The Multiple Gamma Functions and the Log Gamma Integrals

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Volume 2012, Article ID 547459,14pages doi:10.1155/2012/547459

Research Article

The Multiple Gamma-Functions and

the Log-Gamma Integrals

X.-H. Wang

1

_{and Y.-L. Lu}

2

1_{Xi’an International Studies University, Xi’an, Shaanxi 710128, China}

2_{Department Of Mathematics, Weinan Teachers’ College, Shaanxi 714000, China}

Correspondence should be addressed to X.-H. Wang,xiaohan369@gmail.com

Received 15 May 2012; Accepted 30 July 2012

Academic Editor: Shigeru Kanemitsu

Copyrightq2012 X.-H. Wang and Y.-L. Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, which is a companion paper toW, starting from the Euler integral which appears in a generalization of Jensen’s formula, we shall give a closed form for the integral of logΓ1±t. This

enables us to locate the genesis of two new functionsA1/aandC1/aconsidered by Srivastava and

Choi. We consider the closely related function A(a) and the Hurwitz zeta function, which render the task easier than working with theA1/afunctions themselves. We shall also give a direct proof

of Theorem 4.1, which is a consequence ofCKK, Corollary 1.1, though.

1. Introduction

Iffzis analytic in a domainDcontaining the circleC:|z|rand has no zero on the circle,

then the Gauss mean value theorem

logf0 1

2π

_2π

0

logfreiθdθ 1.1

is true. In1, page 207the case is considered wherefzhas a zeroreiθ0 _{on the circle, and}

1.1turns out that the Euler integral

π/2

0

log sinxdx−π

2 log 2 1.2

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LetGdenote the Catalan constant defined by the absolutely convergent series

G

∞

n1

−1n−1

2n−12 L

2, χ4, 1.3

whereχ4is the nonprincipal Dirichlet character mod 4.

As a next step from1.2the relation

_π/4

0

log sintdt−π

4 log 2−

1

2G 1.4

holds true. In this connection, in2we obtained some results onGviewing it as an intrinsic

value to the BarnesG-function. The BarnesG-functionwhich isΓ2−1in the class of multiple

gamma functionsis defined as the solution to the diﬀerence equationcf.2.3

logGz 1−logGz logΓz 1.5

with the initial condition

logG1 0 1.6

and the asymptotic formula to be satisfied

logGz N 2 N 1 z

2 log 2π

1 2

N2 2N 1 B2 z2 2N 1zlogN

−3

4N

2₋_N₋_Nz₋_log_A 1

12 O

N−1,

1.7

N → ∞, whereΓsindicates the Euler gamma functioncf., e.g.,3.

Invoking the reciprocity relation for the gamma function

Γssinπs π

Γ1−s, 1.8

it is natural to consider the integrals of logΓα tor of multiple gamma functionsΓr cf.,

e.g.,4,5. Barnes’ theorem6, page 283reads

_a

0

logΓα tdt−logGα a

Gα −1−αlogΓα

a Γα

alogΓα a−1

2a

2 1

2

log 2π 1−2αa

1.9

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In this paper, motivated by the above, we proceed in another direction to developing

some generalizations of the above integrals considered by Srivastava and Choi 7. For

q-analogues of the results, compare the recent book of the same authors8. Our main result is

Theorem 2.1which gives a closed form for ₀alogΓ1−tdtand locates its genesis. A slight

modification ofTheorem 2.1gives the counterpart of Barnes’ formula1.9which reads.

Corollary 1.1. Except for integral values ofa, one has

a

0

logΓα−tdtlogGα−a

Gα 1−αlog

Γα−a

Γα

alogΓα−a 1

2a

2 1

2

log 2π 1−2αa.

1.10

Srivastava and Choi introduced two functions logA1/aand logC1/aby2.9and2.9

with formal replacement of 1/aby−1/a, respectively. They stateC1/a A−1/a, which is

rather ambiguous as to how we interpret the meaning because2.9is defined fora > 07,

page 347, l.11. They use thisC1/a function to express the integral ₀alogΓ1−tdt, without

giving proof. This being the case, it may be of interest to locate the integral of logΓ1−t 7,

13, page 349, thereby logC1/a7, page 347.

For this purpose we use a more fundamental functionAathanA1/adefined by

logAa −ζ−1, a 1

12, 1.11

whereζs, ais the Hurwitz zeta-function

ζs, a ∞

n0 1

n as, Resσ >1 1.12

in the first instance. For its theory, compare, for instance,3,9, Chapter 3.

We shall prove the following corollary which gives the right interpretation of the

functionC1/a.

