In the appendix we discuss a relation between the volume and Appell’s hy- pergeometric functions of type F

INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1992

HYPERLOGARITHMIC EXPANSION AND THE VOLUME OF A HYPERBOLIC SIMPLEX

K . A O M O T O

Department of Mathematics, Nagoya University Nagoya, Japan

0. Introduction. Hyperlogarithmic functions (or higher logarithmic func- tions) are multivalued analytic functions defined on complex projective varieties, with unipotent monodromy and with regular singularity. It is known that they can be expressed by the use of iterated integrals of suitable logarithmic 1-forms in the sense of K. T. Chen (see [A1], [H1]). Recently these functions have played a considerable role in various problems of geometry and arithmetic (for example, see [H2], [B1], [G2], [V], etc.). These are a special case of hypergeometric functions on a Grassmannian manifold (see [A2], [G1], [V]).

However, there are other kinds of hyperlogarithmic functions which are re- lated to the configuration of hyperplanes and a hyperquadric (see [A3]). The volume of a simplex in a hyperbolic space is a hyperlogarithmic function of basic algebraic invariants, as a simple consequence of the Schl¨ afli formula. However, there remains the problem of divergence in the case where the vertices lie on the boundary.

In this note we want to derive a modified Schl¨ afli formula in such a degenerate case and give a hyperlogarithmic expansion for the volume, by using a technique developed in [A3]. A similar result has been obtained by Kellerhals [K4]. Her method is to decompose a simplex into several orthoschemes and to obtain an explicit formula for each orthoscheme by using the Lobachevski˘ı function L(x).

In the appendix we discuss a relation between the volume and Appell’s hy- pergeometric functions of type F

.

1. The Schl¨ afli formula. A geodesic simplex ∆ in the n-dimensional hyper- bolic space H = {t

− t

− . . . − t

= 1, t

> 0} is defined by the inequalities f

(t) ≥ 0 for n + 1 linear functions f

(t) = u

+ P

ν=1

u

t

, 1 ≤ j ≤ n + 1. Its

[9]

volume V

(∆) is given by the integral

(1.1) V

(∆) = R

f1≥0,...,fn+1≥0

Φ dt

∧ . . . ∧ dt

,

for Φ = exp[−

(t

−t

−. . .−t

)]. This is also equal to 2

Γ ((n+1)/2)V

( b ∆) for a geodesic simplex b ∆ in the disc D = {x

+ . . . + x

< 1} defined by the inequalities b f

(x) ≥ 0 for the inhomogeneous linear functions b f

(x) = u

+ P

ν=1

u

x

. The volume V

( b ∆) is defined by the integral (1.2) V

( b ∆) = R

∆

b

(1 − x

− . . . − x

)

dx

∧ . . . ∧ dx

.

First we assume that b ∆ lies in D. Then u

− P

ν=1

u

< 0. We may normalize it so that u

− P

ν=1

u

= −1. Because of conformal invariance, V

(∆) or equiv- alently V

( b ∆) depends only on the inner products a

= u

u

− P

ν=1

u

u

for 1 ≤ j, k ≤ n+1. a

, j 6= k, can also be expressed as coshj, ki, where hj, ki de- notes the dihedral angle subtended by b ∆ between the hyperplanes F

= { b f

(x) = 0} and F

= { b f

(x) = 0}. We denote by A the symmetric (n + 1) × (n + 1) matrix ((a

))

. Note that a

= −1. We denote by A

1,...,jp

the subdetermi- nant of A with lines i

, . . . , i

and columns j

, . . . , j

for {i

, . . . , i

}, {j

, . . . , j

} ⊂ {1, 2, . . . , n + 1}. We abbreviate A

1,...,jp

to A(i

, . . . , i

).

