Gauss hypergeometric equation as a representation of the formal KZ ...

Acknowledgment
The author expresses his deep gratitude to Professor Kimio Ueno. Professor
Ueno is his academic supervisor and undertook the chief examiner for this
thesis. Without his warm and careful mentorship, this thesis would not attain
completion.
He is also grateful to Professor Kiichiro Hashimoto, Professor Yumiko Hironaka
and Professor Jun Murakami for assuming second readers on examining
this thesis and for giving many useful advices. Furthermore, Professor
Hironaka gave the chances of speaking on the main topic of this thesis to the
author in twice.
He thanks Doctor Jun-ichi Okuda and members of Ueno’s laboratory for
valuable advice and discussions.
Shu Oi
Major in Mathematical Science
Graduate School of Science and Engineering
Waseda University
3-4-1, Okubo Shinjuku-ku
Tokyo 169-8555, Japan
e-mail: shu oi@toki.waseda.jp

Contents
1 Multiple zeta values, the formal KZ equation and Gauss hypergeometric
equation 7
1.1 Multiple zeta values and multiple polylogarithms of one variable 7
1.1.1 Multiple zeta values . . . . . . . . . . . . . . . . . . . 7
1.1.2 Multiple polylogarithms of one variable . . . . . . . . . 8
1.1.3 The free shuffle algebra h generated by letters x, y . . . 9
1.1.4 The double shuffle relation of MZVs . . . . . . . . . . . 11
1.2 The formal KZ equation . . . . . . . . . . . . . . . . . . . . . 12
1.2.1 The moduli space M0,n and the cubic coordinate . . . 12
1.2.2 The formal KZ equation on M0,n . . . . . . . . . . . . 13
1.2.3 The formal KZ equation of one variable . . . . . . . . . 14
1.2.4 The normalized fundamental solution of the formal KZ
equation of one variable and Drinfel’d associator . . . . 15
1.3 Gauss hypergeometric equation . . . . . . . . . . . . . . . . . 17
1.3.1 Gauss hypergeometric equation and its solution . . . . 18
1.3.2 Existing researches about Gauss hypergeometric equation
and MZVs . . . . . . . . . . . . . . . . . . . . . . 19
2 Gauss hypergeometric equation as a representation of the
formal KZ equation 21
2.1 Representations of the formal KZ equation . . . . . . . . . . . 21
2.2 Analytic properties of MPLs . . . . . . . . . . . . . . . . . . . 22
2.3 One dimensional representations of the formal 1KZ equation . 25
2.4 Gauss hypergeometric equation as a representation of the formal
1KZ equation . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 The expression of the hypergeometric function by MPLs . . . 27
2.5.1 Definitions and Notations . . . . . . . . . . . . . . . . 27
2.5.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.3 The image of word in H by the representation ρ0 . . . . 29
2.5.4 Asymptotic properties of ρ0(H0) and Φ0 . . . . . . . . 32
2.5.5 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . 32
iii

iv
2.6 The connection formula between the regular solutions to Gauss
hypergeometric equation at z = 0 and z = 1 . . . . . . . . . . 34
2.6.1 The inverse of the fundamental solution matrix on the
neighborhood of z = 1 . . . . . . . . . . . . . . . . . . 34
2.6.2 The expansion of the connection matrix as a series of
the zeta values . . . . . . . . . . . . . . . . . . . . . . 36
2.6.3 Functional relations obtained from the (1, 1)-element
of the connection formula (2.21) . . . . . . . . . . . . . 39
2.6.4 Various examples of functional relations of MPLs . . . 43
2.7 Functional relations derived from the connection formula between
irregular solutionsChapter. 22.7. FUNCTIONAL RE-
LATIONS DERIVED FROM THE CONNECTION FORMULA
BETWEEN IRREGULAR SOLUTIONSCHAPTER. 22.7 FUNC-
TIONAL RELATIONS DERIVED FROM Z = 0, ∞CHAPTER.
22.7 FUNCTIONAL RELATIONS DERIVED FROM Z =
0, ∞2.7 Functional relations derived from z = 0, ∞ . . . . . . 50
2.7.1 The fundamental solution matrix on the neighborhood
of z = ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.7.2 The functional relations derived from the (1, 1)-element
of the connection formula between z = 0 and ∞ . . . . 56
3 General representations and many variable formal KZ equations
63
3.1 Generalized hypergeometric equations and multiple zeta values 63
3.2 Representations of the formal generalized 1KZ equation and
MZVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.1 The formal generalized KZ equation of one variables . . 66
3.2.2 Representations of the formal generalized 1KZ equation 67
3.3 The formal KZ equation of two variables . . . . . . . . . . . . 68
3.3.1 The formal KZ equation of two variables . . . . . . . . 68
3.3.2 The reduced bar algebra and iterated integral of two
variables . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3.3 The normalized fundamental solution to the formal
2KZ equation . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.4 Decomposition theorem of the normalized fundamental
solution and the generalized harmonic product relation 73
3.3.5 Representations of the formal 2KZ equation . . . . . . 75

Introduction
In this thesis, we discuss relationship between Gauss hypergeometric equation
and multiple zeta values through the viewpoint of representations of the
formal KZ equation of one variable based on [O]. We regard Gauss hypergeometric
equation as a representation of the formal KZ equation and apply
algebraic theory of the formal KZ equation and multiple zeta values. And
then, comparing the results with the connection formulas of the solutions to
Gauss hypergeometric equation, we obtain various relations of the multiple
polylogarithms and multiple zeta values.
Multiple zeta values are real numbers defined as the series
ζ(k1, . . . , kr) =
1
n k1
1 · · · nkr r
n1>n2>···>nr>0
for index (k1, . . . , kr) of positive integers (k1 > 1). They are expansions of
Riemann zeta values (that is, the case of r = 1). Multiple zeta values of the
case of r = 2 are introduced and researched by Euler, and in 1990s, they
are re-discovered by Hoffman and Zagier. Since then, algebraic properties
of them have been studied actively. There exists various relations among
multiple zeta values, and obtaining all the relations among multiple zeta
values and finding the structure of Q-vector space spanned by all multiple
zeta values are great aims on the number theory. For these purposes, Ihara-
Kaneko-Zagier([IKZ]) and Racinet([R]) conjectured that all the Q-linearly
relations of multiple zeta values can be obtained from the regularized double
shuffle relation of multiple zeta values.
Multiple zeta values are regarded as the limit of multiple polylogarithms
zn1 Lik1,...,kr(z) =
n1>n2>···>nr>0
n k1
1 · · · n kr
r
as z tends to 1−0 and considering differential equations satisfied by multiple
polylogarithms is very useful to study multiple zeta values. Moreover, multiple
polylogarithms are very interesting subject in themselves. For instance,
1

2 Introduction
they have various functional relations such as Euler’s inversion formula, Landen’s
formula and five term relations of di-logarithms (that is, the case of
r = 1, k1 = 2).
The formal Knizhnik-Zamolodchikov equation (the formal KZ equation,
for short) , introduced by Drinfel’d([D]), is a differential equation on the
moduli space M0,n of the configuration space of n points on P 1 over the
action of PGL(2, C). The formal KZ equation is derived from the equation
satisfied by correlation functions on the conformal field theory and appears
on the knot theory because their monodromy representation yields the braid
group([Kon],[LM]). For n = 4, the formal KZ equation on M0,4 is regarded
as an equation of one variable on P 1 ;
dG
dz =
X
z
Y
+ G,
1 − z
where X, Y are non-commutative formal elements. Recently algebraic properties
of the formal KZ equation of one variable and its solutions are explored
([D],[G2],[IKZ],[OkU] and so on). Especially, by means of the theory of iterated
integrals and free shuffle algebras, the generating function of all multiple
polylogarithms with indeterminate elements X, Y becomes the fundamental
solution to the formal KZ equation of one variable normalized at z = 0, and
the connection coefficient of certain solutions, referred to as Drinfel’d associator,
is expressed as the generating function of all the multiple zeta values.
Drinfel’d associator plays a very important role on the arithmetic geometry.
In this context, Deligne-Terasoma([DT]) and Furusho([F]) showed that the
pentagon and hexagon relations of Drinfel’d associator leads the double shuffle
relation of multiple zeta values. Hence it is conjectured that the relations
of Drinfel’d associator yields all the Q-linearly relations of multiple zeta values.
This conjecture suggests that all the relations of multiple zeta values
can be interpreted as a connection problem of the formal KZ equation.
Furthermore, one can define the iterated integral of many variables by
using the Orlik-Solomon algebra of hyperplane arrangements and Chen’s reduced
bar construction([C2],[Koh],[OT],[B]) and can discuss the algebraic
structure of the formal KZ equation of many variables and the relationship
to multiple zeta values([B],[OU]). For multiple zeta values, it suffice to consider
the formal KZ equations of one and two variables, however, the formal
KZ equation of three or more variables is also important in the sight of a
study of differential equations. It is thought of as corresponding bigger class
of relations containing multiple L values and so on.

On the other hand, the formal KZ equation of one variable is the universal
Fuchsian equation which has three regular singular points at 0, 1, ∞ on
P 1 . Thus we can obtain specific Fuchsian equation by replacing the formal
elements X, Y with certain square matrix (that is, taking a representation).
On these specific equations, algebraic properties of the formal KZ equation is
inherited restrictively via representations. And then, under representations,
one can consider analytic manipulations (taking limit, infinite sum, specializing
indeterminate elements, and so on) to relations derived from the formal
1KZ equation. Moreover if the specific Fuchsian equation has certain analytic
properties, such as an integral expression of the solution, an connection
formula and so on, we can obtain various non-trivial functional relations of
multiple polylogarithms and relations of multiple zeta values by combining
these analytic properties and the algebraic theory on the formal KZ equation.
However, previously synthesizing studies on Fuchsian equations as a
representation of the formal KZ equation are not made.
The most important and elemental representation of the formal KZ equation
is Gauss hypergeometric equation
z(1 − z) d2f df
+ (γ − (α + β + 1)z) − αβf = 0,
dz2 dz
where α, β, γ are complex parameters. It is a general form of second order
Fuchsian equations which have three regular singular points on P 1 . Furthermore
the integral expression of the solution, which is known as Gauss
hypergeometric function, and the connection formulas are well known explicitly
and completely ([WW],[Ha] and so on). Especially, the connection
coefficients can be expressed as a factor of Gamma functions.
In previous time, Terasoma([T1]) said that values of Selberg type integrals
(they contain the limit of Gauss hypergeometric function as z tends to 1) can
be written as series of multiple zeta values, and Ohno-Zagier([OZ]) showed
that the generating function of sum of multiple zeta values fixed weight,
depth and height satisfies Gauss hypergeometric equation and derived various
relations of multiple zeta values. However, these existing research did not
refer to the viewpoint of a representation of the formal KZ equation and of a
connection problem of a differential equation. Additionally, even though it is
well known that Gauss hypergeometric function can be expressed as a series
of multiple polylogarithms by solving hypergeometric equation by successive
integration([AoK]), but the explicit expression is not given yet.
Compared with these existing researches, on [O], the author suggested the
framework of obtaining relations of multiple polylogarithms by the methods
above via representations of the formal KZ equation and establish the way to
3

4 Introduction
obtain relations systematically. Moreover by regarding Gauss hypergeometric
equation as a representation of the formal KZ equation and by calculating
the representation of the fundamental solution to the formal KZ equation,
the author succeeded in obtaining the expansion of Gauss hypergeometric
function as a series of multiple polylogarithms. This expansion is effected
on the universal covering space of P 1 − {0, 1, ∞} and can apply to the connection
problem of solutions not only at z = 0, 1 but also at z = ∞. The
author derived new functional relations of multiple polylogarithms from the
connection formula of Gauss hypergeometric equation between solutions at
z = 0 and z = 1, and solutions at z = 0 and z = ∞. Since relations of
multiple zeta values derived the limit as z tends to 1 − 0 of them restore
Ohno-Zagier’s result, these results gives the expansion of Ohno-Zagier’s result
to functional relations of multiple polylogarithms and the interpretation
of them on the theory of differential equations. Furthermore since connection
formulas between z = 0, 1 and between z = 0, ∞ are clearly independent algebraically,
the relations of multiple zeta values derived from z = 0, ∞ might
differ essentially from Ohno-Zagier relation.
This thesis is organized as follows. This thesis consists of three chapters.
Chapter 1 is an introduction of basic knowledges and existing results. In
Section 1.1, we define multiple zeta values and multiple polylogarithms, and
show some famous relations of multiple zeta values. We also give the iterated
integral expression of multiple polylogarithms and multiple zeta values by using
the free shuffle algebra. In Section 1.2, we review the formal KZ equation
(especially the formal KZ equation of one variable), its normalized fundamental
solutions and Drinfel’d associator. In Section 1.3, we outline Gauss
hypergeometric equation and introduce existing researches on the equation
and multiple zeta values, in particular Ohno-Zagier’s results.
Chapter 2 is the main part of this thesis. In this chapter, we discuss a
relationship between Gauss hypergeometric equation and multiple polylogarithms
from a viewpoint of a representation of the formal KZ equation based
on [O]. In Section 2.1, we define a representation of the formal KZ equation
and its fundamental solutions, and in Section 2.2 we discuss analytic properties
of multiple polylogarithms (Proposition 1) in order to show convergence
of representations of the fundamental solution. In Section 2.3, we express
one dimensional representations of the formal KZ equation. They are the
simplest examples of representations, but these representations are trivial
and there is no non-trivial relation of multiple zeta values.
In Section 2.4, we redefine Gauss hypergeometric equation as a representation
of the formal KZ equation, and in Section 2.5, we calculate the
representation of the fundamental solution normalized at z = 0 concretely