Corollary 1.2. For 0< a <1,

logC1/a logA1−a−1

4a

2_, _1.13

or

logC1/alogA1−1/a

1

41−a

2₁₋_a_log1₋_a₋1

4a

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2. Barnes Formula

There is a generalization of1.4as well as1.2in the form7, equation28, page 31:

_a

0

log sinπtdtalogsinπa

2π log

G1 a

G1−a, a /∈Z. 2.1

Equation2.1is Barnes’ formula6, page 279which is equivalent to Kinkelin’s 1860 result

10 7, equation26, page 30:

z

0

πtcotπtdtlogG1−z

G1 z zlog 2π. 2.2

Since1.5is equivalent to

Gz 1 GzΓz, 2.3

it follows that

_a

0

log sinπtdtalogsinπa

2π log

Ga

G1−a logΓa. 2.4

Puttinga1/2, we obtain

π−1

_π/2

0

log sinxdx

_1/2

0

log sinπtdt−1

2log 2π logΓ

1 2

−1

2log 2, 2.5

which is1.2.

The counterpart of 2.1 follows from the reciprocity relation 1.8, known as

Alexeievsky’s Theorem7, equation42, page 32.

a

0

logΓ1 tdt 1

2

log 2π−1a−a

2

2 alogΓa 1−logGa 1, 2.6

which in turn is a special case of1.9.

Indeed, in7, page 207, only1.9and the integral of logGt αare in closed form and

the integral of logΓ3t αis not. A general formula is given by Barnes4with constants to

be worked out. We shall state a concrete form for this integral inSection 3, using the relation

7, equation455, page 210between logΓ3t αand the integral ofψ and appealing to a

closed form for the latter in11.

Formula2.6is stated in the following form7, equation12, page 349:

_a

0

logΓ1 tdt 1

2

log 2π−1a−3

4a

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where logAis the Glaisher-Kinkelin constant defined by7, equation2, page 25

logA lim

N→ ∞

_N

n1

nlogn−1

2

N2 N B2logN 1

4N

2

, 2.8

and logA1/ais defined by7, equation9, page 347

logA1/a lim

N→ ∞

_N

n1

n alogn a

−1

2

N2 2a 1N a2 a B2logN a 1

4N

2 a

2N

,

2.9

fora >0.

Comparing2.6and2.7, we immediately obtain

logA1/alogGa 1−alogΓa 1 logA−a

2

4

logGa 1−alogΓa loga−a

2

4 −aloga,

2.10

on using the diﬀerence relationΓa 1 aΓa.

Thus, in a sense we have located the genesis of the function logA1/a. although they

prove2.7by an elementary method7, page 348.

Indeed,A1/aandAaare almost the same:

logA1/a logAa−1

4a

2₋_a_log_a, _2.11

a proof being given below. However, logAais more directly connected withζ−1, afor

which we have rich resources of information as given in9, Chapter 3.

We prove the following theorem which gives a closed form for ₀alogΓ1−tdt, thereby

giving the genesis of the constantC1/a.

Theorem 2.1. Fora /∈ Z, one has

a

0

logΓ1−tdtlogG1−a alogΓ1−a 1

2a

2 1

2

log 2π−1a. 2.12

If 0< a <1, then

_a

0

logΓ1−tdtlogA1−a−logA 1

2a

2 1

2

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Proof. We evaluate the integral

I

a

0

logΓ1 tsinπtdt 2.14

in two ways. First,

I alogπ aloga−a−

_a

0

logΓ1−tdt. 2.15

On the other hand, noting thatIis the sum of2.1and2.7, we deduce that

Ialogsinπa

2π logGa 1 logA−logG1−a

1 2

log 2π−1a−3

4a

2₋_log_A1/a.

2.16

Substituting1.5, we obtain

Ialogsinπa

2π alogΓa logAa−logA1/a

−logG1−a 1

2

log 2π−1a− 3

4a

2_.

2.17

The first two terms on the right of2.17become

alogΓasinπa

2π alog

1

2Γ1−a −a

log 2 logΓ1−a, 2.18

while the 3rd and the 4th terms give, in view of2.11,1/4a2 _a_log_a.

Hence, altogether

I−alog 2−alogΓ1−a−logG1−a aloga−1

2a

2 1

2

log 2π−1a. 2.19

Comparing2.15and2.19proves2.12, completing the proof.

Comparing2.13and7, equation13, page 349

a

0

logΓ1−tdtlogA1−a−logA 3

4a

2 1

2

log 2π−1a, 2.20

we proveCorollary 1.2.

Hence the relation between C1/a and A1/a is1.14, that is, one betweenC1/a and

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At this point we shall dwell on the underlying integral representation for the

derivative ofthe Hurwitz zeta-function, which makes the argument rather simple and lucid

as in12and gives some consequences.