One can show that b ∆ defines a simplex lying in D if and only if (−1)

A(i

, . . . , i

) > 0 for 1 ≤ p ≤ n, and (1.3)

(−1)

A(1, 2, . . . , n + 1) < 0 . (1.4)

We denote by v

, . . . , v

the vertices of b ∆ such that v

∈ b ∆ ∩ F

∩ . . . ∩ F

∩ F

∩ . . . ∩ F

. Then v

is on the boundary ∂D of D if and only if A(1, 2, . . . , j − 1, j + 1, . . . , n + 1) = 0. The Schl¨ afli formula says that, as a function of the variables a

, V

(∆) satisfies the variational formula

(1.5) dV

( b ∆) = − 1 2

X

1≤j,k≤n+1 j6=k

V

( b ∆

) dhj, ki ,

where b ∆

denotes the (n − 2)-dimensional subsimplex b ∆

= b ∆ ∩ F

∩ F

. dhj, ki is equal to the logarithmic 1-form

θ

 ∅ j, k



= 1

2i d log  a

+ ipA(j, k) a

− ipA(j, k)

 .

Further, for I = {i

, . . . , i

} and J = {i

, . . . , i

, i

, i

} we define the loga-

rithmic 1-form

(1.6) θ  I

J



= 1

2i d log A

p+2

+ ipA(I)A(J) A

p+2

− ipA(I)A(J)

!

for p ≤ n − 2, and

(1.7) θ  I

J



= 1

2 d log A

n+1

+ p−A(I)A(J) A

n+1

− p−A(I)A(J)

!

for p = n − 1, n odd.

As a simple consequence of (1.5), V

( b ∆) can be expressed as a hyperlogarithm (sometimes called higher logarithm) (see [A5]):

(1.8) V

( b ∆) = X

∅⊂I1⊂...⊂Iν

A

R

∗

θ  ∅ I



. . . θ I

I

 ,

for a sequence of increasing subsets I

, . . . , I

of {1, 2, . . . , n + 1}, I

= {i

, . . . . . . , i

} . ν is equal to (n + 1)/2 or n/2 according as n is odd or even. The integration on the right hand side means K. T. Chen’s iterated integrals along a path from the base point ∗ to A. As special cases we have

(1.9) V

( b ∆) =

β

R

α

(1 − x

)

dx = 1 2 log

a

+ q

a

− 1 a

− q

a

− 1

for α = −u

/u

, β = −u

/u

and a

= u

u

− u

u

, while (1.10) V

( b ∆) = π − h1, 2i − h2, 3i − h3, 1i .

The following is an immediate consequence of (1.7).

Lemma 1. The hyperbolic distance between v

and v

is given by

(1.11) 1

2 log A

1,...,n−1,n+1

+ pA(1, . . . , n − 1)A(1, . . . , n + 1) A

1,...,n−1,n+1

− pA(1, . . . , n − 1)A(1, . . . , n + 1) . We see at the same time that

(1.12) A

 i

, . . . , i

, i

i

, . . . , i

, i



> 0 . This inequality will be used later for n = 3.

2. Regularization of divergent integrals. When one of the vertices lies

on ∂D, V

(∆) is well defined and continuous in a

, while V

(∆) diverges. The

formula (1.5) holds for n ≥ 4 but not for n = 3. We want to derive a modified

version of the Schl¨ afli formula for V

(∆). To do this, we use the technique of

regularization of divergent integrals which has been frequently used since the

times of J. Hadamard. We consider the integral (2.1) V

( b ∆|µ) = R

∆

b

(1 − |x|

)

dx

∧ . . . ∧ dx

,

for µ > 0. When µ = 0, it reduces to V

( b ∆). (2.1) is no more conformally invariant.

It cannot be expressed as a function of the variables a

for 1 ≤ j, k ≤ n + 1 . We denote by e A the enlarged symmetric (n + 2) × (n + 2) matrix ((a

))

with a

= 1, and a

= a

= u

. Obviously a

is not conformally invariant.

The following variational formula has been proved in [A3] (see the formula (3.7) loc. cit. for λ

, . . . , λ

→ 0):

Lemma 2.1. For an arbitrary n ≥ 1, (2.2) (n − 1 + µ) dV

( b ∆|µ)

= − 1 2

X

1≤j,k≤n+1 j6=k

dhj, ki

 A(j, k) A(0, j, k)



V

( b ∆

|µ)

+ µ

n+1

X

k=1

da

 −1 A(0, k)



1

pA(0, k) V

( b ∆

|µ − 1) , where b ∆

= b ∆ ∩ F

∩ F

and b ∆

= b ∆ ∩ F

. V

( b ∆

|µ) has a definite meaning and gives a function meromorphic in µ at least with a pole at µ = 0.