and derive the expansion of Gauss hypergeometric function as a series of multiple
polylogarithms (Theorem 5). This is the first main result of this thesis.
This expansion is regarded as the iterated integral expression of Gauss hypergeometric
function.
Next, in Section 2.6, we apply the same method of Section 2.5 to the
representation of the fundamental solution normalized at z = 1 (Proposition
12), and consider the connection formula between the regular solution
at z = 0 and z = 1. Then we obtain various functional relations of multiple
polylogarithms (Theorem 15) as the second main result of this thesis.
Furthermore, we have various relations of multiple zeta values, as known as
Ohno-Zagier relation, as the limit of these functional relations. Specializing
these relations, we show also some interesting relations, for instance the sum
formula of multiple polylogarithms (Proposition 17), which is the functional
relation of multiple polylogarithms extending the sum formula of multiple
zeta values.
For the connection formula between singular solutions or solutions on
z = ∞ are generally complicated. However, we describe partially the relations
derived from the connection formulas between the solutions on z = 0
and z = ∞ (Proposition 22 and Corollary 23) in Section 2.7. These relations
contains the expression of the values of Riemann zeta function on even integers
by using Bernoulli numbers and expressions of multiple zeta values such
as ζ(m, 1, . . . , 1) as a polynomial of Riemann zeta values, and are thought of
as different relations from Ohno-Zagier relation.
The main theme of this thesis, a relationship between Gauss hypergeometric
equation and multiple zeta values via a representation of the formal
KZ equation, is the first step of the investigation on the representation theory
of the formal KZ equation. We mention general representations of the
formal KZ equation of one variable and of the formal KZ equation of two
variables in Chapter 3. In Section 3.1, we consider the representations of the
formal KZ equation of one variable with respect to the generalized hypergeometric
equations and review existing works on these equations and multiple
zeta values. In Section 3.2, we introduce the formal generalized KZ equation
of one variable, which is the universal Fuchsian equation with many regular
singular points on P 1 , and their representations. In Section 3.3, we discuss
algebraic properties of the formal KZ equation of two variables according to
[OU], which is a subsequent work of the author after [O], and explain its
representations such as Appell hypergeometric equations. These topics are
under investigation, but very important subjects to apply our research.
5

Chapter 1
Multiple zeta values, the formal
KZ equation and Gauss
hypergeometric equation
In this chapter, we preliminary recall the fundamental properties and known
results on multiple zeta values, multiple polylogarithms, shuffle algebras, the
formal KZ equation and Gauss hypergeometric equation. Algebraic aspects
of multiple polylogarithms and the formal KZ equation of one variable play
essential role in this thesis.
1.1 Multiple zeta values and multiple polylogarithms
of one variable
1.1.1 Multiple zeta values
Multiple zeta values (MZVs, for short) are real numbers associated with the
index of positive integers (k1, k2, . . . , kr), k1 ≥ 2 defined as
ζ(k1, k2, . . . , kr) =
n1>···>nr>0
1
n k1
1 · · · n kr
r
. (1.1)
We call k1 + · · · + kr the weight and r the depth of index (k1, . . . , kr). If
r = 1, the multiple zeta values ζ(k1) are nothing but the values of Riemann
zeta function at positive integers. There are many relations among multiple
zeta values, for instance;
(i) The sum formula (Granville[Gr], Zagier[Z2])
7

8 Chapter. 1
For all k > r ≥ 1,
(ii) The duality formula
k1+···+kr=k
k1≥2,ki≥1
ζ(k1, . . . , kr) = ζ(k). (1.2)
For an index k = (k1, . . . , kr) = (a1 + 1, 1, . . . , 1 , . . . , as + 1, 1, . . . , 1 )
b1−1 times
bs−1 times
(ai, bi ≥ 1), we define the dual index k ′ as
Then we have
(iii) Ohno relation ([Oh])
k ′ = (bs + 1, 1, . . . , 1 , . . . , b1 + 1, 1, . . . , 1 ).
as−1 times
a1−1 times
ζ(k) = ζ(k ′ ). (1.3)
For all index k = (k1, . . . , kr), k1 ≥ 2, we denote the dual index by
k ′ = (k ′ 1, . . . , k ′ r ′). Then
ζ(k1+c1, . . . , kr+cr) =
ζ(k ′ 1+c ′ 1, . . . , k ′ r ′ +c′ r ′) (1.4)
c1+···+cr=l
c ′ 1 +···+c′
r ′=l
holds. These relations include the sum formula and duality formula.
We denote by Zk (k > 1) the Q vector space spanned by MZVs whose
weight is k, and define Z0 = Q, Z1 = 0, Z = ∞ k=0 Zk. It is conjectured
by Zagier and others that Z = ∞ k=0 Zk and dim Zk = dk where the sequence
{dk} is defined by the recurring formula d0 = d2 = 1, d1 = 0, dk =
dk−2 + dk−3. Goncharov([G1]) and Terasoma([T2]) showed dim Zk ≤ dk in
the context of mixed Tate motives.
1.1.2 Multiple polylogarithms of one variable
We define the multiple polylogarithms of one variable (MPLs, for short) as
Lik1,...,kr(z) =
n1>···>nr>0
z n1
n k1
1 · · · n kr
r
. (1.5)
These series converge absolutely on |z| < 1. Moreover, if k1 > 1, they
converge also at z = 1 and the limit values become MZVs,
lim
z→1−0 Lik1,...,kr(z) = ζ(k1, . . . , kr). (1.6)

1.1 MVZs,MPLs 9
Multiple polylogarithms have the iterated integral expressions
z
Lik1,...,kr(z) =
0
where the iterated integral
by
z
z0
dz dz dz dz dz dz
◦ · · · ◦ ◦ · · · ◦ ◦ · · · ◦ ,
z z
1 − z z z
1 − z
(1.7)
k1−1 times
kr−1 times
z
ω1 ◦ · · · ◦ ωr =
z0
ω1 ◦ · · · ◦ ωr, (ωi’s are 1-forms of dz) is defined
z
z0
ω1(z ′ z ′
)
z0
ω2 ◦ · · · ◦ ωr (r > 1)
inductively. By using the iterated integral expression, multiple polylogarithms
Lik1,...,kr(z) can be continued as many-valued analytic function on
P 1 − {0, 1, ∞} (for detail, see Section 2.2).
We can also obtain the iterated integral expression of MZVs:
ζ(k1, . . . , kr) =
1
0
dz dz dz dz dz dz
◦ · · · ◦ ◦ · · · ◦ ◦ · · · ◦ ,
z z
1 − z z z
1 − z
(1.8)
k1−1 times
kr−1 times
where k1 ≥ 2 and the duality formula of MZVs can be viewed as the transformation
of the variable t = 1 − z by the iterated integral expression.
1.1.3 The free shuffle algebra h generated by letters
x, y
Let A be a set of letters and C〈A〉 a non-commutative polynomial algebra
of letters A over C. We denote by 1 the unit of C〈A〉 (empty word) and ◦
the product of C〈A〉 by concatenation (we will omit it if there is no worry
of confusing). We define the shuffle product x on C〈A〉 inductively as
w x 1 = 1 x w = w,
(a1 ◦ w1) x (a2 ◦ w2) = a1 ◦ (w1 x (a2 ◦ w2)) + a2 ◦ ((a1 ◦ w1) x w2)
where w, w1, w2 are words of C〈A〉 and a1, a2 are letters in A. Then S(A) =
(C〈A〉, x, 1) becomes an associative and commutative algebra due to Reutenauer([Re]).
This is referred to as the free shuffle algebra generated by A. The free shuffle
algebra S(A) has a grading S(A) = ∞ s=0 Ss(A) with respect to the length
of words and has a Hopf algebra structure with respect to the coproduct ¯ ∆,
the counit ¯ε and the antipode ¯ S defined as

10 Chapter. 1
¯∆(a1 ◦ · · · ◦ ar) =
r
(a1 ◦ · · · ◦ ai) ⊗ (ai+1 ◦ · · · ◦ ar), (1.9)
i=0
¯ε(a1 ◦ · · · ◦ ar) = 0, (1.10)
¯S(a1 ◦ · · · ◦ ar) = (−1) r ar ◦ · · · ◦ a1
(1.11)
for a1, . . . , ar ∈ A, where we assume that a1 · · · a0 = ar+1 · · · ar = 0. For
details of Hopf algebras, see [K].
For A = {x, y}, we denote by h = S({x, y}) the free shuffle algebra
generated by letters x, y and h0 = C1 + hy (resp. h10 = C1 + xhy) the
subspace of h spanned by elements ended with y (resp. started with x and
ended with y). Clearly h0 and h10 are both x-subalgebra of h. Furthermore h
is regarded as a polynomial algebra of x over h0 (resp. a polynomial algebra
of x, y over h10 ) under x multiplication, namely
∞
h = h 0 x x n (= h 0 [x]) (1.12)
=
n=0
∞
m,n=0
h 10 x x m x y n (= h 10 [x, y]), (1.13)
where an stands for a x · · · x a
= n!a
n times
n . By this isomorphism, we define the
regularization map regi : h → hi (i = 0, 10) as follows;
reg 0 (w) = the constant term of w in the decomposition (1.12), (1.14)
reg 10 (w) = the constant term of w in the decomposition (1.13). (1.15)
The regularization map reg0 satisfies the following properties([IKZ]);
wx n n
= reg 0 (wx n−j ) x x j
for w ∈ h 0 , (1.16)
j=0
reg 0 (wyx n ) = (−1) n (w x x n )y for n ≥ 0, w ∈ h. (1.17)
Under these notations, we identify the letters x and y as 1-forms dz
z and
dz , and the iterated integral
1−z
z
0
as a linear map from h 0 to the algebra of
analytic functions (we define 1 = 1 the constant function). From properties
of iterated integrals, the map
z
0
z
0
w1 x w2 =
is a x-homomorphism, namely
z
0
w1
z
0
w2

1.1 MVZs,MPLs 11
for all words w1, w2 ∈ h 0 . Thus multiple polylogarithms Li can be regarded
as a homomorphism from h 0 to the algebra of analytic functions on P 1 −
{0, 1, ∞}. We denote by
Li(x k1−1 y · · · x kr−1 y; z) = Lik1,...,kr(z) (1.18)
via the iterated integral. We also express ζ(k1, . . . , kr) = ζ(x k1−1 y · · · x kr−1 y)
by MZVs and regard ζ as a homomorphism from h 10 to R.
We extend the domain of Li(•; z) from h 0 to h via Li(x; z) = log z and
x-homomorphism. That is, by virtue of (1.16),
Li(wx n ; z) =
n
j=0
Li(reg 0 (wx n−j ); z) logj z
j!
(1.19)
for w ∈ h 0 . These extended MPLs are also many-valued analytic functions on
P 1 − {0, 1, ∞} and satisfy the following differential recursive relations([Ok]);
d Li(xw; z)
dz
d Li(yw; z)
dz
= 1
Li(w; z),
z
(1.20)
= 1
Li(w; z).
1 − z
(1.21)
1.1.4 The double shuffle relation of MZVs
The iterated integral formulation of MPLs induce the (integral) shuffle product
of MZVs;
ζ(w1 x w2) = ζ(w1) x ζ(w2) (1.22)
for any words w1, w2 ∈ h 10 . Furthermore MZVs have another product-sum
relation referred to as the harmonic product (or series shuffle product) defined
by the following due to Hoffman([Ho]).
We denote by χk = x k−1 y and w = χk1 · · · χkr the word of h 0 . The
harmonic product ∗ of h 0 is defined by
w ∗ 1 = 1 ∗ w = w, (1.23)
(χk1w1) ∗ (χk2w2) = χk1(w1 ∗ (χk2w2)) + χk2((χk1w1) ∗ w2)
+ (χk1+k2)(w1 ∗ w2) (1.24)
inductively for any words w, w1, w2 ∈ h 0 . Then (h 0 , ∗, 1) becomes an associative
and commutative algebra and the relation

12 Chapter. 1
ζ(w1 ∗ w2) = ζ(w1)ζ(w2) (1.25)
holds. This is a generalization of the relations via the series expression
ζ(k1)ζ(l1) =
=
1
n n1>0
k1
1
n1>m1>0
m1>0
1
m l1
1
+
n1=m1>0
+
m1>n1>0
= ζ(k1, l1) + ζ(k1 + l1) + ζ(l1, k1).
1
n k1
1 m l1
1
We provide an interpretation of the harmonic product as iterated integrals
in Section 3.3.4.
Combining the shuffle and harmonic product of MZVs, one can obtain
the non-trivial Q-linear relations among MZVs referred to as the regularized
double shuffle relation([IKZ]);
ζ(reg 10 (w1 x w2 − w1 ∗ w2)) = 0 (1.26)
for w1 ∈ h 0 , w2 ∈ h 10 . Ihara-Kaneko-Zagier([IKZ]) and Racinet([R]) conjectured
that the regularized double shuffle relations contain the all Q-linear
relations of MZVs.
1.2 The formal KZ equation
In this section, we introduce the formal KZ equation. First we discuss the
formal KZ equation on the moduli space M0,n. Next, we define the formal KZ
equation of one variable as the case of M0,4. Since the formal KZ equation
of one variable plays essential roles in Chapter 2, we review the algebraic
properties of the formal KZ equation of one variable carefully according to
[OkU]. We note that the case of M0,5 (the formal KZ equation of two
variables) appears in Section 3.3.
1.2.1 The moduli space M0,n and the cubic coordinate
Let (P 1 ) n ∗ be a configuration space of n points on P 1
(P 1 ) n ∗ = {(x1, . . . , xn) ∈ (P 1 ) n | xi = xj (i = j)}. (1.27)
We denote by M0,n = PGL(2, C)\(P 1 ) n ∗ the moduli space of (P 1 ) n ∗ over the
actions