Proof of2.11. Consider that

ζs, a− 1

12 −

1

2a

2_log_a₋1

4a

2₋1

2aloga

− B2

2 loga−

1 3!

_∞

0

B3tt a−2

dt

2.21

9,3.15, page 59, where the last integral may be also expressed as

−1

2!

_∞

0

B2tt a−1_dt, _2.22

and whereBktis thekth periodic Bernoulli polynomial. Then

−ζ_{−1, a}

0≤n≤x

n alogn a−1

2x a

2_logx _a

1

4x a

2 _B1xx _a₋1

2B2xx a O

x−1logx;

2.23

whence in particular, we have the generic formula forζ−1, aand consequently for logAa

through1.11:

logAa lim

N→ ∞

_∞

n0

n alogn a−1

2logN a

×N a2 _N _a _B2 1

4N a

2

.

2.24

This may be slightly modified in the form

logAa lim

N→ ∞

_N

n0

n alogn a

−1

2

N2 2a 1N a2 a B2logN a 1

4N

2 1

2aN

1

4a

2 _a_log_a.

2.25

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The merit of usingAais that by way ofζ−1, a, we have a closed form for it:

logAa 1

2a

2_log_a₋1

4a

2 1

2aloga

B2

2 loga

1 2!

_∞

0

B2tt a−1

dt.

2.26

In the same way, via another important relation7, equation23, page 94,

logGa −

ζ−1, a− 1

12

−logA−1−alogΓa. 2.27

Equation2.21gives a closed form for logGa, too. We also have from1.11and2.27

logAa logGa 1−alogΓa logA

logGa 1−alogΓa logA. 2.28

There are some known expressions not so handy as given by2.27. For example,7,

page 25and7, equation440, page 206, one of which reads

G

G1 z

∞

n1

n

z n−1

z n

1 2

log 2π−1−1 γz, 2.29

with γ designating the Euler constant. Equation2.29 is a basis of 2.2 cf. proof of 2,

Lemma 1.

Remark 2.2. The Glaisher-Kinkelin constantAis connected withA1andA1as follows:

logAlogA1 logA1 1

4. 2.30

This can also be seen from Vardi’s formula7,31, page 97:

logA−ζ−1 1

12, 2.31

which is1.11witha1.

We may also give another direct proof ofCorollary 1.2.

Proof ofCorollary 1.2(another proof). logC1/ais the limit of the expression

SN

N

k1

k−1 αlogk−1 α−

1

2N

2

α−1

2

N 1

2B2α

×logN−1 α 1

4N

2 N

2 α−1,

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whereα1−a. LetNM 1. Then

SN

M

k0

k αlogk α−

1

2M 1

2

α−1

2

M 1 1

2B2α

×logM α 1

4M 1

2 M 1

2 α−

M 1

2 .

2.33

Hence, simplifying, we find that

SN

M

k1

k αlogk α−

1

2M

2

α 1

2

M 1

2

α2 α B2

×logM α 1

4M

2 1

2αM αlogα−

α−12

4 1

4α

2_.

2.34

Hence

logC1/alogAα αlogα−α−1

2

4 1

4α

2_, 2.35

which is1.14. This completes the proof.

As an immediate consequence ofCorollary 1.2, we prove2.36as can be found in7,

pages 350–351.

A1/a

πa

sinπa

−a_G1 _a

G1−aC1/a, 0< a <1. 2.36

Proof of2.36. From2.28,1.5, and1.8, we obtain

logAa−logA1−a logG1 a

G1−a−alog

π

sinπa. 2.37

On the other hand, by2.11and1.13, we see that the left-hand side of2.37is

logA1/a

C1/a aloga, 2.38

whence we conclude that

logA1/a

C1/a log

G1 a

G1−a−alog

πa

sinπa. 2.39

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3. Polygamma Function of Negative Order

In this section we introduce the functionAkq 13:

Akqkζ1−k, q, 3.1

which is closely related to the polygamma function of negative order and states some simple

applications. We recall some properties ofAkq:

A2q 1A2 q 2qlogq,

A2

1 2

−ζ₋₁₋ 1

12log 2,

A2

1 4

−1

4ζ

₋₁ G

2π,

3.2

A2

3 4

−1

2ζ

₋₁₋_A2 1

4

. 3.3

Equation3.3is2, equation2.31, which is used in proving2, Theorem 2and can

be read oﬀfrom the distribution property9, equation3.72, page 76as follows:

4

a1

ζs,a

4

4sζs. 3.4

Diﬀerentiation gives

4

n1

ζs,a

4

4slog 4ζs ζs. 3.5

Puttings−1, we obtain

ζ−1 ζ

−1,1

2

ζ

−1,1

4

ζ

−1,3

4

4−1log 4ζ−1 ζ−1, 3.6

which we solve inζ−1,3/4:

ζ

−1,3

4

1

4

2 log 2ζ−1 ζ−1

−ζ−1−1

2A2

1 2

−ζ

−1,1

4

.

