On the Barnes function

Victor AdamchikCarnegie Mellon University

adamchik@cs.cmu.edu

Abstract. The multiple Barnes function, defined as a generalization of the Euler gamma

function, is used in many applications of pure and applied mathematics and theoreticalphysics. This paper presents new integral representations as well as special values of theBarnes function. Moreover, the Barnes function is expressed in a closed form by meansof the Hurwitz zeta function. These results can be used for numeric and symbolic compu-tations of the Barnes function.

Preamble

In 1899, Barnes [1,2,3] introduced and studied the generalization of the Euler gamma functiondefined by the following functional equation:

G+z� 1/ *+z/G+z/, z± C

G +1/ 1

where * is the gamma function. For the integer positive values of z, the Barnes G function issimply a product of factorials:

G+n� 1/ Çk 1

n�1

k� Çk 1

n

*+k/

From the above functional equation and the Weierstrass canonical product for the gammafunction,

*+z/ 1cccccz

exp+�zJ/Èm 1

�

exp+ zcccccm /

cccccccccccccccccccccccc

1� zcccccm

Barnes derived

G+z� 1/ +2S/zs2 expL

NMMM�

z� z2 +1� J/cccccccccccccccccccccccccccccccccccccc

2\

^]]]È

k 1

�

-1�

zccccc

k1k exp

LNMMM

z2

cccccccccc2k

� z\

^]]]

where J is the Euler-Mascheroni constant. The above product is an entire function in a wholecomplex plane and can serve as an explicit definition of the Barnes function. Originally, the Gfunction was introduced (in a different form) by Kinkelin [5] (see also Glaisher [6]) in hisresearch on the asymptotics behavior of this product at n��:

(1)11 22 ...nn

n�nccccccccccccccccccccccccccG+n� 1/

Kinkelin also applied the theory of the G function to the class of trigonometric integrals

Ã0

z

log trig+x/ dx

where trig(x) is any trigonometric or hyperbolic function. All such integrals can be expressed infinite terms of the G function.

2

Alexeiewsky [7] generalized the Kinkelin product (1) to

(2)11p

22p

... nnp

exp+ ]�+� p, n� 1/ � ]�+� p/ /

which is related to the multiple Barnes function.

These days the G function remains the active topic of research (see [8--15].) The theory of the

Barnes function has been related to certain spectral functions in mathematical physics, to thestudy of determinants of Laplacians, and to the Hecke L-functions. In [10] and [15] the Gfunction is expressed in terms of the Hurwitz zeta function by

(3)logG+z� 1/� zlog*+z/ ]�+�1/ � ]�+�1, z/, R+z/ ! 0

where ]�+t, z/ = dcccccccd t ]+t, z/ and ]�+�1/ is related to Glaisher's constant A:

log A 1ccccccccc12

� ]�+�1/ J � log+2S/cccccccccccccccccccccccccccccccccccccc

12�]�+2/ccccccccccccccc2S2

In [15] Adamchik obtained the closed form representation for the class of integrals involvingcyclotomic polynomials and nested logarithms in terms of the Hurwitz zeta function. In theview of (3) the results can be translated into the G function notation, for example

Ã0

1 +x4 � 6x2 � 1/ccccccccccccccccccccccccccccccccccccccccccc+x2� 1/3 log log- 1

ccccc

x1 dx 4 log

LNMMMM

G+ 3cccc4 /

cccccccccccccccccc

G+ 1cccc4 /

\^]]]]� 3 log*- 1

ccccc4

1� log*- 3ccccc4

1

Choi et al. [13, 14] and Adamchik [15-17] considered a class of sums involving the Riemannzeta function which can be evaluated by means of the G function. Here is a series representa-tion for the G function for �z� � 1,