1.2 The formal KZ equation 13
xi ↦→ axi
a
of
c
b
d
∈ PGL(2, C).
+ b
(1.28)
cxi + d
We introduce the simplicial coordinate system
(y1, . . . , yn−3) of M0,n via
yi = xi − xn−2
xi − xn
xn−1 − xn
xn−1 − xn−2
and the cubic coordinate system (z1, . . . , zn−3) via
z1 = y1, zi = yi
yi−1
(1 ≤ i ≤ n − 3) (1.29)
(2 ≤ i ≤ n − 3) (1.30)
due to Brown([B]). Since the cross ratio yi is an invariant under the action
of linear fractional transformation, the simplicial and the cubic coordinate
define the coordinate system of M0,n. Roughly speaking, the simplicial coordinate
means fixing the three points yn−2 = 0, yn−1 = 1 and yn = ∞, and
the cubic coordinate means blowing up on the origin to be a normal crossing
point of the divisors.
1.2.2 The formal KZ equation on M0,n
We consider the Pfaffian system on (P 1 ) n ∗
dG = ΩG, Ω =
1≤i

14 Chapter. 1
due to Arnold([A]). By using the IPBR and the AR, one can show that the
equation (1.31) is integrable and has PGL(2, C) invariance ([B], [OU]). Thus
it is an integrable system on M0,n.
Since the IPBR is homogeneous, X has a grading with respect to the
degree of Lie polynomials and the universal enveloping algebra U(X) =
⊕ ∞ s=0 Us(X) has also a grading with respect to the length of words. We denote
by I the unit of U(X) and U(X) the completion of U(X) with respect to this
grading. U(X) (resp. U(X)) has a Hopf algebra structure (resp. topological
Hopf algebra structure) by the coproduct ∆, the counit ε and the antipode
S defined as
for A ∈ X.
∆(A) = 1 ⊗ A + A ⊗ 1 algebra morphism, (1.34)
ε(A) = 0, algebra morphism (1.35)
S(A) = −A anti-algebra morphism (1.36)
1.2.3 The formal KZ equation of one variable
For n = 4, the formal KZ equation (1.31) is represented via the cubic coordinate
system as
dG = ΩG, Ω = ζ1Z1 + ζ11Z11, (1.37)
where z = z1, Z1 = X12, Z11 = −X13, ζ1 = dz
z and ζ11 = dz . We call this
1−z
the formal KZ equation of one variable (the formal 1KZ equation, for short).
The IPBR (1.32) for {Z1, Z11} is trivial, thus the infinitesimal pure braid
Lie algebra X is a free Lie algebra generated by Z1, Z11. The universal enveloping
algebra U(X) is a ring of non-commutative polynomials of the variables
Z1, Z11 namely U(X) = C〈Z1, Z11〉, and U(X) = C〈〈Z1, Z11〉〉 is an
algebra of the non-commutative formal power series.
We denote by B the dual Hopf algebra of U(X). This is a free shuffle
algebra generated by ζ1, ζ11. We remark that the AR (1.33) for {ζ1, ζ11} is
only ζ1 ∧ ζ11 = 0 and B is a 0-th cohomology of the reduced bar complex of
the exterior algebra generated by ζ1, ζ11 (see also Section 3.3.2).
Through the identification ζ1 = x, ζ11 = y, B is nothing but the free
shuffle algebra h introduced in Section 1.1.3. In what follows, we use the
notation h, x = dz
z
and y = dz
1−z instead of B, ζ1 and ζ11. We also denote by
X = Z1, Y = Z11 and H = U(X). In this notation, we can express the formal
KZ equation of one variable (1.37) as a H-valued Fuchsian equation
dG
dz =
X Y
+ G. (1.38)
z 1 − z

1.2 The formal KZ equation 15
We note that the Hopf algebra structure of H is given by (1.34),(1.35)
and (1.36) and the Hopf algebra structure of h is given by (1.9),(1.10) and
(1.11).
1.2.4 The normalized fundamental solution of the formal
KZ equation of one variable and Drinfel’d
associator
We consider a solution L(z) to the formal 1KZ equation (1.38) which have a
decomposition
L(z) = ˆ L(z)z X
(1.39)
where ˆ L(z) is holomorphic at z = 0 and ˆ L(0) = I. This is referred to as
the fundamental solution to (1.38) normalized at z = 0. Since the equation
(1.38) is Fuchsian, if there exists, the normalized fundamental solution is the
unique solution of (1.38) which has a asymptotic behavior L(z)z −X → I as
z tends to 0.
According to Okuda([Ok]), we define H-valued analytic function H0(z) as
H0(z) =
w:word of h
Li(w; z)W, (1.40)
where W is the word of H determined as replacing x, y in w by X, Y . (1.20)
and (1.21) imply that H0(z) satisfies the formal 1KZ equation (1.38). Furthermore,
by virtue of (1.16), one can calculate as
H0(z) =
=
=
=
=
w:word of h
Li(w; z)W =
∞
s
w:word of h0 s=0 j=0
∞
∞
w:word of h0 i=0 j=0
w:word of h
w:word of h
s∈Z≥0
w:word of h 0
Li(wx s ; z)W X s
Li(reg 0 (wx s−j ); z) logj (z)
j! W Xs−j X j
Li(reg 0 (wx i ); z) logj (z)
j! W Xi X j
Li(reg 0 (w); z)W
Li(reg 0 (w); z)W
∞
j=0
z X .
log j (z)
j! Xj

16 Chapter. 1
Since Li(reg 0 (w); z) is regular at z = 0 for all word w ∈ h and
Li(reg 0 (w); 0) =
1 (w = 1)
0 (w = 1)
(1.41)
holds, w:word of h Li(reg0
(w); z)W is regular at z = 0 and ˆ L(0) = I.
Therefore H0(z) is the unique fundamental solution to (1.38) normalized
at z = 0. The inverse of H0(z) is given as
(H0(z)) −1 =
w:word of h
Li( ¯ S(w); z)W (1.42)
by using the antipode ¯ S of h. Indeed one can show easily that
w1:word of h
=
w1,w2:word of h
=
w:word of h (w)
=
= I,
w:word of h (w)
Li(w1; z)W1
w2:word of h
Li(w1 x ¯ S(w2); z)W1W2
Li(w ′ x ¯ S(w ′′ ); z)W
Li(¯ε(w)1; z)W
Li( ¯ S(w2); z)W2
where the sum in the third expression stands for the Sweedler’s notation
¯∆(a) =
(a) a′ ⊗ a ′′ .
We remark that the fundamental solution normalized at z = 0 can be
constructed not to use the regularization map reg 0 as follows (for detail, see
[OU]). We define ˆ Ls(z) recursively as
ˆL0(z) = I, (1.43)
z
ˆLs+1(z)
1
=
z [X, ˆ Ls(z)] + 1
1 − z Y ˆ
Ls(z) dz. (1.44)
0
Thus we can show 1
z [X, ˆ Ls(z)] is holomorphic at z = 0 for all s by induction
on s, then ˆ Ls(z) is well-defined for all s and ˆ L(z) = ∞
s=0 ˆ Ls(z) is the
fundamental solution normalized at z = 0. In term of iterated integral,

1.2 The formal KZ eqaution 17
it is expressed as
ˆL(z) =
z
(k1,...,kr)
0
x k1−1 kr−1
◦ y · · · ◦ x ◦ y
× ad(X) k1−1 kr−1
µ(Y ) · · · ad(X) µ(Y )(I) (1.45)
=
(k1,...,kr)
Li(x k1−1 ◦ y · · · ◦ x kr−1 ◦ y; z)
× ad(X) k1−1 µ(Y ) · · · ad(X) kr−1 µ(Y )(I), (1.46)
where (k1, . . . , kr) runs over all indexes of positive integers and ad (resp.
µ) stands for an adjoint ad(A)(W ) = [A, W ] (resp. a left multiplication
µ(A)(W ) = AW ) for A ∈ {X, Y }, W ∈ H.
We also consider the fundamental solution normalized at z = 1, which is
the unique solution with an expression
L (1) (z) = ˆ L (1) (z)(1 − z) Y , (1.47)
where ˆ L (1) (z) is holomorphic at z = 1 and ˆ L (1) (1) = I. One can show the
function H1(z) below is the fundamental solution normalized at z = 1:
H1(z) =
w:word of h
Li(σ(w); 1 − z)W, (1.48)
where σ is an ◦-involution of h defined by σ(x) = −y, σ(y) = −x.
The connection coefficient between H0(z) and H1(z) is referred to as
Drinfel’d associator and denoted by ΦKZ(X, Y ) = (H1(z)) −1 H0(z). Drinfel’d
associator ΦKZ(X, Y ) is expressed as a generating function of MZVs;
ΦKZ(X, Y ) =
w:word of h
due to Le-Murakami([LM]) and Okuda-Ueno([OkU]).
ζ(reg 10 (w)) (1.49)
1.3 Gauss hypergeometric equation
In Section 1.3.1, we briefly review Gauss hypergeometric equation and Gauss
hypergeometric functions. Details for Gauss hypergeometric equation can
be found on [Ha], [Ki], [WW] and other texts. Next in Section 1.3.2, we
introduce some existing researches about Gauss hypergeometric equation and
MZVs.

18 Chapter. 1
1.3.1 Gauss hypergeometric equation and its solution
Gauss hypergeometric equation is a second order Fuchsian differential equation
expressed as
z(1 − z) d2f df
+ (γ − (α + β + 1)z) − αβf = 0, (1.50)
dz2 dz
where α, β, γ are complex parameters. The equation has three regular singular
points 0, 1, ∞ on P 1 .
In what follows, we assume that the parameter α, β, γ, α−β and γ −α−β
are not integers. Under this assumption, the linearly independent solutions
ϕ (i)
0 (z), ϕ (i)
1 (z) (i = 0, 1, ∞) on the neighborhoods of z = 0, 1, ∞ are given
as follows;
ϕ (0)
0 (z) = F (α, β, γ; z) =
∞
n=0
(α)n(β)n
(γ)nn! zn , (1.51)
ϕ (0)
1 (z) = z 1−γ F (α + 1 − γ, β + 1 − γ, 2 − γ; z), (1.52)
ϕ (1)
0 (z) = F (α, β, α + β + 1 − γ; 1 − z), (1.53)
ϕ (1)
1 (z) = (1 − z) γ−α−β F (γ − α, γ − β, γ − α − β + 1; 1 − z), (1.54)
ϕ (∞)
0 (z) = z −α F (α, α + 1 − γ, α − β + 1; 1/z), (1.55)
ϕ (∞)
1 (z) = z −β F (β, β + 1 − γ, β − α + 1; 1/z), (1.56)
where we define the branch of these complex power by the principal values
and (α)n stands for the Pochhammer symbol (α)n = (α + n − 1)(α + n −
2) · · · (α + 1)α. The function F (α, β, γ; z) is Gauss hypergeometric function.
It has Euler’s integral expression
F (α, β, γ; z) =
Γ(γ)
Γ(α)Γ(γ − α)
1
t
0
α−1 (1 − t) γ−α−1 (1 − zt) −β dt (1.57)
and is continued analytically to P1 − {0, 1, ∞} as a many-valued function.
We also regard to all ϕ (i)
j ’s (i = 0, 1, ∞, j = 0, 1) as many-valued functions
on P1 − {0, 1, ∞}.
The connection matrices of these solutions are given by the following

1.3 Gauss hypergeometric equation 19
formula via the Euler’s integral expression of hypergeometric function;
(ϕ (0)
0 , ϕ (0)
1 ) = (ϕ (1)
0 , ϕ (1)
1 )C 01 ,
C 01 ⎛
Γ(γ)Γ(γ − α − β)
⎜
=
Γ(γ − α)Γ(γ − β)
⎝
Γ(γ)Γ(α + β − γ)
Γ(α)Γ(β)
(ϕ (0)
0 , ϕ (0)
1 ) = (ϕ (∞)
0 , ϕ (∞)
1 )C 0∞ ,
C 0∞ ⎛
e
⎜
= ⎝
−πiα Γ(γ)Γ(β − α)
Γ(β)Γ(γ − α)
e−πiβ Γ(γ)Γ(α − β)
Γ(α)Γ(γ − β)
Γ(2 − γ)Γ(γ − α − β)
⎞
Γ(1 − α)Γ(1 − β)
Γ(2 − γ)Γ(α + β − γ)
Γ(α + 1 − γ)Γ(β + 1 − γ)
⎟
⎠ , (1.58)
eπi(γ−α−1) Γ(2 − γ)Γ(β − α)
Γ(β + 1 − γ)Γ(1 − α)
eπi(γ−β−1) ⎞
⎟
Γ(2 − γ)Γ(α − β)
⎠ .
Γ(α + 1 − γ)Γ(1 − β)
(1.59)
1.3.2 Existing researches about Gauss hypergeometric
equation and MZVs
The first result for the relation between Gauss hypergeometric equation and
MZVs is due to Ohno-Zagier([OZ]). For index k = (k1, . . . , kr), k1 > 1, we
call the number #{i|ki > 1} the height of k and denote by G(k, n, s; z) the
sum of MPLs which have fixed weight k, depth n and height s. Ohno-Zagier
showed that the generating function
Φ(z) =
n≥s≥1,k≥s+n
G(k, n, s; z)u k−n−s v n−s w s−1
satisfies Gauss hypergeometric equation and can express as
Φ(z) =
(1.60)
1
(1 − F (α − u, β − u, 1 − u, z)), (1.61)
uv − w
where α + β = u + v, αβ = w.
Taking the limit of (1.61) as z tends to 1 by using
Γ(γ)Γ(γ − α − β)
lim F (α, β, γ; z) =
z→1 Γ(γ − α)Γ(γ − β)
(1.62)
and the expansion of Gamma function by Riemann zeta values (see Section
2.6.2), they showed the relations known as Ohno-Zagier relation
Φ(1) =
1
(1 − exp(
uv − w
∞
n=2
ζ(n)
n Sn(u, v, w))), (1.63)