The following lemma can be seen by a computation.

Lemma 2.2. A(0, i) < 0, A(0, i, j) > 0, A(0, i, j, k) < 0 for any i, j, k ∈ {1, 2, 3, 4} and A(0, 1, 2, 3, 4) = 0.

When β = 1 in (1.9), V

( b ∆|µ) has a Laurent expansion at µ = 0:

(2.3) V

( b ∆|µ) = − 1 µ +



log 2 − 1

2 log 1 + α 1 − α



+ O(µ) ,

with α = −a

/p−A(0, 2), where the constant term (denoted by C.T. V

( b ∆|µ)) represents the regular part of the divergent integral V

( b ∆):

(2.4) reg V

( b ∆) = C.T. V

( b ∆|µ) = log 2 − 1

2 log 1 + α 1 − α . When α = β = −1, we have

(2.5) V

( b ∆|µ) = − 2

µ + 2 log 2 + O(µ) , whence reg V

( b ∆) = 2 log 2.

3. Modified Schl¨ afli formula for n = 3. Because of symmetry, we only have

to consider the following 4 cases: (i) v

∈ ∂D, (ii) v

, v

∈ ∂D, (iii) v

, v

, v

∈ ∂D

and (iv) v

, v

, v

, v

∈ ∂D.

(1) Assume that v

lies on ∂D and v

, v

, v

∈ D. This is equivalent to saying that A(1, 2, 3) = 0, i.e. h1, 2i + h2, 3i + h3, 1i = π. Then V

( b ∆

), V

( b ∆

) and V

( b ∆

) diverge, while V

( b ∆

) V

( b ∆

), V

( b ∆

) are well defined. For j, k = 1, 2, 3, as µ tends to 0, the coefficient of hj, ki on the right hand side of (2.2) has a Laurent expansion

(3.2)

 A(j, k) A(0, j, k)



V

( b ∆

|µ)

= − 1 µ +



log 2 − 1

2 log 1 + α 1 − α + 1

2 log A(j, k) A(0, j, k)



+ O(µ) , i.e.

(3.3) C.T.

 A(j, k) A(0, j, k)



V

( b ∆

|µ)



= log 2 − 1

2 log 1 + α 1 − α + 1

2 log A(j, k) A(0, j, k) . Here α denotes −A

/p−A(j, k)A(0, j, k, 4). We set

W

= A(j, k)(1 + α) A(0, j, k)(1 − α) . Then by taking the constant term of (2.2) in µ, we have

2dV

( b ∆) = dh1, 2i log W

+ dh2, 3i log W

+ dh3, 1i log W

(3.4)

+ dh1, 4i log W

+ dh2, 4i log W

+ dh3, 4i log W

, for i, j = 1, 2, 3, since h1, 2i + h2, 3i + h3, 1i = π, i.e.

(3.5) W

= A(0, i, j)

A(i, j) · p−A(i, j)A(0, i, j, 4) − A

p−A(i, j)A(0, i, j, 4) + A

and

(3.6) W

= A

− p−A(i, 4)A(i, j, k, 4) A

+ p−A(i, 4)A(i, j, k, 4)

for the complement {j, k} = {1, 2, 3, 4} − {i, 4}. Since dh1, 2i = −dh1, 3i − dh2, 3i, (3.4) can be expressed as

2dV ( b ∆) = dh1, 3i log W

/W

+ dh2, 3i log W

/W

(3.7)

+ dh1, 4i log W

+ dh2, 4i log W

+ dh3, 4i log W

.

We want to express the quantities W

/W

and W

/W

in terms of the

variables a

, 1 ≤ j, k ≤ 4. By a conformal change of variables we may assume

that v

= (0, 0, 1) ∈ ∂D ∩ F

∩ F

∩ F

and that f

= x

, f

= u

x

+ u

x

, (3.8)

f

= u

x

+ u

x

+ u

x

+ u

, for j = 3, 4 ,

where 1 = u

+ u

= u

+ u

= u

+ u

+ u

− u

and u

+ u

= 0.

We can further assume that u

> 0, u

< 0, u

< 0 and u

> 0. We then have a

= a

= 0 and

Lemma 3.1.