2 logG+z� 1/ zlog+2S/ � J z2� z+z� 1/� 2Æ

k 2

�

+�1/k ]+k/zk�1

cccccccccccccccck� 1

and for Glaisher's constant

log A log 2cccccccccccccccc

12�Å

k 1

�

LNMM]+2k� 1/ � 1cccccccccccccccccccccccccccccccccccccccc

36\^]]

+4k� 7/ +7k� 8/ccccccccccccccccccccccccccccccccccccccccccccccccccc

+k� 1/ +k � 2/

3

Barnes [4] (followed by Vigneras [8] and Vardi [10]) generalized the G function to the multi-ple gamma function Gn by the recurrence formula

(4)

Gn�1+z� 1/ Gn�1+z/ccccccccccccccccccccccc

Gn+z/, z± C , n± ´

G2+z/ G+z/

G1+z/ 1

cccccccccccccccc*+z/

Here G+z/ is the Barnes function and *+z/ is the Euler gamma function. Vigneras and Vardiconsidered a slightly different form of the multiple gamma function defined as a reciprocal to

the Barnes function *n+z/ 1

cccccccccccccGn+z/. Vardi derived an implicit representation for *n+z/ in terms

of the multiple zeta function. In this paper we derive a closed form of Gn in finite terms of theHurwitz function and special polynomials. Here are two particular cases rewritten in terms ofthe derivatives of the Hurwitz function when n 3

(5)2 logG3+z� 1/ � z2 log*+z/ +1� 2z/ logG2+z� 1/ � ]�+�2/ � ]�+�2, z/and n 4

(6)

6 logG4+z� 1/ � z3 log*+z/ 6 +1� z/ logG3+z� 1/ �

+3z2� 3z� 1/ logG2+z� 1/ � ]�+�3/ � ]�+�3, z/

We also present various integral and series representations as well as some special values ofthe Barnes function. These results can be used in numeric and symbolic computations of the Gfunction. I believe the multiple gamma function is of direct interest for computer algebraresearchers and users, because of its significant applications in physics, number theory, combina-torics and applied mathematics. The Barnes function deserves to be implemented in computeralgebra systems.

1. G function of the rational argument

There are a few known special cases when the G function is expressible in a closed form. The

first one is due to Barnes [1]:

logG- 1ccccc21 log 2

cccccccccccccccc24

�logScccccccccccccccc

4�

3 logAcccccccccccccccccccccc

2�

1ccccc8

the other two are due to Choi and Srivastava [14]:

4

logG - 1ccccc41 � G

cccccccccc4S

�3ccccc4

log*- 1ccccc41� 9

ccccc8

log A�3ccccccccc32

logG - 3ccccc41 logG- 5

ccccc41 � G

cccccccccc2S

�log +2S2/ccccccccccccccccccccccccccccc

8

where G is Catalan's constant. In a view of closed form representation for the derivatives of theHurwitz functions, obtained in [18], and the formula (3), and it is easy to derive the additionalspecial cases of logG+p/ for p 1cccc3 , 1cccc6 , 2cccc3 , 5cccc6 .

Proposition 1.

(7)logG- 1ccccc31 log 3

cccccccccccccccc72

�S

ccccccccccccccccccccc18

r����3�

2ccccc3

log*- 1ccccc31� 4

ccccc3

log A�\+1/+

1cccc3 /

cccccccccccccccccccccccccc

12r����

3 S�

1ccccc9

(8)logG- 2ccccc31 log 3

cccccccccccccccc72

�S

ccccccccccccccccccccc18

r����3�

1ccccc3

log*- 2ccccc31� 4

ccccc3

log A�\+1/+ 2

cccc3 /cccccccccccccccccccccccccc

12r����

3 S�

1ccccc9

(9)logG- 1ccccc61 � log 12

ccccccccccccccccccc144

�S

ccccccccccccccccccccc20

r����3�

5ccccc6

log*- 1ccccc61� 5

ccccc6

log A�\+1/+ 1

cccc6 /cccccccccccccccccccccccccc

40r����

3 S�

5ccccccccc72

(10)logG- 5ccccc61 � log 12

ccccccccccccccccccc144

�S

ccccccccccccccccccccc20

r����3�

1ccccc6

log*- 5ccccc61� 5

ccccc6

log A�\+1/+ 5

cccc6 /cccccccccccccccccccccccccc

40r����

3 S�

5ccccccccc72

where \+1/+z/ �2

cccccccc

�z2 log*+z/ is the polygamma function.