20 Chapter. 1
where Sn(u, v, w) = u n +v n −α n −β n , α+β = u+v, αβ = w. They obtained
also many relations of MZVs by specializing the Ohno-Zagier relation,for
instance the sum formula, ζ(2, . . . , 2)
=
l times
π2s and Le-Murakami formula.
(2s+1)!
Terasoma([T1]) said that the values of Selberg integral, which contains
the special values of Gauss hypergeometric function at z = 1, can be written
as a linear combination of MZVs. Ohno-Zagier’s result gives an example of
Terasoma’s assertion. In chapter 2, we show again the expansion of Φ(z) and
interpret as an iterated integral expression of Gauss hypergeometric equation.
Moreover we also interpret Ohno-Zagier’s result as a connection problem of
the formal KZ equation and extend to functional relations of MPLs.
The second result is the formula
ζ(3, 1, . . . , 3, 1)
=
2n times
2π4n
. (1.64)
(4n + 2)!
This formula is conjectured by Zagier and proved by Borwein-Bradley-Broadhurst-
Lisoněk ([BBBL]). They show that the generating function of Li3,1,...,3,1(z) is
expressed as
∞
Li3,1,...,3,1
n=0 2n times
(z)t 4n t −t t −t
= F ( , , 1; z)F ( , , 1; z). (1.65)
1 + i 1 + i 1 − i 1 − i
Indeed one can show that the both sides of the equation above are eliminated
by the action of
(1 − z) d
2
z
dz
d
2 − t
dz
4 . (1.66)
Taking the limit of (1.65) as z tends to 1 and using
we obtain (1.64).
lim F (a, −a, 1, z) =
z→1
1
Γ(1 − a)Γ(1 + a)
sin πa
= , (1.67)
πa

Chapter 2
Gauss hypergeometric equation
as a representation of the
formal KZ equation
This chapter is the main part of this thesis. We discuss Gauss hypergeometric
equation as a representation of the formal 1KZ equation, according
to [O]. We will obtain two main results. The first one is the expansion of
Gauss hypergeometric equation to a series of parameters whose coefficients
are multiple polylogarithms on the universal covering space of P 1 −{0, 1, ∞}.
The second result is system of functional relations of MPLs, which restore
the Ohno-Zagier relations of MZVs as the limit as z tends to 1. These relations
are followed from the connection formula between the regular solutions
of Gauss hypergeometric equation at z = 0 and z = 1. Furthermore, we will
lead various relations of MPLs with respect to the connection formula of the
solutions between z = 0 and z = ∞.
2.1 Representations of the formal KZ equation
Let X = C{X, Y } be a free Lie algebra of letters X, Y and ρ : X → M(n, C)
be a representation of X. We denote a corresponding representation of U(X)
by the same symbol ρ. We call the M(n, C)-valued differential equation
dG
dz =
ρ(X) ρ(Y )
+ G (2.1)
z 1 − z
21

22 Chapter. 2
a representation of the formal 1KZ equation by ρ and the formal sum
ρ(H0(z)) =
Li(w; z)ρ(W ) (2.2)
w
a representation of the fundamental solution H0(z) of the formal 1KZ equation.
In general, a representation of the fundamental solution does not converge,
but if the formal sum (2.2) converges absolutely and uniformly, it
gives the fundamental solution to the representation of the formal 1KZ equation
(2.1). Furthermore a representation of the fundamental solution can be
viewed as a solution expressed by the iterated integral.
2.2 Analytic properties of MPLs
In preparation for proving convergence of representations of the fundamental
solution, we discuss an analytic continuation of the multiple polylogarithms
of one variable to the universal covering space of P 1 − {0, 1, ∞}.
Proposition 1. Let U be the universal covering space of P 1 − {0, 1, ∞} and
K be a compact subset of U. There exists a constant MK depending only on
K such that, for any word w ∈ h,
| Li(w; z)| < MK ∀z ∈ K. (2.3)
A basic idea to prove this proposition is due to Lappo-Danilevsky on [L]
p.159-163.
Let π : U → P1 − {0, 1, ∞} be the canonical projection and the real
interval I be a simply-connected subset of U such that arg(z) = arg(1−z) = 0
on I. That is, on z ∈ I, Li(w; z) has an expansion (1.5) for all word w ∈ h0 and log z has an expansion log z = − ∞ (1−z)
n=1
n
. We note that, on z ∈ I,
n
Li(w; z) converges to 0 as z tends to 0 for any word w = 1 and log z converges
to 0 as z tends to 1.
Let z, p be points of U and Cz p be a path on U from p to z. For any word
w ∈ h, we define MPLs with prescribing an initial point and a path by an
inductive way such as
Lip,Cz (xw; z) =
p
Lip,Cz (yw; z) =
p
C z p
C z p
Lip,C z p
Lip,C z p
(w; z)
dz, (2.4)
z
(w; z)
dz, (2.5)
1 − z
Lip,Cz (1; z) = 1. (2.6)
p
These MPLs satisfy the following properties.

2.2 Analytic properties of MPLs 23
Lemma 2. ([L])
(i) For any word w ∈ h and points p, z ∈ U, the value of Lip,Cz (w; z) does
p
not depend on choice of a path Cz p on U (in other words, the value of
Lip,Cz (w; z) depends only a homotopy class of the integral contour on
p
P1 − {0, 1, ∞}).
(ii) Let δ be the distance between π(C p z ) and {0, 1};
δ = dist(π(C p z ), {0, 1}) = inf
z1∈π(C p z )
z2∈{0,1}
and σ be the length of the path π(C z p). Then we have
| Lip,Cz 1
(w; z)| < p |w|!
|z1 − z2|, (2.7)
σ
|w|
, (2.8)
δ
where |w| stands for the length of the word w (that is, the number of
letters in w).
The next lemma, which is also due to [L], follows from the coproduct
structure of h (1.9) as a Hopf algebra.
Lemma 3. Let w = a1a2 · · · ar be a word of h 0 , where each ai denotes the
letter x or y (i = 1, . . . , r, ar = y), and C z p be a path from p to z on U.
Choosing a point q on Cz p, we divide the path Cz p as Cz p = Cz q ◦ Cq p. Then
(w; z) satisfies
Lip,C z p
Lip,Cz (w; z) =
p
r
i=0
Liq,C z q (a1 · · · ai; z) Li p,C q p (ai+1 · · · ar; q). (2.9)
Here we use a convention such as Liq,C z q (a1 · · · ai; z) = 1 for i = 0 and
Li p,C q p (ai+1 · · · ar; q) = 1 for i = r.
Clearly, for a word w ∈ h 0 and p ∈ I, the usual MPL Li(w; z) defined in
Section 1.1.2 is characterized as
Li(w; z) = lim
ε→0
ε∈I
Liε,C z p ◦[ε,p](w; z), (2.10)
where [ε, p] stands for the path from ε to p on the interval I.
One can prove the following lemma easily.
Lemma 4. For any z ∈ (0, 1
2 ] ⊂ R and any word w ∈ h0 , we have
| Li(w; z)| ≤ 1. (2.11)

24 Chapter. 2
Proof. We prove this lemma by induction on the length of the word w. If
|w| = 0, Li(w; z) = Li(1; z) = 1 clearly satisfies the claim. If |w| = 1, we
have | Li(y; z)| = | − log(1 − z)| ≤ | log( 1
1
)| ≤ 1 on z ∈ (0, 2 2 ].
We assume | Li(w; z)| ≤ 1 for a word w = xk1−1 kr−1 0 y · · · x y ∈ h . Thus
we obtain
| Li(xw; z)| =
≤
≤
and
| Li(yw; z)| =
on z ∈ (0, 1
2 ].
≤
n1>···>nr>1
n1>···>nr>1
n1>···>nr>1
z
0
z
0
z n1
n k1+1
1 n k2
2 · · · nkr r
r
|z| n1
n k1+1
1 n k2
2 · · · nkr ≤
1 n1
2
n k1
1 n k2
2 · · · n kr
r
n1>···>nr>1
= Li(w; 1
) ≤ 1
2
dz
Li(w; z)
1 − z ≤
z
dz
|Li(w; z)|
0 1 − z
dz
= − log(1 − z) ≤ 1
1 − z
Using these lemmas, we now prove the proposition.
n k1+1
1
1 n1
2
n k2
2 · · · n kr
r
Proof of Proposition 1. Let p(K) and l(K) be constants depending only on
K such as
p(K) = min dist(π(K), {0, 1}), 1
, (2.12)
2
⎛
⎞
l(K) = max
z∈K
⎝ inf
C : path from p(K) to z on U
dist(π(C),{0,1})≥p(K)
length(π(C)) ⎠ . (2.13)
In the right hand side of (2.13), we identify the number p(K) ∈ (0, 1]
with a
2
point in I ⊂ U.
By virtue of Lemma 3 and (2.10), for any word w = a1a2 · · · ar ∈ h0 , we
obtain
Li(w; z) =
r
i=0
Lip(K),C z p(K) (a1 · · · ai; z) Li(ai+1 · · · ar; p(K)). (2.14)

2.2 Analytic properties of MPLs 25
Consequently, by Lemma 2 and Lemma 4, we have
| Li(w; z)| ≤
<
r
i=0
r
i=0
| Lip(K),C z p(K) (a1 · · · ai; z)|| Li(ai+1 · · · ar; p(K))| (2.15)
1
i!
i l(K)
< exp
p(K)
Furthermore, for any word w ∈ h 0 ,
| Li(wx s ; z)| ≤
s
j=0
< exp
l(K)
.
p(K)
| Li(reg 0 (wx s−j | log z|j
); z)|
l(K)
exp
p(K)
j!
max | log z|
z∈K
l(K)
Thus the estimation (2.3) holds for MK = exp
p(K)
.
+ max | log z| .
z∈K
(2.16)
2.3 One dimensional representations of the
formal 1KZ equation
Let ρ be a one dimensional representation of X and we denote by ρ(X) =
α, ρ(Y ) = β ∈ C. The representation of the formal 1KZ equation reads
dG
dz =
α β
+ G. (2.17)
z 1 − z
Clearly the equation has the solution z α (1 − z) −β on U uniquely up to scalar
multiplication.
On the other hand, we try to calculate the representation of the fundamental
solution as follows:
ρ(H0(z)) =
Li(w; z)ρ(W )
=
=
w
∞
p,q=0 w: word of h
w consists p x’s and q y’s
∞
Li(x p x y q ; z)α p β q
p,q=0
Li(w; z)α p β q

26 Chapter. 2
=
=
where we use x p = xp
∞ log
p,q=0
p (z)
Li 1,...,1 (z)α
p!
q times
p β q
∞
log p (z)
p! αp
∞
(− log(1 − z)) q
β
q!
q
p=0
= z α (1 − z) −β ,
q times
p! and Li 1,...,1
q=0
(z) = (− log(1 − z)) q . The representation
ρ(H0(z)) clearly converge for |α|, |β| < 1 by virtue of Proposition 1. Then a
one dimensional representation of the formal 1KZ equation is trivial.
2.4 Gauss hypergeometric equation as a representation
of the formal 1KZ equation
Next, we consider two dimensional representations of the formal 1KZ equation.
Let α, β and γ be complex parameters and ρ0 : X → M(2, C) be a
representation defined by
ρ0(X) =
0 β
0 0
, ρ0(Y ) =
. (2.18)
0 1 − γ
α α + β + 1 − γ
Then the representation of the formal KZ equation by ρ0 is the Gauss hypergeometric
equation (1.50). Indeed one can rewrite the equation (1.50) to
the system;
d v1 1 0 β
=
+
dz v2 z 0 1 − γ
1
0 0
v1
, (2.19)
1 − z α α + β + 1 − γ v2
where v1 = f and v2 = 1 df
z . By Section 1.3, we can write the fundamental
β dz
solution matrix of (2.19) on a neighborhood of z = i (i = 0, 1, ∞) as
where ϕ (i)
j
Φi =
1
β
ϕ (i)
0
d z dz ϕ(i) 0
1
β
ϕ (i)
1
d z dz ϕ(i) 1
, (2.20)
are linearly independent solutions defined at (1.51) ∼ (1.56). The
connection formulas of each fundamental solution matrices are given by
Φ −1
1 Φ0 = C 01 , (2.21)
Φ −1
∞ Φ0 = C 0∞ , (2.22)

