∞ xdx π z − ψ z − 1 , z  2 2πx 2 2 =log ( ) arg( ) = 0 (e − 1)(x + z ) 2z 2 20 Victor S. Adamchik

4 Binet-like representation

In this section we derive the Binet integral representation for the Barnes function.

Proposition 4.1. The Barnes G-function admits the Binet integral repre- sentation:

z2 log z log G(z +1)=z log Γ(z)+ − B2(z) − log A − 4 2 ∞ e−zx x (4.26) 1 − 1 − 1 − dx,  z > . 2 −x ( ) 0 0 x 1 − e x 2 12

Proof. Recall the well-known integral for the Hurwitz function [8]

∞ xs−1e−zx ζ s, z 1 dx,  s > ,  z > ( )= −x ( ) 1 ( ) 0 (4.27) Γ(s) 0 1 − e

The integral in the right hand-side of (4.27) can be analytically continued to the larger domain (s) > −2 by subtracting the truncated Taylor series of 1 x 1−e−x at =0.Since 1 1 1 x = + + + O(x3) (4.28) 1 − e−x x 2 12 and

1 ∞ 1 1 x s z−s z1−s xs−1e−zx + + dx = z−1−s + − , Γ(s) 0 x 2 12 12 2 1 − s (s) > 1, (z) > 0 we therefore can rewrite (4.27) as

z1−s z−s s ζ(s, z)=− + + z−s−1 + 1 − s 2 12 ∞ x 1 xs−1e−zx 1 − 1 − 1 − dx,  s > − . −x ( ) 2 Γ(s) 0 1 − e x 2 12 Contributions to the Theory of the Barnes Function 21

Differentiating the above formula with respect to s and computing the limit at s = −1, we obtain

2  1 z log z ζ (−1,z)= − + B2(z)+ 12 4 2 ∞ e−zx x 1 − 1 − 1 − dx,  z > , 2 −x ( ) 0 0 x 1 − e x 2 12 where B2(z) is the second Bernoulli polynomial. The convergence of the above integral is ensured by (4.28). Finally, using (1.3), we conclude the proof.

5 Asymptotic expansion of Γn(z)

The asymptotic expansion for Γn(z)whenz →∞follows straightforwardly from formulas (3.22) and (3.23) by expanding log(1 + x2) and arctan(x)into the Taylor series, and then performing formal term-by-term integration. Lemma 5.1. For (z) > 0 z2 3 log z log G(z +1)= log z − − − zζ(0) + ζ(−1) − 2 2 12 n (5.29) B2k+2 1 + O( ),z→∞. k k z2k z2n+2 k=1 4 ( +1) Proof. We start with the integral representation (3.21). By recalling the Taylor series for log(1 + x)

n x − k x k+1 − n x yn+1 ( 1) − ( 1) dy, log(1 + z )= k (z ) zn+1 y z k=0 +1 0 + we find that ∞ x x2/z2 log(1 + ) dx 2 πx = 0 e − 1 n − k ∞ x2k+3 − n ∞ tdt t2 yn+1 ( 1) 1 dx − ( 1) dy k z2k+2 e2 πx − z2n+2 e2 πt − y z2 k=0 +1 0 1 0 1 0 + 22 Victor S. Adamchik which, by appealing to (formula 1.6.4 in [7]) ∞ t2k−1 B dt − k+1 2k ,k∈ N, 2 πt =( 1) (5.30) 0 e − 1 4 k reduces to

∞ x x2/z2 log(1 + ) dx 2 πx = 0 e − 1 n+1 ∞ t/z 2n+3 B2k+2 tdt y − − 2(−1)n dy k k z2k e2 πt − y2 k=1 4 ( +1) 0 1 0 1+

Substituting this into (3.21), we complete the proof.

Lemma 5.2. For (z) > 0

3 2 z 11 z 3  2logΓ3(z +1)=− log z − + log z − + ζ (0) − 3 6 2 2 log z − z ζ(0) + 2ζ(−1) + ζ(−1) + ζ(−2) − 12 (5.31) n B n B 2k+2 − 2k+2 k k z2k k k − z2k−1 + k=1 4 ( +1) k=1 2( +1)(2 1) 1 O( ),z→∞. z2n+1

Proof. Employing the same technique as in getting (5.29), we begin with the integral representation of the triple gamma function:

z3 2logΓ3(z +1)=− (log z − H3)+logG(z +1)+ 2 (5.32) ∞ x2 x/z z2ζ − zζ − ζ − arctan( ) dx (0) 2 ( 1) + ( 2) +2 2 πx 0 e − 1 Contributions to the Theory of the Barnes Function 23 that follows directly from Proposition 3.1 with n = 1. Expanding arctan into the Taylor series

n x (−1)k−1 x 2k−1 (−1)n x y2n arctan( )= ( ) + dy z k − z z2n−1 y2 z2 k=1 2 1 0 + and making use of (5.30), we readily obtain ∞ x2 x/z n B arctan( ) dx − 2k+2 e2 πx − = k k − z2k−1 + 0 1 k=1 4( + 1)(2 1) − n ∞ t2 dt t y2n ( 1) dy. 2n−1 2 πt 2 2 z 0 e − 1 0 y + z The conclusion of the lemma follows immediately by substituting this into formula (5.32) and using Lemma 5.1.