Similar representations can be derived for the multiple G function.

2. Connection to the polylogarithm

From the Lerch functional equation for the Hurwitz zeta function and formula (3), we caneasily derive

(11)logLNMM

G+1� z/ccccccccccccccccccccccccc

G+1� z/\^]] � S i

cccccccccc

2B2+z/ � zlog- S

cccccccccccccccccccccc

sin+S z/ 1 � icccccccccc

2SLi2+e2 S i z/, 0� z� 1

where B2+z/ is the second Bernoulli polynomial, and Li2+z/ is the polylogarithm.The identitycan be rewritten in an alternative form by means of the Clausen function Cl2+z/:

(12)logLNMM

G+1� z/ccccccccccccccccccccccccc

G+1� z/\^]] zlog- S

cccccccccccccccccccccc

sin+S z/ 1� 1cccccccccc

2SCl2+2S z/, 0� z� 1

where

Cl2+x/ �Im+Li2+e�i x//

5

Obviously enough, the G function is related to the Dirichlet L-series. Here is one of theformulas:

(13)logL

N

MMMMG+ 7

cccc6 /cccccccccccccccccc

G+ 5cccc6 /

\

^

]]]] 1ccccc

6log+2S/ � 3

r����3

cccccccccccccccccc

8SL�3+2/

3. Binet-like representation

The Binet formula for the gamma function is

log*+z/ -z� 1ccccc2

1 logz � z� log,r�������2S 0 � Ã

0

� e�z x

ccccccccccccccx

- 1cccccccccccccccccccccc1� e�x

�1cccccx

�1ccccc2

1 dx

In this section we derive a similar representation for the Barnes function. Recall the well-

known integral for ] +s, z/:

(14)]+s, z/ 1ccccccccccccc*+s/ Ã

0

� xs�1 e�z x

cccccccccccccccccccccccccccc1� e�x

Åx, R+s/ ! 1, R+z/ ! 0

The integral can be analytically continued to the domain R+s/ ! �1. To do so, we perform thestandard procedure of removing the integrand singularity by subtracting the truncated Taylorseries at x 0:

]+s, z/ � z1�s

ccccccccccccccc1� s

�

z�s

cccccccccc2

�sccccccccc12

z�s�1

�1

ccccccccccccc*+s/ Ã

0

�

xs�1 e�z x - 1cccccccccccccccccccccc

1� e�x�

1cccccx�

xccccccccc12

�1ccccc2

1 dx

Differentiating the above formula with respect to s and computing the limit at s �1, we get

]�+�1, z/ 1ccccccccc

12�

z2

ccccccc

4�

logzccccccccccccccc

2B2+z/ � Ã

0

� e�z x

cccccccccccccc

x2- 1

cccccccccccccccccccccc

1� e�x�

1ccccc

t�

tccccccccc

12�

1ccccc

21 dx

where B2+z/ is the second Bernoulli polynomial. From here it immediately follows

Proposition 2. The Barnes G function admits the Binet integral representation:

(15)logG+z� 1/ z log*+z/ � log A�

z2

ccccccc4�

logzccccccccccccccc

2B2+z/ �

Ã0

� e�z x

ccccccccccccccx2

- 1cccccccccccccccccccccc

1� e�x�

1ccccc

t�

1ccccc

2�

tccccccccc

121 dx, R+z/ ! 0

In a similar way we derive the representation for Glaisher's constant:

(16)log A

1� log+2S/ccccccccccccccccccccccccccccccccccc

12�

1ccccccccccccc

2S2 Ã0

� x logxcccccccccccccccccccc

ex� 1

dx

6

4. The multiple G function

For the multiple G function defined by the functional equation (4), Vardi [10] obtained thefollowing formula

(17)logGn+z/ �lims�0

LNMM�]n+s, z/cccccccccccccccccccccccccc

�s\^]] �Å

k 1

n

+�1/kL

NMM

z

k� 1\

^]] Rn�1�k

where

(18)Rn Åk 1

n

lims�0

LNMM�]k+s, 1/ccccccccccccccccccccccccccc

�s\^]]

and ]n+s, z/ is the multiple zeta function:

(19)]n+s, z/ Åk1 0

�

...Åkn 0

� 1ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

+k1 � k2 � ...� kn� z/s Åk 0

� 1cccccccccccccccccccccc

+k � z/sL

NMM

k� n� 1

n� 1\

^]]

The aim of this section is to find a closed form representation for Gn+z/ in terms of the Hurwitzzeta function. For clarity of exposition, we first consider polynomials

(20)Pk, n+z/ Åi k�1

n

+�z/i�k�1 LNMM

i � 1

k\

^]] $

n

i(

which can be rewritten in the alternative form

(21)Pk, n+z/ Åi 1

n LNMM

z

n� i\

^]]

+n� 1/�cccccccccccccccccccccccccccc+i � 1/�

$i

k� 1(

where % ni ) are unsigned Stirling numbers of the first kind. The polynomials Pk, n+z/ satisfy the

functional equation

Pk, n�1+z/ � n Pk, n+z/� Pk, n�1+z� 1/ 0

Pk, n +z/ 0, k � n

7

Here are the main properties of these polynomials:

(22)

Pn�1,n+z/ 1 Pk, n+1/ $n� 1

k(

Pn�2,n+z/ z+1� n/ �

n+n� 1/cccccccccccccccccccccccc

2P0,n+z/

L

NMMn� z� 1

n� 1\

^]] +n� 1/�

P0,n-1ccccc2

1 +2n� 3/��ccccccccccccccccccccccccccccccc

2n�1

Lemma 1. The multiple zeta function ]n+s, z/ defined by (19) may be expressed by means of

the Hurwitz function

(23)]n+s, z/ 1cccccccccccccccccccccccc+n� 1/� Å

j 0

n

Pj, n+z/ ]+s� j, z/

Proof. Recall the definition of unsigned Stirling numbers of the first kind

L

NMM

k� n� 1

n� 1\

^]]

1cccccccccccccccccccccccc

+n� 1/� Åi 0

n

ki�1 $n

i(

Expanding ki�1 ++k� z/ � z/i�1 by the binomial theorem, implies that

L

NMM

k� n� 1

n� 1\

^]]

1cccccccccccccccccccccccc

+n� 1/�Æi 0

n

Åj 0

i�1L

NMM

i � 1

j\

^]] +k� z/ j +�z/i� j�1 $

n

i(

Interchanging the order of summation and making use of (20), we obtain

L

NMM

k� n� 1

n� 1\

^]]

1cccccccccccccccccccccccc

+n� 1/�Åj 0

n

Pj, n+z/ +k� z/ j

We complete the proof by substituting this into (19).

Lemma 2. The multiple zeta function Rn defined by (18) may be expressed by means of the

derivatives of the zeta function

(24)Rn 1

cccccccccccccccccccccccc

+n� 1/� Åk 0

n�1

]�+�k/ $n

k� 1(

Proof. First we make use of the Lemma 1 and the identity (22) for Pj, n+1/

lims�0

LNMM�]k+s, 1/ccccccccccccccccccccccccccc

�s\^]]