2.4 Gauss hypergeometric eq. as rep. 27
where C 01 , C 0∞ are connection coefficients defined by (1.58) and (1.59).
It is well known fact that any second order Fuchsian equation which has
three regular singular points on P 1 can be transformed to Gauss hypergeometric
equation.
2.5 The expression of the hypergeometric function
by MPLs
In this section, we give a concrete expression of ρ0(H0(z)) the representation
of the fundamental solution to the formal 1KZ equation and expand Gauss
hypergeometric function F (α, β, γ; z) as a series of the multiple polylogarithms
of one variable.
2.5.1 Definitions and Notations
We now redefine the weight, depth and height of indexes by using the term
of the shuffle algebra h. For a word w in h, we define the weight |w|, the
depth d(w) and the height h(w) of w as follows;
|w| := the number of letters in w, (2.23)
d(w) := the number of y which appears in w, (2.24)
h(w) := (the number of yx which appears in w) + 1. (2.25)
Denote by gi(k, n, s) (i = 0, 10) the sum of all words in h i with fixed
weight k, depth n and height s, namely
gi(k, n, s) =
w∈h i
|w|=k, d(w)=n
h(w)=s
w. (2.26)
Set g10(k, n, s; z) = 0 if k < n + s, n < s or k, n, s ≤ 0, and g0(k, n, s; z) = 0
if k < n + s − 1, n < s or k, n, s ≤ 0. We note that, if a word w =
x k1−1 y · · · x kr−1 y belongs to h 10 , we have h(w) = #{i|ki ≥ 2}. This is the
original definition of the height given by [OZ] appeared in Section 1.3.2.
Hence one obtains the expression
g10(k, n, s) =
k1+···+kn=k
k1≥2, k2,...,kn≥1
#{i|ki≥2}=s
x k1−1 yx k2−1 y · · · x kn−1 y. (2.27)

28 Chapter. 2
We denote by Gi(k, n, s; z) the sum of MPLs associated to gi(k, n, s);
Gi(k, n, s; z) := Li(gi(k, n, s); z) =
Li(w; z). (2.28)
Especially, we have the following formula;
G10(k, n, s; z) =
k1+···+kn=k
k1≥2, k2,...,kn≥1
#{i|ki≥2}=s
w∈h i
|w|=k, d(w)=n
h(w)=s
Lik1,...,kn(z). (2.29)
Now the equation (1.20) implies the following differential relation;
z d
dz G10(k, n, s; z) = G0(k − 1, n, s; z). (2.30)
For any word W in H, we also define the weight |W |, the depth d(W ) and
the height h(W ) of W in a similar fashion;
|W | := the number of letters in W,
d(W ) := the number of Y which appears in W ,
h(W ) := (the number of Y X which appears in W ) + 1.
2.5.2 Main Result
Theorem 5. Assume that |1 − γ|, |α + 1 − γ|, |β + 1 − γ| and
|α + β + 1 − γ| < 1.
Then we have the expression of F (α, β, γ; z), the
2
hypergeometric function, as follows;
F (α, β, γ; z) = 1 + αβ
G10(k, n, s; z)
k,n,s>0
k≥n+s
n≥s
× (1 − γ) k−n−s (α + β + 1 − γ) n−s ((α + 1 − γ)(β + 1 − γ)) s−1 . (2.31)
The series in the right hand side converges absolutely and uniformly on any
compact subset K of the universal covering space U of P 1 − {0, 1, ∞}.
This expression is equal to Ohno-Zagier’s result (1.61) in Section 1.3.2
through u = 1 − γ, v = α + β + 1 − γ, w = (α + 1 − γ)(β + 1 − γ) in |z| < 1
and |α|, |β|, |1 − γ| are sufficiently small. Thus this theorem can be regarded
as an expansion of Ohno-Zagier’s result to the universal covering space of
P 1 − {0, 1, ∞} and an interpretation as an iterated integral expression of
Gauss hypergeometric function.

2.5 The expression of hypergeometric function 29
We prove this theorem in the following. Put p = 1 − γ, q = α + β + 1 − γ
for convenience.
2.5.3 The image of word in H by the representation ρ0
We consider the representation ρ0 : X → M(2, C) given by (2.18)
ρ0(X) =
0 β
0 p
, ρ0(Y ) =
0 0
α q
. (2.32)
As previously mentioned, the representation of the formal 1KZ equation
(1.38) by ρ0
d ρ0(X)
G =
dz z + ρ0(Y
)
G
1 − z
(2.33)
is the hypergeometric equation (1.50) and the representation of the fundamental
solution H0(z)
ρ0(H0(z)) =
Li(w; z)ρ0(W ) (2.34)
w
gives , if it converges absolutely, the fundamental solution matrix which has
the asymptotic property ρ0(H0)z −ρ0(X) → I (z → 0). In order to compute
the series of the right hand side of (2.34) and to show its convergence, we
prepare the following lemma.
Lemma 6. For any non-empty word W ∈ H, the representation ρ0(W ) is
given by
where
ρ0(W ) = p |W |−d(W )−h(W ) q d(W )−h(W ) (αβ + pq) h(W )−1 M, (2.35)
⎧
αβ βq
αp pq
0 0
⎪⎨
αp pq
M =
0 βq
0 pq
0 0
⎪⎩
0 pq
(if W ∈ XHY ),
(if W ∈ Y HY or W = Y ),
(if W ∈ XHXor W = X),
(if W ∈ Y HX).
(2.36)

30 Chapter. 2
Proof. This is proved by straightforward computation as follows.
First, by easily induction, we have
ρ0(X n ) = p n−1 ρ0(X),
ρ0(Y n ) = q n−1 ρ0(Y ),
ρ0((XY ) n ) = (αβ + pq) n−1
For W = X a1 Y b1 · · · X as Y bs (ai, bi ≥ 1), we have
αβ βq
.
αp pq
ρ0(W ) = (p a1−1 q b1−1 ρ0(XY )) · · · (p as−1 q bs−1 ρ0(XY ))
= p P s
i=1 (ai−1) q P s
i=1 (bi−1) ρ0(XY ) s
= p P s
i=1 (ai−1) q P s
i=1 (bi−1) (αβ + pq) s−1
αβ βq
.
αp pq
Under the rewriting W = X a1 Y b1 · · · X as Y bs = X k1−1 Y · · · X kr−1 Y (k1 ≥
2, ki ≥ 1), s = #{i|ki > 1}, s i=1 (bi − 1) = #{i|ki = 1}, s i=1 (ai
− 1) =
ki=1 (ki − 2) holds. Thus we obtain
ρ0(X k1−1 Y · · · X kr−1 Y )
= p ¯P (ki−2) q #{i|ki=1} (αβ + pq) #{i|ki>1}−1
for k1 > 1, k2 . . . , kr > 0, where ¯ (ki − 2) stands for
Now we have also
ρ0(X k1−1 Y · · · X kr−1 Y )
¯
(ki − 2) :=
(ki − 2).
ki=1
= p ¯P (ki−2) q #{i|ki=1}−1 (αβ + pq) #{i|ki>1}
ρ0(X k1−1 Y · · · X kr−1 Y X s )
⎧
⎪⎨
=
⎪⎩
p ¯P (ki−2)+s−1 q #{i|ki=1} (αβ + pq) #{i|ki>1}
0 0
α q
p ¯P (ki−2)+s−1 q #{i|ki=1}−1 (αβ + pq) #{i|ki>1}+1
0 β
0 p
0 0
0 1
αβ βq
,
αp pq
(k1 = 1),
(s > 0, k1 > 1),
(s > 0, k1 = 1).

2.5 The expression of hypergeometric function 31
On the other hand, by definition of height, we have
⎧
#{i|ki > 1} (k1 > 1, s = 0),
⎪⎨
#{i|ki > 1} + 1 (k1 = 1, s = 0),
h(X k1−1 Y · · · X kr−1 Y X s ) =
Thus these results yield the lemma.
#{i|ki ⎪⎩
> 1} + 1 (k1 > 1, s > 0),
#{i|ki > 1} + 2 (k1 = 1, s > 0).
By putting δ = max(|αβ|, |αp|, |βq|, |pq|, 1), we obtain the following corollaries.
Corollary 7. For any word W ∈ H, there exists a constant δ depending only
on α, β, p, q such that
||ρ0(W )|| ≤ δ|p| |W |−d(W )−h(W ) |q| d(W )−h(W ) |αβ + pq| h(W )−1 , (2.37)
where ||A|| denotes the maximal norm of a matrix A = (aij)1≤i,j≤2, namely
||A|| = maxi,j |aij|.
Corollary 8. If |1 − γ|, |α + 1 − γ|, |β + 1 − γ|, |α + β + 1 − γ| < 1,
the 2
representation ρ0(H0(z)) converges absolutely and uniformly on any compact
subset K of the universal covering space U of P1 − {0, 1, ∞}.
Proof. Let K be a compact subset of U. By using Proposition 1 and the fact
that the number of words with weight k in h is 2k , we can show that there
exists a constant MK depending only on K such that
Li(w; z) k
< 2 MK ∀z ∈ K. (2.38)
w∈h: word, |w|=k
d(w)=n, h(w)=s
By Corollary 7 we have
||ρ0(H0(z))|| = ||
Li(w; z)ρ0(W )|| (2.39)
≤ 1 + δ
w
k,n,s w
|w|=k, d(w)=n
h(w)=s
≤ 1 + δMK
< 1 + 4δMK
k,n,s
k,n,s>0
k≥n+s
n≥s
Li(w; z) |p| |W |−d(W )−h(W ) |q| d(W )−h(W ) |αβ + pq| h(W )−1
2 k |p| k−n−s |q| n−s |αβ + pq| s−1
(2|1 − γ|) k−n−s (2|α + β + 1 − γ|) n−s
× (2|α + 1 − γ|)(2|β + 1 − γ|) s−1 .

32 Chapter. 2
Hence the series ρ0(H0(z)) converges absolutely and uniformly on K if
|1 − γ|, |α + 1 − γ|, |β + 1 − γ|, |α + β + 1 − γ| < 1
2 .
2.5.4 Asymptotic properties of ρ0(H0) and Φ0
From the discussion above, it follows that the representation ρ0(H0(z)) is the
fundamental solution matrix of the hypergeometric equation on U. Hence
there exists a linear relation between ρ0(H0(z)) and Φ0, which is also a fundamental
solution matrix in a neighborhood of z = 0 defined by (2.20), as
follows.
Lemma 9.
1 1
Φ0(z) = ρ0(H0(z)) . (2.40)
0
Proof. Let D be a domain in C defined by D = {z ∈ C | |z| < 1} − {ℜz ≤
0, ℑz = 0} and specify branches of all MPLs, which appears in ρ0(H0(z)), by
the expansion (1.5) and a branch of log z by the principal value on D (that
is, D is a domain in U which includes the interval I and π(D) is simplyconnected).
It is enough to prove that both sides of (2.40) have the same
asymptotic property as z tends to 0 in D.
By definition, ρ0(H0(z))z −ρ0(X) → I as z → 0 holds. On the other hand,
since the formula
holds, we have
1 1
Φ0(z)
0
−1
p z
β
−ρ0(X)
=
z −ρ0(X)
1
=
F00(z)
1
β zF ′ 00(z)
p
β
β
p (z−p − 1)
0 z −p
β
p (F01(z) − F00(z))
1
p (pF01(z) + zF ′ 01(z) − zF ′ 00(z))
(2.41)
as z → 0, where F00 and F01 are regular functions on |z| < 1 introduced
through ϕ (0)
0 = F00, ϕ (0)
1 = z1−γ −1
1 1
F01. Hence Φ0(z) p and ρ0(H0(z))
0 β
are the same fundamental solution matrix to Gauss hypergeometric equation.
2.5.5 Proof of Theorem 5
Now we prove Theorem 5 by using the preceding lemmas.
→ I

2.5 The expression of hypergeometric function 33
Proof of Theorem 5. From Corollary 8, if |1 − γ|, |α + 1 − γ|, |β + 1 − γ|, |α +
β + 1 − γ| < 1
2 , any matrix element of ρ0(H0(z)) converges absolutely and
uniformly on any compact subset K of U. Hence we see that ϕ (0)
0 (z) =
1 1
ρ(H0(z)) p satisfies the same property, here Aij stands for the
0 β 11
(i, j) element of a matrix A. This is computed as follows;
ϕ (0)
0 (z) =
I +
Li(w; z)ρ0(W )
w=1
= 1 +
k,n,s>0
k≥n+s
n≥s
= 1 + αβ
|w|=k,d(w)=n
h(w)=s
k,n,s>0
k≥n+s
n≥s
= 1 + αβ
k,n,s>0
k≥n+s
n≥s
11
Li(w; z)p k−n−s q n−s (αβ + pq) s−1 αβ
G10(k, n, s; z)p k−n−s q n−s (αβ + pq) s−1
G10(k, n, s; z)(1 − γ) k−n−s (α + β + 1 − γ) n−s
× ((α + 1 − γ)(β + 1 − γ)) s−1 . (2.42)
In the discussion above, we have assumed that 1 − γ, α, β = 0. However,
the formula (2.42) makes sense even if 1 − γ → 0, α → 0 and β → 0.
Corollary 10. The solution ϕ (0)
1 (z) to Gauss hypergeometric equation, which
has the exponent 1 − γ at z = 0, is given as follows;
ϕ (0)
1 (z) = z 1−γ
1 + (α + 1 − γ)(β + 1 − γ)
k,n,s
G10(k, n, s; z)
× (γ − 1) k−n−s (α + β + 1 − γ) n−s (αβ) s−1
. (2.43)
Remark. This corollary is proved immediately by using of Theorem 5 and
the formula ϕ (0)
1 (z) = z1−γF (α+1−γ, β+1−γ, 2−γ; z). However one can also
prove this as an algebraic way to compute ϕ (0)
1 1
1 (z) = ρ0(H0(z))
.
0
p
β
12