6 Implementation

In this section we discuss a numeric computational scheme for the Barnes function. Let us consider an integral representation dated back to Alexejew- sky [6] and Barnes [9]:

z z(1 − z) z log Γ(x)dx = + log(2π)+z log Γ(z) − log G(z +1), 0 2 2 where (z) > −1. Employing integration by parts, we derive

z(1 − z) z z log G(z +1)= + log(2π)+ xψ(x) dx, (z) > −1 (6.33) 2 2 0

The integral representation (6.33) provides an efficient numeric procedure for arbitrary precision evaluation of the G-function. This representation demonstrates that the complexity of computing G(z) depends at most on the computational complexity of the polygamma function. The later can be numerically computed by the Spouge approximation [30] (a modification of 24 Victor S. Adamchik the Lanczos approximation). For the numerical integration in (6.33) we shall use the Gaussian quadrature scheme. The restriction (z) > −1 in (6.33) can be easily removed by analyticity of the polygamma. For example, by resolving the singularity of the integrand at the pole x = −1, we continue log G(z + 1) to the wider area (z) > −2: z(1 − z) z z 1 log G(z +1)= + log(2π)+log(z +1)+ xψ(x) − dx 2 2 0 x +1 Another method of the analytic continuation of the G-function is to use the following well-known identity

ψ x ψ −x − π πx − 1 ( )= ( ) cot( ) x which, upon substituting it into (6.33), yields

z log G(1 − z)=logG(1 + z) − z log(2π)+ πx cot(πx) dx. (6.34) 0 The identity (6.34) holds everywhere in a complex plane of z, except the real axes, where the integrand has simple poles. Therefore, combining (6.33) with (6.34) yields the following definition of the Barnes function

G(−n)=0,n∈ N, (z − 1)(z − 2) G(z)=(2π)(z−1)/2 exp − + xψ(x) dx , (6.35) 2 γ

arg(z) = π, where the contour of integration γ is a line between 0 and z − 1thatdoes not cross the negative real axis; for example, γ could be the following path {0,i,i+ z − 1,z− 1}. It remains to define G(z) for z ∈ R−. Wedothisbyusingthereflexion formula (2.14), which upon periodicity of the Clausen function (2.15) can be rewritten as follows:

z+1 sin(πz) G −z − z/2−1G z ∗ ( )=( 1) ( +2) π (6.36) 1 − exp Cl2 2π(z − z ) ,z∈ R . 2π Contributions to the Theory of the Barnes Function 25

7 Glaisher’s constant

In this section we discuss numeric computational schemes for Glaisher’s con- stant: 1 A =exp − ζ(−1) (7.37) 12 In light of the functional equation for the zeta function πs ζ(s)=2sπs−1Γ(1 − s)ζ(1 − s)sin 2 we can rewrite (7.37) as

 γ ζ (2) 1/12 A =exp − (2π) . (7.38) 12 2π2 Exactly this formula is used in Mathematica V4.2 for computing Glaisher’s constant. Another computational scheme for Glaisher’s constant is based on the Barnes G-function identity: 1 log 2 log π 2 1 log A = + − − log G . (7.39) 12 36 6 3 2 G 1 Here ( 2 ) can be computed via integral (6.33). A more efficient algorithm is provided by the following identity

∞ A log 2 1 ζ k − 3 − 6 , log = + (2 +1) 1 28 + k k (7.40) 12 36 k=1 1+ 2+ which allows to approximate Glaisher’s constant as

N log 2 1 3 6 1 log A = + ζ(2k +1)− 1 28 + − + O k k N 12 36 k=1 1+ 2+ 4

This approximation is suitable for arbitrary precision computation - it re-  p  p quires 2 log210 terms to achieve decimal-digit accuracy. We close this section with a proof of formula (7.40). Recall that [8] 26 Victor S. Adamchik

∞ xs−1 ζ s − 1 dx,  s > . ( ) 1= x x ( ) 1 (7.41) Γ(s) 0 e (e − 1)

Upon replacing ζ(2k +1)− 1 in (7.40) by its integral representation (7.41) and inverting the order of summation and integration, we obtain

∞ log 2 1 ∞ dx t2k 3 6 log A = + 28 + − . (7.42) ex ex − k k k 12 36 0 ( 1) k=1 (2 )! 1+ 2+

Next,notingthat ∞ t2k t − k =cosh() 1 k=1 (2 )! and

∞ t2k t 2 x2s−1 x − dx, k k s = t2s (cosh( ) 1) k=1 (2 )! ( + ) 0 we compute the inner sum

t4 ∞ t2k 3 − 6 − t2 − t4 k 28 + k k = 36 + 3 14 + 2 k=1 (2 )! 1+ 2+ 36 + 15 t2 +14t4 cosh(t) − 36 t +3t3 sinh(t)

In the next step, we substitute this back to (7.42) and, employed by (4.27), evaluate the integral. Unfortunately, integration cannot be done term by term. The way around is to multiply the integrand by tλ,λ>0 and consider the limit as λ → 0. This yields Contributions to the Theory of the Barnes Function 27

∞ tλ−4 tλ−4 tλ−4 tλ−3 36 36 − 72 − 36 t + 2 t t t t t + 0 e − 1 e (e − 1) e (e − 1) e − 1

36 tλ−3 15 tλ−2 15 tλ−2 6 tλ−2 3 tλ−1 + + + − + e2 t (et − 1) et − 1 e2 t (et − 1) et (et − 1) et − 1 3 tλ−1 14 tλ 14 tλ 28 tλ + + − dt = e2 t (et − 1) et − 1 e2 t (et − 1) et (et − 1)

36 Γ(λ − 3) − 925−λ Γ(λ − 3) − 36 Γ(λ − 2) − 924−λ Γ(λ − 2) −

3Γ(λ) 21 Γ(λ − 1) − 15 21−λ Γ(λ − 1) − 3Γ(λ) − +14Γ(λ +1)− 2λ 7Γ(λ +1) +36Γ(λ − 1) ζ(λ − 1). 2λ

Computing the limit as λ → 0, we find that the right hand-side of the above expression evaluates to

3 − 3log2− 36 ζ(−1).

This completes the proof of (7.40). 28 Victor S. Adamchik

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