1cccccccccccccccccccccccc+n� 1/�

Åj 0

n

Pj, n+1/ ]�+� j/ 1cccccccccccccccccccccccc

+n� 1/� Åj 0

n

]�+� j/ $n� 1

j(

8

Further, in a view of (18), we have

Rn Åk 1

n 1cccccccccccccccccccccccc+k � 1/�

Åj 0

k

]�+� j/ $ k� 1

j( ]�+0/ �Å

j 1

n

]�+� j/Æk j

n1

cccccccccccccccccccccccc

+k� 1/�$

k� 1

j(

Taking into account that the inner sum on the right hand side is (it follows from (21) withz 1)

Åk j

n+n� 1/�cccccccccccccccccccccccc+k � 1/�

$k� 1

j( $

n

j � 1(

we complete the proof.

Proposition 3. The multiple Barnes function Gn+z/ may be expressed by means of the deriva-

tives of the zeta functions

(25)logGn+z/ 1

cccccccccccccccccccccccc+n� 1/� Åj 0

n�1

Pj, n+z/ +]�+� j/ � ]�+� j, z//

where the polynomials Pj,n+z/ are defined by (20).

5. Integral representations via polygammas

In [15] Adamchik derived a closed form solution to the integral

Ã0

z

xn\+x/ dx, R+z/ ! 0, n ± ´

in finite terms of the Hurwitz zeta function:

(26)

Ã0

z

xn \+x/dx +�1/n�1 ]�+�n/ � +�1/ncccccccccccccccccccn� 1

Bn�1 Hn �

Åk 0

n

+�1/k LNMM n

k\^]] zn

�k -]�+�k, z/�Bk�1+z/ Hkcccccccccccccccccccccccccccccccccc

k� 11

where Bnand Hn are Bernoulli and Harmonic numbers respectively. If n 1 the integral (26)leads to the following representation for the Barnes G function:

(27)logG+z� 1/ z� z2

cccccccccccccccccc

2� zlog,r�������

2S 0 � Ã0

z

x\+x/ dx, R+z/ ! �1

This representation demonstrates that the complexity of computing G+z/ depends at most onthe complexity of computing the polygamma function. The restriction R+z/ ! �1 can be easily

9

removed by analyticity of the polygamma. For example, by resolving the singularity of theintegrand at the pole x �1, we continue logG+z� 1/ to the wider area R+z/ ! �2:

(28)logG+z� 1/ z� z2

cccccccccccccccccc2

� zlog,r�������2S 0 � log +z� 1/ � Ã

0

z

x\+x/� 1ccccccccccccccccx� 1

dx

Another way to continue (27) into the left half-plane is to use the identity

\+x/ \+�x/ � S cot+S x/ � 1cccccx

which upon substituting it into (27) yields

(29)logG+1� z/ logG+1� z/ � zlog+2S/ �Ã0

z

S xcot+S x/ dx

The identity (29) holds everywhere in a complex plane of z, except the real axes, where theintegrand has simple poles. Therefore, in a view of (29) we can continue (27) to © / ¸�. Note,if z is negative we can still use the representation (27), but with a contour of integrationdeformed in such a way that it does not cross poles.

For n 2, the formula (26) yields

(30)Ã

0

z

x2 \+x/ dx �2 logG3+z� 1/ � logG2+z� 1/ �z2 ]�+0/ � 2z]�+�1/ � z

ccccccccc12

+6z2 � 3z� 1/and for n 3 :

(31)Ã

0

z

x3\+x/ dx 6 logG4+z� 1/ � 6 logG3+z� 1/ � logG2+z� 1/ �

z3 ]�+0/ � 3z2 ]�+�1/ � 3z]�+�2/� z2

ccccccccc24

+11z2 � 4z� 1/The general formula can also be derived. Skipping the technical details, we have

Proposition 4. Let n be a positive integer and and R+z/ ! �1, then

(32)

Ã0

z

xn\+x/ dx �Å

k 1

n

+�1/k k� � nk! logGk�1+z� 1/ �

Æk 0

n�1

+�1/k L

NMM n

k\^]] zn

�k -]�+�k/ �Bk�1+z/ Hkcccccccccccccccccccccccccccccccc

k� 11� +�1/n Hn

Bn�1 � Bn�1+z/ccccccccccccccccccccccccccccccccccccccccccc

n� 1

where � nk! are Stirling numbers of the second kind.