34 Chapter. 2
2.6 The connection formula between the regular
solutions to Gauss hypergeometric
equation at z = 0 and z = 1
In this section, we investigate various functional relations of the multiple
polylogarithms of one variable by considering the (1, 1)-element of the connection
formula (2.21), and obtain the relations of the multiple zeta values
as known as Ohno-Zagier relation([OZ]) by taking the limit as z → 1.
For the purpose, we first express the inverse element of Φ1 (2.20) as a series
of MPLs, and expand the gamma functions which appear in the connection
matrix C 01 (2.21) as a series of Riemann zeta values.
2.6.1 The inverse of the fundamental solution matrix
on the neighborhood of z = 1
Let G be a solution to the hypergeometric equation (2.33). Thus the trans-
posed inverse matrix t G −1 satisfies
d t −1
G =
dt
tρ0(Y )
t
+
tρ0(X) tG−1 , (2.44)
1 − t
where t = 1 − z. So we define a representation ρ1 : X → M(2, C) such as
ρ1(X) = t
0
ρ0(Y ) =
0
α
, ρ1(Y ) =
q
t
0
ρ0(X) =
β
0
.
p
(2.45)
The matrix-valued function t Φ −1
1
representation of the formal 1KZ equation (1.38) by ρ1
d
G =
dt
is a fundamental solution matrix of the
ρ1(X)
t + ρ1(Y )
1 − t
G. (2.46)
On the other hand, the representation ρ1(H0(t)) is also a fundamental solution
matrix of (2.46). It is nothing but ρ0(H0(z)) up to changing the variable
z → t and the parameters (α, β, p, q) → (β, α, q, p). Similarly as Lemma 9,
we obtain
Lemma 11. There exists a linear relation
t −1
1
Φ1 = ρ1(H0(t))
0
αβ
(α+β−γ)q
β
α+β−γ
. (2.47)

2.6 Connection formula between z = 0, 1 35
Proof. It is suffice to show
t Φ −1
1
1
0
αβ
(α+β−γ)q
β
α+β−γ
−1
t −ρ1(X) → I
as t tends to 0. We express solutions to Gauss hypergeometric equation
ϕ (1)
0 , ϕ (1)
1 as ϕ (1)
0 = F10(t), ϕ (1)
1 = t 1−q F11, where F10(t), F11(t) are holomorphic
at t = 0. By easy calculation, we have
t −1
Φ1 = 1
1 −
det Φ1
det Φ1 = 1
αβ
1
0
t −ρ1(X) =
β (1 − t)((1 − q)t−qF11 + t1−qF ′ 1
11)
β (1 − t)F ′ 10
−t1−qF11 F10
β (1 − t)t−q ((1 − q)F10F11 + tF10F ′ 11 − tF ′ 10F11),
−1
α 1 − q =
0 − 1−q
,
β
,
(α+β−γ)q
β
α+β−γ
α 1 q (t−q − 1)
0 t−q where F ′ stands for d F . Thus we obtain
dt
t −1
lim Φ1 t→0
1
0
αβ
(α+β−γ)q
β
α+β−γ
(1 − t)t
= lim
t→0
−q
det Φ1
−1
t −ρ1(X)
− 1
β ((1 − q)F11 + tF ′ 11)
,
α
βq ((1 − q)F11 + tF11) − 1−q
α
q tF11 − 1−q
β F10
−tF11
1
= lim
t→0 − 1
1
α
− (1 − q)F11
β βq
((1 − q)F10F11
β (1 − q)F11 − 1−q
β2 F ′ 10
0 − 1−q
β F10
= β
1−q α(1−q) 1−q
− + β βq β
1 − q
2
αβ
q = I.
0
1−q
β
β 2 F ′ 10
By this lemma, the (1, 1) and (2, 1)-elements of t Φ −1
1 lead to the following
proposition immediately.
Proposition 12. Assume that |1 − γ|, |α + 1 − γ|, |β + 1 − γ| and
|α + β + 1 − γ| < 1.
The (1, 1) and (1, 2)-elements of Φ−1
2 1 are expressed as

36 Chapter. 2
follows;
(Φ −1
1 )11 = 1 + αβ
k,n,s>0
k≥n+s
n≥s
G10(k, k − n, s; 1 − z) (2.48)
× (1 − γ) k−n−s (α + β + 1 − γ) n−s ((α + 1 − γ)(β + 1 − γ)) s−1 ,
(Φ −1
1 )12 = β
k,n,s>0
k≥n+s
n≥s
G0(k − 1, k − n, s; 1 − z) (2.49)
× (1 − γ) k−n−s (α + β + 1 − γ) n−s ((α + 1 − γ)(β + 1 − γ)) s−1 .
The series of the right hand sides converge absolutely and uniformly on any
compact subset K of U.
Proof. Lemma 11 says that (Φ −1
1 )11 = ρ1(H0)11, (Φ −1
1 )12 = ρ1(H0)12 and
ρ1(H0(t)) is equal to ρ0(H0(z)) up to changing the variable z → t and the
parameters (α, β, p, q) → (β, α, q, p). Thus we obtain
(Φ −1
1 )11 = 1 + αβ
G10(k, n, s; t)q k−n−s p n−s (αβ + pq) s−1
k,n,s>0
k≥n+s
n≥s
= 1 + αβ
k,n,s>0
k≥n+s
n≥s
= 1 + αβ
k,n,s>0
k≥n+s
n≥s
G10(k, k − n, s; t)q n−s p k−n−s (αβ + pq) s−1
G10(k, k − n, s; 1 − z)
× (1 − γ) k−n−s (α + β + 1 − γ) n−s ((α + 1 − γ)(β + 1 − γ)) s−1 .
The second equation for (Φ −1
1 )12 is followed by
(Φ −1
1 )12 = ρ1(H0)12 = 1
d
t
α dt ρ1(H0)11 = 1
α
t d
dt (Φ−1
1 )11.
2.6.2 The expansion of the connection matrix as a series
of the zeta values
To compare both sides of the connection formula (2.21), we expand the
as a series
(1, 1)-element of the connection matrix (C 01 )11 = Γ(γ)Γ(γ−α−β)
Γ(γ−α)Γ(γ−β)

2.6 Connection formula between z = 0, 1 37
of Riemann zeta values. We set p = 1 − γ, q = α + β + 1 − γ and r =
(α + 1 − γ)(β + 1 − γ).
The following formula for the gamma function is well known classically
([WW]);
1
Γ(1 − z)
where c := limn→∞( n 1
k=1 k
= exp(−cz −
∞
n=2
− log n) is Euler constant.
ζ(n)
n zn ), (2.50)
For a sequence of complex numbers a = (a1, a2, . . .) and a non-negative
integer n, we introduce Schur polynomial Pn(a) through the following generating
function;
∞
exp( anz n ) =
n=1
that is ⎧
⎪⎨ P0(a) = 1,
⎪⎩ Pn(a) =
k1+2k2+3k3+···=n
We also define the integers N (n)
i,j as
∞
Pn(a)z n , (2.51)
n=0
a k1
1
k1!
a k2
2
k2!
a k3
3
· · · (n ≥ 1).
k3!
⎧
⎪⎨
N
⎪⎩
(n)
i,j = 0 (i < 0 or j < 0),
N (0)
0,0 = 1 (i = j = 0),
a n + b n =
N (n)
i,j (a + b)i (ab) j
(otherwise).
i,j
(2.52)
(2.53)
Since an + bn is a symmetric polynomial of a and b, this definition is welldefined
and we have N (n)
i,j = 0 if i + 2j = n. We denote by Ni,j = N (i+2j)
i,j .
Under these notations, one can show the following lemma and proposition.
Lemma 13. In the algebra of formal power series C[[a, b]], we have
∞
Aia i
∞
Aib i
∞
l
=
i=0
i=0
i=0
i=0
k,l=0
i=j≥0
i=0
AiA2l+k−iNk,l−i
Proof.
∞
Aia i
∞
Aib i
= +
+
i>j≥0
j>i≥0
=
A 2 l (ab) l +
l≥0
l≥0
n>1
(a + b) k (ab) l . (2.54)
AiAja i b j
AlAl+n(ab) l (a n + b n )

38 Chapter. 2
= A 2 0 +
+
l≥0
l≥1
= A 2 0 +
A 2 l (ab) l
i,j
(i,j)=(0,0)
l≥1
A 2 l (ab) l
+
AlAl+2jN0,j(ab) l+j
l≥0
j>0
+
l≥0
i>0
= A 2 0 +
l≥1
+ l−1
l≥1
j=0
+
i>0
l≥0
= A 2 0 +
=
l≥1
+
i>0
l≥0
∞
l
i,l=0
j=0
j≥0
A 2 l (ab) l
AlAl+i+2j(ab) l Ni,j(a + b) i (ab) j
AlAl+i+2jNi,j(a + b) i (ab) l+j
AjA2l−jN0,l−j(ab) l
l
j=0
l
j=0
l
j=0
AjA2l+i−jNi,l−j(a + b) i (ab) l
AjA2l−jN0,l−j(ab) l
AjA2l+i−jNi,l−j(a + b) i (ab) l
AjA2l+i−jNi,l−j
(a + b) i (ab) l
Proposition 14. For the sequence ζ = (0, ζ(2) ζ(3) ζ(4)
, , , · · · ) and the com-
2 3 4
plex numbers p = 1 − γ, q = α + β + 1 − γ and r = (α + 1 − γ)(β + 1 − γ),
we obtain the following expansion;
Γ(γ)Γ(γ − α − β)
Γ(γ − α)Γ(γ − β)
∞
k l m
i + j
=
i
k,l,m=0
i=0 j=0 µ=0
Pk−i(ζ)Pl−j(ζ)Pµ(−ζ)Pi+j+2m−µ(−ζ)Ni+j,m−µ
× p k q l r m . (2.55)

2.6 Connection formula between z = 0, 1 39
Proof.
Γ(γ)Γ(γ − α − β) Γ(1 − p)
=
Γ(γ − α)Γ(γ − β) ec(1−p) Γ(1 − q)
ec(1−q) ec(1−(α+1−γ)) e
Γ(1 − (α + 1 − γ))
c(1−(β+1−γ))
Γ(1 − (β + 1 − γ)
∞ ζ(n)
= exp(
n
n=2
pn ∞ ζ(n)
) exp(
n
n=2
qn )
∞ ζ(n)
× exp(−
n
n=2
(α + 1 − γ)n ∞ ζ(n)
) exp(−
n
n=2
(β + 1 − γ)n )
∞
= Pn(ζ)p n
∞
Pn(ζ)q n
∞
Pn(−ζ)(α + 1 − γ) n
∞
Pn(−ζ)(β + 1 − γ) n .
n=0
n=0
n=0
Here by making use of Lemma 13, we have
Γ(γ)Γ(γ − α − β)
Γ(γ − α)Γ(γ − β)
∞
= Pn(ζ)p n
∞
Pn(ζ)q n
n=0
n=0
×
k + l
k
m
k,l,m
µ=0
= the right hand side of (2.55).
n=0
Pµ(−ζ)Pk+l+2m−µ(−ζ)Nk+l,m−µ
p k q l r m
2.6.3 Functional relations obtained from the (1, 1)-element
of the connection formula (2.21)
Theorem 15. For |1 − γ|, |α + β + 1 − γ|, |(α + 1 − γ)(β + 1 − γ)| < 1,
the 2
equation
ϕ (0)
0 (z)ϕ (1)
0 (z) − 1 d
z
β dz ϕ(0) 0 (z) 1
α
(1 − z) d
dz ϕ(1)
0 (z) =
Γ(γ)Γ(γ − α − β)
Γ(γ − α)Γ(γ − β) (2.56)
yields functional relations of multiple polylogarithms. Especially by expanding
(2.56) as a series of (1 − γ), (α + β + 1 − γ) and ((α + 1 − γ)(β + 1 − γ)),

40 Chapter. 2
we have
¯G10(k + l + 2m, l + m, m; z) + ¯ G10(k + l + 2m, k + m, m; 1 − z)
k,l,m
+
k ′ +k ′′ =k
l ′ +l ′′ =l
m ′ +m ′′ =m
=
k
k,l,m
¯G10(k ′ + l ′ + 2m ′ , l ′ + m ′ , m ′ ; z) ¯ G10(k ′′ + l ′′ + 2m ′′ , k ′′ + m ′′ , m ′′ ; 1 − z)
+ ¯ G0(k ′ +l ′ +2m ′ −1, l ′ +m ′ , m ′ ; z)G0(k ′′ +l ′′ +2m ′′ +1, k ′′ +m ′′ +1, m ′′ +1; 1 − z)
i=0
l
j=0 µ=0
m
i + j
i
× (1 − γ) k (α + β + 1 − γ) l ((α + 1 − γ)(β + 1 − γ)) m
Pk−i(ζ)Pl−j(ζ)Pµ(−ζ)Pi+j+2m−µ(−ζ)Ni+j,m−µ
where ¯ Gi(k, n, s; z) = Gi(k, n, s; z) − Gi(k, n, s + 1; z)
(i = 0, 10).
Proof. We try to expand the (1, 1)-element of (2.21)
× (1 − γ) k (α + β + 1 − γ) l ((α + 1 − γ)(β + 1 − γ)) m ,
(2.57)
(Φ −1
1 )11(Φ0)11 + (Φ −1
1 )12(Φ0)21 = (C 01 )11 (2.58)
as series in p, q, r. According to Proposition 14, (C01 )11 is the right hand side
of (2.57). On the other hand, from Theorem 5 and Proposition 12, we had
already
(Φ0)11 = ϕ (0)
0 (z) = 1 + αβ
G10(k, n, s; z)p k−n−s q n−s r s−1 ,
k,n,s>0
k≥n+s
n≥s
(Φ0)21 = 1 d
z
β dz ϕ(0) 0 (z) = α
G0(k − 1, n, s; z)p
k,n,s>0
k≥n+s
n≥s
k−n−s q n−s r s−1 ,
(Φ −1
1 )11 = ϕ (1)
0 (z) = 1 + αβ
G10(k, k − n, s; 1 − z)p k−n−s q n−s r s−1 ,
(Φ −1
1 )12 = − 1
α
k,n,s>0
k≥n+s
n≥s
d
(1 − z)
dz ϕ(1) 0 (z) = β
k,n,s>0
k≥n+s
n≥s
G0(k − 1, k − n, s; 1 − z)p k−n−s q n−s r s−1 .