W

= a

A(1, 2, 4) A(1, 2, 3, 4) . P r o o f. Since

(3.9) 0 = A(0, 1, 2, 3, 4)

= A(1, 2, 3, 4) − a

A(1, 2, 4) + 2a

a

A 1, 2, 3 1, 2, 4



− a

A(1, 2, 3) , we have

(3.10) a

= A(1, 2, 4)a

− A(1, 2, 3, 4) 2a

A

from the equality A(1, 2, 3) = 0. Since A(1, 2)A(1, 2, 3, 4) − A

= 0, we have A(0, 1, 2, 4) = A(1, 2, 4) − A(1, 2)a

(3.11)

= − A(1, 2){a

A(1, 2, 4) + A(1, 2, 3, 4)}

4a

A

, i.e.

p−A(1, 2)A(0, 1, 2, 4) = − A(1, 2){a

A(1, 2, 4) + A(1, 2, 3, 4)}

2a

A

. (3.12)

Note that a

> 0, A(1, 2) > 0, A(1, 2, 4) < 0, A

> 0 and A(1, 2, 3, 4) < 0.

Again from (3.10),

(3.13) p−A(1, 2)A(0, 1, 2, 4) − A 4, 1, 2 0, 1, 2



= −a

A(1, 2, 4)A(1, 2) A

. In the same way

(3.14) p−A(1, 2)A(0, 1, 2, 4) + A 4, 1, 2 0, 1, 2



= − A(1, 2, 3, 4)A(1, 2) a

A

, whence Lemma 3.1 is proved.

Lemma 3.2.

W

= a

A(1, 2)A(1, 3, 4)

A(1, 3)A(1, 2, 3, 4) ,

(3.15)

W

= a

A(1, 2)A(2, 3, 4) A(2, 3)A(1, 2, 3, 4) . (3.16)

P r o o f. First remark a

= u

> 0, A

= u

u

u

(u

+ u

) < 0, A

= u

u

< 0 and A

= −u

u

(u

u

− u

u

) > 0. By the Jacobi identity

0 = A(0, 1, 2, 3, 4)A(0, 1, 3) (3.17)

= A(0, 1, 3, 4)A(0, 1, 2, 3) − A 0, 1, 3, 2 0, 1, 3, 4



= − A(0, 1, 3, 4)A(1, 2)a

− A 0, 1, 3, 2 0, 1, 3, 4



since A(0, 1, 2, 3) = −a

A(1, 2), whence

(3.18) A(0, 1, 3, 4) = − A

a

A(1, 2) . From (1.3) and the above,

(3.19) p−A(1, 3)A(0, 1, 3, 4) = s

A(1, 3) A(1, 2)

A

a

, where A

equals

(3.20) 1

2 A 1, 3, 2 1, 3, 4



− a

A

A

−

A

A(1, 2, 4) A

, in view of the Jacobi identities A

A

= −A(1, 2)A

and A(1, 2)A(1, 3) − A

= 0. Hence (3.21) p−A(1.3)A(0, 1, 3, 4) + A 4, 1, 3

0, 1, 3



= s

A(1, 3) A(1, 2)

A

a

+ a

A

A(1, 2)

= − A

A(1, 2)



− A(1, 3) a

− a



= A

A(0, 1, 3) a

A(1, 2) , since A

= A

A

− A(1, 2)A

and A(0, 1, 3) = A(1, 3) + a

. Sim- ilarly

(3.22) p−A(1, 3)A(0, 1, 3, 4) − A 4, 1, 3 0, 1, 3



= −a

A(1, 2)A(1, 3, 4)

A

.

Now (3.21) and (3.22) imply

(3.23) p−A(1, 3)A(0, 1, 3, 4) − A

p−A(1, 3)A(0, 1, 3, 4) + A

= − a

A(1, 2)

A(1, 3, 4) A

A(0, 1, 3)

= A(1, 2)A(1, 3, 4)a

A(1, 2, 3, 4)A(0, 1, 3) , which proves (3.15); (3.16) follows by symmetry.

Corollary.

W

/W

= A(1, 2)A(1, 3, 4) A(1, 3)A(1, 2, 4) , (3.24)

W

/W

= A(1, 2)A(2, 3, 4) A(2, 3)A(1, 2, 4) . (3.25)

As a result we have

Proposition (modified Schl¨ afli formula).