10

We can resolve the equation (32) with respect to Gk�1+z� 1/. In particular, combining (27)with (30) leads us to the following integral representation for the triple Barnes function:

(33)2 logG3+z� 1/ Ã

0

z

x+1� x/\+x/ dx� z+1� z/ ]�+0/�2 z]�+�1/ � z

ccccccccc12

+6z2� 9z� 5/and combining (27), (30) and (31) yields:

(34)6 logG4+z� 1/ Ã

0

z

x +x� 1/ +x� 2/\+x/ dx� z+z� 1/ +z� 2/ ]�+0/ �3z+z� 2/ ]�+�1/ � 3 z]�+�2/ � z

ccccccccc24

+11z3 � 40z2� 41z� 18/Formulas (33) and (34) are suitable for numeric computation of G3+z/ and G4+z/ for any

z± © s¸�.

6. Implementation remarks

We have presented here many available results on the Barnes function as well as new results onnumeric and symbolic computations. In this section we discuss a numeric computationalscheme for the Barnes G function. The integral representation via polygamma functions consid-ered in the previous section seems to be an efficient numeric procedure for evaluating Gfunction. Based on (27), we define the double Barnes function G+z/ G2+z/ as

(35)G+z/ +2S/

z�1cccccccccc2

expL

NMMM�

+z� 1/ +z� 2/ccccccccccccccccccccccccccccccccccccccccc

2�Ã

0

z�1

x\+x/ dx\

^]]], arg+z/ � S

This representation is valid for z± © s¸�. If z is a negative real, we have

(36)

G+�n/ 0, n± ´

G+z/ +2S/

z�1cccccccccc2

expL

NMMM� +z� 1/ +z� 2/

ccccccccccccccccccccccccccccccccccccccccc

2� Ã

J

x\+x/ dx\

^]]], arg+z/ S

where the contour of integration J is a line between 0 and z� 1 that does not cross the nega-tive real axis; for example, J could be the following path: �0, i, i � z� 1, z� 1�.

11

Another method of defining the Barnes G function for negative reals is to use the formula (12).This gives

(37)G+�z/ +�1/Gzs2W�1 G+z� 2/

LNMM�sin+S z/�cccccccccccccccccccccccccc

S

\

^]]

z�1

exp- 1cccccccccc

2S

Cl2+2S +z� GzW//1

where GzW is the floor function, and Cl2+z/ is the Clausen function. Here is a picture of the Gfunction over the interval (-3,3):

-3 -2 -1 1 2 3

-0.2

0.2

0.4

0.6

0.8

1

In a similar manner, taking into consideration the formulas (33) and (34), we can define G3 andG4 respectively.

There are many applications, particularly in number theory, where the logarithm of the Barnesfunction often appears. However, because of a branch cut of logarithm, the function logG+z/(where G is defined by (35)) includes spurious discontinuities for complex argument. Thefollowing the picture of Im+logG+2� i y//, where y± �1, 10� demonstrates this:

2 4 6 8 10

-3

-2

-1

1

2

3

12

Therefore, we define an additonal function LogG+z/ (the logarithm of the Barnes function)which is an analytic function throughout the complex z plane

(38)LogG+z/ � +z� 1/ +z� 2/cccccccccccccccccccccccccccccccccccccccc

2�

z� 1cccccccccccccccc

2log+2S/ � Ã

0

z�1

x\+x/ Åx

If z is a negative real, we understand the path of integration as in (36). Here is the picture ofIm+LogG+2� i y//, where y± �1, 10� :

2 4 6 8 10

10

20

30

40

50

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13

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