2.6 Connection formula between z = 0, 1 41
By αβ = r − pq, we have
(Φ0)11 = 1 + (r − pq)
= 1 + (r − pq)
= 1 +
= 1 +
= 1 +
−
−
∞
k,l,m=0
∞
k,l,m=0
∞
k,l,m=0
m=0
∞
k,l,m=0
k,l=0
∞
k,l,m=0
k,n,s>0
k≥n+s
n≥s
∞
k,l,m=0
G10(k, n, s; z)p k−n−s q n−s r s−1
G10(k + l + 2m + 2, l + m + 1, m + 1; z)p k q l r m
G10(k + l + 2m + 2, l + m + 1, m + 1; z)p k q l r m+1
G10(k + l + 2m + 2, l + m + 1, m + 1; z)p k+1 q l+1 r m
G10(k + l + 2m, l + m, m; z)p k q l r m
G10(k + l + 2m, l + m, m + 1; z)p k q l r m
¯G10(k + l + 2m, l + m, m; z)p k q l r m ,
where G10(k + l, l, 0; z) = G10(l + 2m, l + m, m + 1; z) = G10(k + 2m, k +
m, m + 1; z) = 0. In the same way, we have also
Thus we obtain
(Φ −1
1 )11 = 1 +
(Φ −1
1 )11(Φ0)11 =
×
1 +
1 +
∞
k,l,m=0
∞
k,l,m=0
∞
k,l,m=0
¯G10(k + l + 2m, k + m, m; z)p k q l r m .
¯G10(k + l + 2m, l + m, m; z)p k q l r m
¯G10(k + l + 2m, k + m, m; 1 − z)p k q l r m

42 Chapter. 2
and
+
= 1 +
¯G10(k + l + 2m, l + m, m; z)
k,l,m
+ ¯
G10(k + l + 2m, k + m, m; 1 − z p k q l r m
k,l,m k ′ +k ′′ =k
l ′ +l ′′ =l
m ′ +m ′′
¯G10(k
=m
′ + l ′ + 2m ′ , k ′ + m ′ , m ′ ; z)
× ¯ G10(k ′′ + l ′′ + 2m ′′ , k ′′ + m ′′ , m ′′
; 1 − z) p k q l r m
(Φ −1
1 )12(Φ0)21 = αβ
G10(k + l + 2m + 1, l + m + 1, m + 1; z)
k,l,m
× G10(k + l + 2m + 1, k + m + 1, m + 1; 1 − z)
=
⎜
⎝
k,l,m
=
⎛
k,l,m
−
⎞
⎟
k ′ +k ′′ =k
l ′ +l ′′ =l
m ′ +m ′′ k
=m−1
′ +k ′′ =k−1
l ′ +l ′′ =l−1
m ′ +m ′′ ⎟
⎠
=m
G10(k ′ + l ′ + 2m ′ + 1, l ′ + m ′ + 1, m ′ + 1; z)
× G10(k ′ + l ′ + 2m ′ + 1, k ′ + m ′ + 1, m ′ + 1; 1 − z)p k q l r m
k,l,m k ′ +k ′′ =k
l ′ +l ′′ =l
m ′ +m ′′ =m
=
k,l,m k ′ +k ′′ =k
l ′ +l ′′ =l
m ′ +m ′′ =m
¯G0(k ′ + l ′ + 2m ′ − 1, l ′ + m ′ , m ′ ; z)
G0(k ′′ + l ′′ + 2m ′′ + 1, k ′′ + m ′′ , m ′′ ; 1 − z)p k q l r m
G0(k ′ + l ′ + 2m ′ + 1, l ′ + m ′ , m ′ ; z)
¯G0(k ′′ + l ′′ + 2m ′′ − 1, k ′′ + m ′′ , m ′′ ; 1 − z)p k q l r m
Therefore (Φ −1
1 )11(Φ0)11 + (Φ −1
1 )12(Φ0)21 is calculated to the left hand side of
(2.57).
Since limz→1 G10(k, n, s; z) = G10(k, n, s; 1), limz→1 G10(k, n, s; 1−z) = 0,
and limz→1 G0(k, n, s; z)G0(k ′ , n ′ , s ′ ; 1 − z) = 0, the limit of the equation
(2.57) as z → 1 and 0 implies the following corollary.
.

2.6 Connection formula between z = 0, 1 43
Corollary 16.
k,l,m
=
k,l,m
=
k
k,l,m
¯G10(k + l + 2m, l + m, m; 1)(1 − γ) k (α + β + 1 − γ) l ((α + 1 − γ)(β + 1 − γ)) m
(2.59)
¯G10(k + l + 2m, k + m, m; 1)(1 − γ) k (α + β + 1 − γ) l ((α + 1 − γ)(β + 1 − γ)) m
i=0
l
j=0 µ=0
m
i + j
i
Pk−i(ζ)Pl−j(ζ)Pµ(−ζ)Pi+j+2m−µ(−ζ)Ni+j,m−µ
× (1 − γ) k (α + β + 1 − γ) l ((α + 1 − γ)(β + 1 − γ)) m .
The equation between the first and the second lines is the duality formula
(1.3) for fixed weight, depth and height. The equation between the first (or
the second) and the third lines is Ohno-Zagier relation originally shown by
[OZ].
2.6.4 Various examples of functional relations of MPLs
In what follows, computing the coefficient of p k q l r m of the formula (2.57)
for some lower l and m, or by specializing the parameters, we show various
concrete relations of the multiple polylogarithms of one variable.
The case of m = 0 and l = 1
By easy calculation we have
(the coefficient of p k q 1 r 0 of the left hand side of (2.57))
= ¯ G10(k + 1, 1, 0; z) + ¯ G10(k + 1, k, 0; 1 − z)
+
k ′ +k ′′
¯G10(k
=k
′ + 1, 1, 0; z) ¯ G10(k ′′ , k ′′ , 0; 1 − z)
+ ¯ G0(k ′ , 1, 0; z)G0(k ′′ + 1, k ′′
+ 1, 1; 1 − z)
k ′ +k ′′
¯G10(k
=k
′ , 0, 0; z) ¯ G10(k ′′ + 1, k ′′ , 0; 1 − z)
+ ¯ G0(k ′ − 1, 0, 0; z)G0(k ′′
+ 2, 2, 1; 1 − z)
+
= −G10(k + 1, 1, 1; z) − G10(k + 1, k, 1; 1 − z)
−
k ′ +k ′′ =k
G0(k ′ , 1, 1; z)G0(k ′′ + 1, k ′′ + 1, 1; 1 − z)
= − Lik+1(z) − Li2, 1,...,1
k−1 times
(1 − z) −
k
Lii(z) Li1,...,1
i=1
k−i+1 times
(1 − z).

44 Chapter. 2
On the other hand,
(the coefficient of p k q 1 r 0 of the right hand side of (2.57)) (2.60)
=
k
i=0
1
i + j
Pk−i(ζ)P1−j(ζ)Pi+j(−ζ) =
i
j=0
= the coefficient of z k ∞ ζ(n)
in exp(
n
n=2
zn ) d
dz exp(−
∞
n=2
= the coefficient of z k ∞
in − ζ(i)z i−1
= −ζ(k + 1).
Consequently we obtain
Lik+1(z) + Li2, 1,...,1
k−1 times
(1 − z) +
i=2
k
k
(i + 1)Pk−i(ζ)Pi+1(−ζ)
i=0
Lii(z) Li1,...,1
i=1
k−i+1 times
ζ(n)
n zn )
(1 − z) = ζ(k + 1). (2.61)
This is known as Euler’s inversion formula for polylogarithms.
The case of m = 0 and l = 2
Similarly we have
(the coefficient of p k q 2 r 0 of the left hand side of (2.57)) (2.62)
= ¯ G10(k + 2, 2, 0; z) + ¯ G10(k + 2, k, 0; 1 − z)
+
k ′ +k ′′
¯G10(k
=k
′ + 2, 2, 0; z) ¯ G10(k ′′ , k ′′ , 0; 1 − z)
+ ¯ G0(k ′ + 1, 2, 0; z)G0(k ′′ + 1, k ′′
+ 1, 1; 1 − z)
k ′ +k ′′
¯G10(k
=k
′ + 1, 1, 0; z) ¯ G10(k ′′ + 1, k ′′ , 0; 1 − z)
+ ¯ G0(k ′ , 1, 0; z)G0(k ′′ + 2, k ′′
+ 1, 1; 1 − z)
k ′ +k ′′
¯G10(k
=k
′ , 0, 0; z) ¯ G10(k ′′ + 2, k ′′ , 0; 1 − z)
+ ¯ G0(k ′ − 1, 0, 0; z)G0(k ′′ + 3, k ′′
+ 1, 1; 1 − z)
+
+
= −G10(k + 2, 2, 1; z) − G10(k + 2, k, 1; 1 − z)
−
k ′ +k ′′ =k
+
k ′ +k ′′ =k
G0(k ′ + 1, 2, 1; z)G0(k ′′ + 1, k ′′ + 1, 1; 1 − z)
G10(k ′ + 1, 1, 1; z)G10(k ′′ + 1, k ′′ , 1; 1 − z)
− G0(k ′ , 1, 1; z)G0(k ′′ + 2, k ′′ + 1, 1; 1 − z)

2.6 Connection formula between z = 0, 1 45
and
= − Lik+1,1(z) − Li3, 1,...,1 (1 − z)
k−1 times
−
k ′ +k ′′ =k
+
Lik ′ ,1(z) Li 1,...,1(1
− z)
k ′′ +1 times
k ′ +k ′′ Lik
=k
′ +1(z) Li2, 1,...,1
k ′′ −1 times
−
(1 − z)
k ′ +k ′′ Lik ′(z) Li2,1,...,1
=k
k ′′ (1 − z)
times
= − Lik+1,1(z) − Li3, 1,...,1(1
− z) − Li1(z) Li2, 1,...,1
k−1 times
k−1 times
−
k
(1 − z)
Lii,1(z) Li1,...,1
i=1
k−i+1 times
(1 − z),
(the coefficient of p k q 2 r 0 of the right hand side of (2.57)) (2.63)
k 2
i + j
=
Pk−i(ζ)Pl−j(ζ)Pi+j(−ζ)
i
i=0 j=0
= the coefficient of z k in 1
2 exp(
∞ ζ(n)
n zn ) d2
∞
exp(−
dz2 i=1
n=2
k + 1 1 k−1
= − ζ(k + 2) + ζ(i + 1)ζ(k − i + 1).
2 2
Therefore, we have
Lik+1,1(z) + Li3, 1,...,1(1
− z) + Li1(z) Li2, 1,...,1
k−1 times
k−1 times
+
k
Lii,1(z) Li1,...,1
i=1
k−i+1 times
= k + 1 1 k−1
ζ(k + 2) − ζ(i + 1)ζ(k − i + 1).
2 2
i=1
n=2
ζ(n)
n zn )
(1 − z) (2.64)
(1 − z)
Especially taking the limit as z tends to 1, we get the formula shown by
Euler([Z1]);
ζ(k + 1, 1) −
k + 1 1 k−1
ζ(k + 2) + ζ(i + 1)ζ(k − i + 1) = 0. (2.65)
2 2
i=1

46 Chapter. 2
The sum formula for MPLs
By comparing the coefficients of α 1 in both sides of (2.57) as a series in
p = (1 − γ), q ′ = (β + 1 − γ), we obtain the following proposition;
Proposition 17. For any positive integers k > n > 0,
G10(k, n, s; z) +
G10(k, k − n, s; 1 − z)
s
+
k ′ +k ′′ =k
n ′ +n ′′ =n
s ′
s
G0(k ′ , n ′ , s ′ ; z)
G0(k ′′ , k ′′ − n ′′ , s ′′ ; 1 − z) = ζ(k). (2.66)
s ′′
Particularly, the limit of the formula (2.66) for z → 1 is the sum formula
of the multiple zeta values (1.2)
Proof. Put q0 = (β + 1 − γ). We have
G10(k, n, s; 1) = ζ(k). (2.67)
s
d
(the right hand side of (2.57))
dα α→0
= d
exp(
dα
ζ(n)
n (α + β + 1 − γ)n ) exp(− ζ(n)
n (α + 1 − γ)n )
× exp( ζ(n)
n (1 − γ)n ) exp(− ζ(n)
n (β + 1 − γ)n )
n−1 n−1
= ζ(n)(α + β + 1 − γ) − ζ(n)(α + 1 − γ)
× exp( ζ(n)
n (α + β + 1 − γ)n ) exp(− ζ(n)
n (α + 1 − γ)n )
× exp( ζ(n)
n (1 − γ)n ) exp(− ζ(n)
n (β + 1 − γ)n
)
n−1 n−1
= ζ(n)(β + 1 − γ) − ζ(n)(1 − γ)
= −
ζ(k + 1)p k +
ζ(l + 1)q l 0.
k
l
On the other hand, we have
¯Gi(k, n, m; z) = 0, (2.68)
m
m ¯ Gi(k, n, m; z) =
Gi(k, n, m; z) (2.69)
m
m
α→0
α→0