2dV

( b ∆) = dh1, 3i log(W

/W

) + dh2, 3i log(W

/W

) (3.26)

+ dh1, 4i log W

+ dh2, 4i log W

+ dh3, 4i log W

,

where W

/W

and W

/W

are given by (3.24)–(3.25) and W

are given by (3.6).

(2) Suppose that v

, v

∈ ∂D and v

, v

∈ D. Then A(1, 2, 3) = A(1, 2, 4) = 0, or equivalently h1, 2i + h2, 3i + h3, 1i = h1, 2i + h2, 4i + h4, 1i = π. One can choose as independent variables h1, 3i, h2, 3i, h1, 4i and h3, 4i, so that

2dV

( b ∆) = dh1, 3i log(W

W

/W

) + dh2, 3i log(W

W

/W

) (3.27)

+ dh1, 4i log(W

/W

) + dh3, 4i log(W

) .

We must express each coefficient on the right hand side as a function of a

, 1 ≤ j, k ≤ 4. As functions of µ,

(3.28) C.T.

 A(1, 2) A(0, 1, 2)



V

( b ∆

|µ) − 2 log 2 = − log A(1, 2) A(0, 1, 2) = 0 , i.e. W

= 1, since it is assumed that a

= a

= 0. As for W

, W

, Lemma 3.2 is valid. For W

and W

, similarly,

W

= a

A(1, 2)A(1, 4, 3) A(1, 4)A(1, 2, 3, 4) , (3.29)

W

= a

A(1, 2)A(2, 4, 3) A(2, 4)A(1, 2, 3, 4) . (3.30)

On the other hand, W

equals (3.6). (3.10) reduces to 2a

a

A

=

−A(1, 2, 3, 4). Hence

W

W

/W

= − 1 4

A(1, 2)A(1, 3, 4)A(2, 3, 4) A(2, 4)A(1, 3)A(1, 2, 3, 4) , (3.31)

W

W

/W

= − 1 4

A(1, 2)A(2, 3, 4)

A(2, 4)A(2, 3)A(1, 2, 3, 4) , (3.32)

W

/W

= A(2, 4)A(1, 3, 4) A(1, 4)A(2, 3, 4) , (3.33)

W

= A

− p−A(3, 4)A(1, 2, 3, 4)

3,4,1

3,4,2

+ p−A(3, 4)A(1, 2, 3, 4) , (3.34)

since A

= −A(1, 2)A(1, 2, 3, 4).

(3) We assume that v

, v

, v

∈ ∂D, and v

∈ D. Then A(1, 2, 3) = A(1, 2, 4)

= A(1, 3, 4) = 0, or equivalently h1, 2i + h2, 3i + h3, 1i = h1, 2i + h2, 4i + h4, 1i = h1, 3i + h3, 4i + h4, 1i = π. One can choose as independent variables h1, 2i, h1, 3i and h1, 4i. (3.7) reduces to

(3.35) 2dV

( b ∆) = dh1, 2i log(W

/(W

W

))

+dh1, 3i log(W

/(W

W

)) + dh1, 4i log(W

/(W

W

)) . By using the relation 2a

a

= −A(1, 2, 3, 4)/A

, (3.16) and (3.30), we deduce (3.36) below. (3.37) and (3.38) are obtained by symmetry.

W

/(W

W

) = − 4 A(2, 3)A(2, 4)A(1, 2, 3, 4) A(1, 2)A(2, 3, 4)

, (3.36)

W

/(W

W

) = − 4 A(2, 3)A(3, 4)A(1, 2, 3, 4) A(1, 3)A(2, 3, 4)

, (3.37)

W

/(W

W

) = − 4 A(3, 4)A(2, 4)A(1, 2, 3, 4) A(1, 4)A(2, 3, 4)

. (3.38)

(4) Case where all the vertices v

, v

, v

, v

∈ ∂D. Then A(i, j, k) vanishes for any i, j, k, or equivalently hi, ji + hj, ki + hk, ii = π. One can choose the vertices as v

= (ξ

, ξ

, ξ

), v

= (0, 1, 0), v

= (0, 0, −1), and v

= (0, 0, 1) respectively. The point (ξ

, ξ

, ξ

) in the unit sphere is related to the complex number z = x + iy by stereographic projection:

(3.39) ξ

= 2y

1 + |z|

, ξ

= 2x

1 + |z|

, ξ

= 1 − |z|

1 + |z|

. Then from (2.2) and (2.5),

(3.40) dV

( b ∆) = X

1≤i<j≤4

dhi, jiW

,

where W

equals A(0, i, j)/A(i, j). Actually W

=

W

=

W

= 1, W

=

1 + |z|

, W

= (1 + |z|

)/|z|

and W

= 2(1 + |z|

)/|1 − z|

. Moreover, h1, 2i =

arg z, h2, 3i = arg z(z − 1) and h3, 1i = arg(1 − z). (3.4) becomes (3.41) dV

( b ∆) = 2(log |z|d arg(z − 1) − log |z − 1|d arg z) , i.e. V

( b ∆) is the Bloch–Wigner function represented by

(3.42) V

( b ∆) = 1 i



dilog z − dilog z + log |z| log 1 − z 1 − z

 .

This function and its polylogarithmic extension have been investigated by many authors (see [M1], [M2], [G2], [W], [Z]).

Summarizing all the results in Sections 1 and 3, we have

Theorem. For v

, . . . , v

∈ ∂D ∪ D, V

( b ∆) has a hyperlogarithmic (higher logarithmic) expansion:

(3.43) V

( b ∆) = X

∅⊂I1⊂...⊂Iν−2

A

R

∗

θ  ∅ I

 θ I

I



. . . θ I

I



V

( b ∆

) for n = 2ν − 1, and

(3.44) V

( b ∆) = X

∅⊂I1⊂...⊂Iν−1

A

R

∗

θ  ∅ I

 θ I

I



. . . θ I

I



V

( b ∆

)

for n = 2ν, where V

( b ∆

) and V

( b ∆

) are given by (3.26), (3.27), (3.35), (3.41) respectively. V

( b ∆

) is given by (1.8). I

= {i

, . . . , i

} denotes a subset of {1, 2, . . . , n + 1}.

4. Appendix. Appell’s hypergeometric integrals of type F

and the hyperbolic volume. The integral

(4.1) J (λ) = R

∆

Φf

f

f

f

dt

∧ dt

∧ dt

∧ dt

= 1

p−A(1, 2, 3, 4)

R

y1≥0,y2≥0,y3≥0,y4≥0

exp[−

yBy]

×y

y

y

y

dy

∧ dy

∧ dy

∧ dy

= 1 2

1

p−A(1, 2, 3, 4) Γ  λ

− λ

+ λ

+ λ

2

 Γ (λ

)

× R

η1≥0,η2≥0

η

η

(b

+ b

η

+ b

η

)

×(b

η

+ b

η

+ b

η

η

)

dη

∧ dη

with 2λ

= λ

+ λ

+ λ

− λ

, where B = ((b

))

denotes the inverse A

. By the definition we have the homogeneity

(4.2) J (λ|{b

%

%

}) = %

%

%

%

J (λ|{b

}) ,

for%

∈ C

. One can choose %

such that %

%

b

= −1, %

%

b

= %

%

b

=

%

%

b

= 1. For b

= −1, b

= b

= b

= 1, J (λ|{b

}) has an integral expression similar to Appell’s hypergeometric function of type F

(see [K1]):

(4.3) F

(α, β, γ, γ

| u, v) = X

l≥0,m≥0

(α)

(β)

(γ)

(γ

)

u

v

l!m!

for u = −b

, v = −b

, α = λ

, β = λ

, γ = 1 + (λ

+ λ

− λ

− λ

)/2 and γ

= λ

− λ

+ 1 respectively. They both satisfy the following holonomic system of partial differential equations (E ) (see [K3], Chap. XI):

(4.4) u(1 − u)R − v

T − 2uvS

+{γ − (α + β + 1)u}P − (α + β + 1)vQ − αβJ = 0 , (4.5) v(1 − v)T − u

R − 2uvS

+{γ

− (α + β + 1)v}Q − (α + β + 1)uP − αβJ = 0 for R = ∂

J/∂u

, S = ∂

J/∂u∂v, T = ∂

J/∂v

, P = ∂J/∂u and Q = ∂J/∂v.