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Substituting3.2andζ−1 −B2/2−1/12 and simplifying, we conclude that

ζ

−1,3

4

−1

4ζ

₋₁₋_ζ_−1,1

4

3.8

and that

A2

3 4

2ζ

−1,3

4

−1

2ζ

₋₁₋_2ζ_−1,1

4

, 3.9

whence3.3.

Using these, we deduce from2.37the following.

Example 3.1.

logA1/4 5

64 1

2log 2−

1

8logA−

G

2π. 3.10

Proof. By1.11and3.1, forq >0,

logAq−1

2A2

q 1

12. 3.11

Since logA1/4−logA3/4 −1/2A21/4 −A23/4, it follows from 3.3that

the left-hand side of2.37is

−A2

1 4

−1

4ζ

_−1, _3.12

which is

2 logA

1 4

−1

6 1 4

logA− 1

12

3.13

where we used2.31.

The right-hand side of2.37, logG5/4/G3/4−1/4 logπ/sinπ/4, becomes

−G/2π, in view of known values ofG7, page 30.

Hence, altogether,2.37witha1/4 reads

− G

2π 2 logA

1 4

1

4logA−

3

16. 3.14

Invoking2.11, this becomes3.10.

We note that3.14gives a proof of the third equality in3.2. Both2.36and3.10

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4. The Triple Gamma Function

For general material, we refer to7, page 42. As can been seen on7, page 207, the important

integral ₀zlogΓ3t adtis not in closed form. Recently, Chakraborty-Kanemitsu-Kuzumaki

5, Corollary 1.1have given a general expressions for all the integrals in logΓr, by appealing

to Barnes’ original results.

In this section, we shall give a direct derivation of a closed form by combining 7,

455, page 210and11, Corollary 3 withλ3. The first reads

2

_z

0

logΓ3t adt−

_z

0

t3ψt adt 2zlogΓ3z a

−22a−3logΓ3z a

logΓ3a

3a2−9a 7logGz a

logGa

−a−13logΓz a

logΓa

3

8z

4 1

3

1−log 2πz3

−3

4a

2 7

4a−

9 8

1

42a−3log 2π logA

z2,

a2−3

2a

1 4

1

2a−2−3a 2log 2π 23−2alogA

z,

4.1

while the second readscf. also15

z

0

t3logψt adt−

3

r0

C3r, alogΓr 1a z

Γr 1a

−3

l1

−1l

3

l

ζl−3 B4_l4−la₋_l

zl 11

24z

4_,

4.2

whereC3r, aare defined by

C3r, a −1r

r! 3

mr

3 m

−1m_{Sm, na}₋₁3−m _4.3

and whereSm, nare the Stirling numbers of the second kind7, page 58. To express the

values ofζl−3, we appeal to7

iζ0 −1/2log 2π7,20, page 92,

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and2.31. After some elementary but long calculations, we arrive at

z

0

t3logψt adt−3! logΓ4a z

Γ4a −6a−2log

Γ3a z

Γ3a

−3a2−9a 7logΓ2a z

Γ2a −a−13logΓa

z Γa

11

24z

4

−1

2log 2π

1

3B1a

z3−

1

4−3 logA

1

4B2a

z2

3

logB 1

3B3a

z.

4.4

Combining we have the following.

Theorem 4.1see5, Example 2.3. Except for the singularities of the multiple gamma function,

one has

z

0

logΓ3t adt3 logΓ4a z

Γ4a zlogΓ3z a

a−3logΓ3a z

Γ3a −

1

24z

4₋1

6

a−3

2−

1

2log 2π

z3

1 8

−2a2 _6a₋10

3 2a−3log 2π−8 logA

z2

−1

2

a3−5

2a

2 _2a₋1

4 1 2

a2−3a 2log 2π

22a−3logA 3 logB

z.

4.5

This theorem enables us to put many formulas in7in closed form including, for

instance,7,698, page 245. Compare5.

Acknowledgment

The authors would like to express their hearty thanks to Professor S. Kanemitsu for his enlightening supervision and encouragement.

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2 X.-H. Wang, “The Barnes G-function and the Catalan constant ,” Kyushu Journal of Mathematics. In press.

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9 S. Kanemitsu and H. Tsukada, Vistas of Special Functions, World Scientific, River Edge, NJ, USA, 2007. 10 V. H. Kinkelin, “ ¨Uber eine mit der Gamma Funktion verwandte Transcendente und deren Anwend ung auf die Integralrechnung,” Journal f ¨ur die reine und angewandte Mathematik, vol. 19, pp. 122–158, 1860.

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