The change of variables

(4.6) u = w

w

, v = (1 − w

)(1 − w

) ,

which we call the Burchnall–Chaundy transformation or simply B.C. transforma- tion has an integral representation associated with a line configuration (see [B2], and [K2] for an extension):

(4.7) F

(α, β, γ, γ

| w

w

, w

w

) = Γ (γ)Γ (γ

)

Γ (α)Γ (β)Γ (γ − α)Γ (γ

− β)

×

1

R

0 1

R

0

x

y

(1 − x)

(1 − y)

(1 − xw

)

(1 − yw

)

×(1 − w

x − w

y)

dx ∧ dy , where we put w

= 1 − w

and w

= 1 − w

. However, we do not know whether J (λ) itself is given by a similar representation through the B.C. transformation.

The holonomic system (E ) has an alternative expression, i.e., the Gauss–Manin connection by using the additional integrals ϕ(i, j) and e ϕ(1, 2, 3, 4). Indeed, we e put

ϕ(i, j) = e R

Φ dτ f

f

, (4.8)

ϕ(1, 2, 3, 4) = e R

Φ dτ

f

f

f

f

. (4.9)

Then as functions of the variables (a

))

, ϕ(∅), e ϕ(i, j), e ϕ(1, 2, 3, 4) satisfy e

a variational formula in closed form (Gauss–Manin connection (E

)) (see [A3],

Proposition 2.4

):

(4.10) d ϕ(∅) = e 1 2

X

i6=j

dhi, jiλ

λ

ϕ(i, j) , e (4.11) A(i, j)d ϕ(i, j) e

= dA k, i, j l, i, j



λ

λ

ϕ(1, 2, 3, 4) + da

ϕ(∅) e

+ λ



− dA  i, j k, j



ϕ(k, j) + dA e i, j k, i

 ϕ(k, i) e



+ λ



− dA i, j l, j



ϕ(l, j) + dA e i, j l, i

 ϕ(l, i) e

 , (4.12) A(1, 2, 3, 4)d ϕ (1, 2, 3, 4) = e 1

2 X

i6=j

(−1)

dA k, i, j l, i, j

 ϕ(i, j) e + 1

2 dA(1, 2, 3, 4){−1 + λ

+ λ

+ λ

+ λ

} ϕ(1, 2, 3, 4) , e with the fundamental relations

(4.13) 0 = λ

ϕ(1, 2, 3, 4) − e

4

X

k=1,k6=j

b

ϕ(j, k) , e

for each j, 1 ≤ j ≤ 4. Hence ϕ(1, 2, 3, 4), e ϕ(1, 4), e ϕ(2, 4) and e ϕ(3, 4) are expressed e by linear combinations of ϕ(1, 2), e ϕ(2, 3) and e ϕ(3, 1): e

2λ

b

ϕ(1, 4) e (4.14)

= (λ

+ λ

+ λ

− 1)b

ϕ(2, 3) + (λ e

+ λ

− λ

− λ

) ϕ(1, 3) e + (λ

+ λ

− λ

− λ

)b

ϕ(1, 2) , e etc.

The volume V

( b ∆) given by the formula

(4.15) y R

η1≥0,η2≥0

(1 + b

η

+ b

η

)

(η

+ η

− η

η

)

dη

∧ dη

is a special case of the hypergeometric integrals of Appell’s type F

for α = β = γ = γ

= 1. The equations (E

) reduce to (3.41).

The B.C. transformation gives

(4.16) w, w = 1 + b

− b

± pB(1, 2, 3, 4) 2

for b

= 0, b

= −1, b

= b

= b

= 1 and b

= − 1 − ξ

2(1 + ξ

) , b

= − 1 − ξ

2(1 + ξ

) .

B(1, 2, 3, 4) equals 1 + b

+ b

+ 2b

+ 2b

− 2b

b

= −ξ

/(1 + ξ

)

= y

.

On the other hand,

(4.17) z = ξ

+ iξ

1 + ξ

= 1 − 1 w . Hence the B.C. transformation

(4.18) ww = −b

, (1 − w)(1 − w) = −b

is the composite of the linear fractional transformation (4.17) and the correspon- dence (3.39) between the configuration matrix B and the point z ∈ C which rep- resents the vertex v

.

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