2.6 Connection formula between z = 0, 1 47
by definition for i = 0, 10 and for all k, n. By using them, one can calculate
the left hand side of (2.57) as follows.
d
(the left hand side of (2.57))
dα α→0
=
¯G10(k + l + 2m, l + m, m; z) + ¯ G10(k + l + 2m, k + m, m; 1 − z)
k,l,m
+
k ′ +k ′′ =k
l ′ +l ′′ =l
m ′ +m ′′ =m
¯G10(k ′ + l ′ + 2m ′ , l ′ + m ′ , m ′ ; z) ¯ G10(k ′′ + l ′′ + 2m ′′ , k ′′ + m ′′ , m ′′ ; 1 − z)
+ ¯ G0(k ′ +l ′ +2m ′ −1, l ′ +m ′ , m ′ ; z)G0(k ′′ +l ′′ +2m ′′ +1, k ′′ +m ′′ +1, m ′′ +1; 1 − z)
The first term of (2.70) is calculated as
k,l,m
=
k,l,m
× (lp k+m q l+m−1
0
¯G10(k + l + 2m, l + m, m; z)(lp k+m q l+m−1
0
l ¯ G10(k + l + 2m, l + m, m; z)p k+m q l+m−1
0
+
k,l,m
m ¯ G10(k + l + 2m, l + m, m; z)p k+m−1 q l+m
0
=
(l + 1 − m) ¯ G10(k + l + 1, l + 1, m; z)p k q l 0 +
k,l,m
+ mp k+m−1 q l+m
0 ). (2.70)
+ mp k+m−1 q l+m
0 )
k,l,m
m ¯ G10(k + l + 1, l, m; z)p k q l 0
=
−
G10(k + l + 1, l + 1, m; z) +
G10(k + l + 1, l, m; z)
k,l
m
In the same way, we have
k,l,m
¯G10(k + l + 2m, k + m, m; 1 − z)(lp k+m q l+m−1
0
m
+ mp k+m−1 q l+m
0 )
p k q l 0.
=
−
G10(k + l + 1, k, m; 1 − z) +
G10(k + l + 1, k + 1, m; 1 − z)
k,l
m
as the second term of (2.70). For the third term of (2.70), we obtain
k,l,m k ′ +k ′′ =k
l ′ +l ′′ =l
m ′ +m ′′ =m
m
¯G10(k ′ + l ′ + 2m ′ , l ′ + m ′ , m ′ ; z) ¯ G10(k ′′ + l ′′ + 2m ′′ , k ′′ + m ′′ , m ′′ ; 1 − z)
× (lp k+m q l+m−1
0
+ mp k+m−1 q l+m
0 )
p k q l 0

48 Chapter. 2
=
k ′ ,k ′′ ,l ′ ,l ′′
m ′ ,m ′′
=
+
k ′ ,k ′′ ,l ′ ,l ′′
¯G10(k ′ + l ′ + 2m ′ , l ′ + m ′ , m ′ ; z) ¯ G10(k ′′ + l ′′ + 2m ′′ , k ′′ + m ′′ , m ′′ ; 1 − z)
k ′ ,k ′′ ,l ′ ,l ′′
m ′ ,m ′′
+
=
m ′ ,m ′′
m ′
× (l ′ + l ′′ )p k′ +m ′ +k ′′ +m ′′
q l′ +m ′ +l ′′ +m ′′ −1
0
¯G10(k ′ + l ′ + 2m ′ , l ′ + m ′ , m ′ ; z) ¯ G10(k ′′ + l ′′ + 2m ′′ , k ′′ + m ′′ , m ′′ ; 1 − z)
m ′ ,m ′′
× (m ′ + m ′′ )p k′ +m ′ +k ′′ +m ′′ −1 q l ′ +m ′ +l ′′ +m ′′
0
(l ′ + l ′′ + 1 − m ′ − m ′′ ) ¯ G10(k ′ + l ′ , l ′ , m ′ ; z) ¯ G10(k ′′ + l ′′ + 1, k ′′ , m ′′ ; 1 − z)
(m ′ + m ′′ ) ¯ G10(k ′ + l ′ , l ′ , m ′ ; z) ¯ G10(k ′′ + l ′′ + 1, k ′′ + 1, m ′′ ; 1 − z)
(l ′ + l ′′ + 1)
× p k′ +k ′′
q l′ +l ′′
0
¯G10(k ′ + l ′ , l ′ , m ′ ; z)
¯G10(k ′′ + l ′′ + 1, k ′′ , m ′′ ; 1 − z)
k ′ ,k ′′ ,l ′ ,l ′′
m ′
m ′′
+
(−m ′ ) ¯ G10(k ′ + l ′ , l ′ , m ′ ; z)
¯G10(k ′′ + l ′′ + 1, k ′′ , m ′′ ; 1 − z)
= 0.
+
m ′′
¯G10(k ′ + l ′ , l ′ , m ′ ; z)
(−m ′′ ) ¯ G10(k ′′ + l ′′ + 1, k ′′ , m ′′ ; 1 − z)
m ′
m ′′
+
m ′
m ′ G10(k ¯ ′ + l ′ , l ′ , m ′ ; z)
m ′′
¯G10(k ′′ + l ′′ + 1, k ′′ + 1, m ′′ ; 1 − z)
+
¯G10(k ′ + l ′ , l ′ , m ′ ; z)
m ′′ G10(k ¯ ′′ + l ′′ + 1, k ′′ + 1, m ′′ ; 1 − z)
m ′
m ′′
Lastly for the fourth term of (2.70),
× p k′ +k ′′
q l′ +l ′′
0
¯G0(k ′ +l ′ +2m ′ −1, l ′ +m ′ , m ′ ; z)G0(k ′′ +l ′′ +2m ′′ +1, k ′′ +m ′′ +1, m ′′ +1; 1 − z)
k,l,m k ′ +k ′′ =k
l ′ +l ′′ =l
m ′ +m ′′ × (lp =m
k+m q l+m−1
0 + mp k+m−1 q l+m
0 )
¯G0(k ′ +l ′ +2m ′ −1, l ′ +m ′ , m ′ ; z)G0(k ′′ +l ′′ +2m ′′ +1, k ′′ +m ′′ +1, m ′′ +1; 1 − z)
=
k ′ ,k ′′ ,l ′ ,l ′′
m ′ ,m ′′
=
+
k ′ ,k ′′ ,l ′ ,l ′′
m ′ ,m ′′
× (l ′ + l ′′ )p k′ +m ′ +k ′′ +m ′′
q l′ +m ′ +l ′′ +m ′′ −1
0
¯G0(k ′ +l ′ +2m ′ −1, l ′ +m ′ , m ′ ; z)G0(k ′′ +l ′′ +2m ′′ +1, k ′′ +m ′′ +1, m ′′ +1; 1 − z)
k ′ ,k ′′ ,l ′ ,l ′′
m ′ ,m ′′
× (m ′ + m ′′ )p k′ +m ′ +k ′′ +m ′′ −1 l
q ′ +m ′ +l ′′ +m ′′
0
(l ′ + l ′′ + 1 −m ′ −m ′′ ) ¯ G0(k ′ + l ′ − 1, l ′ , m ′ ; z)G0(k ′′ + l ′′ + 2, k ′′ + 1, m ′′ + 1; 1 − z)
+ (m ′ + m ′′ ) ¯ G0(k ′ + l ′ − 1, l ′ , m ′ ; z)G0(k ′′ + l ′′ + 2, k ′′ + 2, m ′′
+ 1; 1 − z)
× p k′ +k ′′
q l′ +l ′′
0

2.6 Connection formula between z = 0, 1 49
=
k ′ ,k ′′ ,l ′ ,l ′′
=
+
m ′
+
m ′
+
m ′
+
k ′ ,k ′′ ,l ′ ,l ′′
m ′
+
m ′
(l ′ + l ′′ + 1)
¯G0(k ′ + l ′ − 1, l ′ , m ′ ; z)
G0(k ′′ + l ′′ + 2, k ′′ + 1, m ′′ + 1; 1 − z)
m ′
m ′′
(−m ′ ) ¯ G0(k ′ + l ′ − 1, l ′ , m ′ ; z)
G0(k ′′ + l ′′ + 2, k ′′ + 1, m ′′ + 1; 1 − z)
m ′
¯G0(k ′ + l ′ − 1, l ′ , m ′ ; z)
(−m ′′ )G0(k ′′ + l ′′ + 2, k ′′ + 1, m ′′ + 1; 1 − z)
m ′′
m ′ G0(k ¯ ′ + l ′ − 1, l ′ , m ′ ; z)
G0(k ′′ + l ′′ + 2, k ′′ + 2, m ′′ + 1; 1 − z)
m ′′
¯G0(k ′ + l ′ − 1, l ′ , m ′ ; z)
m ′′ G0(k ′′ + l ′′ + 2, k ′′ + 2, m ′′ + 1; 1 − z)
−
=
k,l
Thus
m ′′
× p k′ +k ′′
q l′ +l ′′
0
G0(k ′ + l ′ − 1, l ′ , m ′ ; z)
G0(k ′′ + l ′′ + 2, k ′′ + 1, m ′′ + 1; 1 − z)
m ′
m ′
G0(k ′ + l ′ − 1, l ′ , m ′ ; z)
G0(k ′′ + l ′′ + 2, k ′′ + 2, m ′′ + 1; 1 − z)
k ′ +k ′′ =k+1
l ′ +l ′′ m
=l
′
−
k ′ +k ′′ =k
l ′ +l ′′ =l+1
m ′
m ′′
× p k′ +k ′′
q l′ +l ′′
0
G0(k ′ + l ′ − 1, l ′ , m ′ ; z)
G0(k ′′ + l ′′ + 1, k ′′ + 1, m ′′ ; 1 − z)
m ′′
G0(k ′ + l ′ − 1, l ′ , m ′ ; z)
G0(k ′′ + l ′′ + 1, k ′′ + 1, m ′′ ; 1 − z)
× p k q l 0.
the coefficient of p k q l 0 in the left hand side of (2.70)
=
G10(k + l + 1, l, m; z) −
G10(k + l + 1, l + 1, m; z)
m
m
+
G10(k + l + 1, k + 1, m; 1 − z) −
G10(k + l + 1, k, m; 1 − z)
m
+
k ′ +k ′′ =k+1
l ′ +l ′′ m
=l
′
−
k ′ +k ′′ =k
l ′ +l ′′ =l+1
m ′
m ′
m
G0(k ′ + l ′ − 1, l ′ , m ′ ; z)G0(k ′′ + l ′′ + 1, k ′′ + 1, m ′′ ; 1 − z)
G0(k ′ + l ′ − 1, l ′ , m ′ ; z)G0(k ′′ + l ′′ + 1, k ′′ + 1, m ′′ ; 1 − z)

50 Chapter. 2
holds. Therefore we have
G10(k, n, s; z) +
G10(k, k − n, s; 1 − z) (2.71)
s
+
k ′ +k ′′ =k
n ′ +n ′′ =n
s ′
s
G0(k ′ , n ′ , s ′ ; z)
G0(k ′′ , k ′′ − n ′′ , s ′′ ; 1 − z)
=
G10(k, n − 1, s; z) +
G10(k, k − (n − 1), s; 1 − z)
s
+
k ′ +k ′′ =k
n ′ +n ′′ =n−1
s ′
s ′′
s
G0(k ′ , n ′ , s ′ ; z)
G0(k ′′ , k ′′ − n ′′ , s ′′ ; 1 − z)
s ′′
(k > n, n > 1)
and
G10(k, 1, s; z) +
G10(k, k − 1, s; 1 − z) (2.72)
s
+
k ′ +k ′′ =k
n ′ +n ′′ =1
These formulas yield the proposition.
s ′
s
G0(k ′ , n ′ , s ′ ; z)
G0(k ′′ , k ′′ − n ′′ , s ′′ ; 1 − z) = ζ(k).
s ′′
2.7 Functional relations derived from the connection
formula between irregular solutions
In this section, we consider functional relations of the multiple polylogarithms
derived from the connection formula between solutions to the hypergeometric
equation on neighborhoods of z = 0 and ∞. The construction of solutions
shown in Section 2.5 and Section 2.6 can be applied whether the solution is
holomorphic or not, then we can obtain the functional relations with respect
to the (1, 2), (2, 1) and (2, 2)-element of the connection formula (2.21) between
z = 0 and z = 1 and the connection formula (2.22) between z = 0 and
z = ∞ in a similar way as above. But, in general, these relations are too
complicated to be described explicitly.
In what follows, we give the functional relations derived from the limit
of the (1, 1)-element of (2.22) as β tends to 0. The results include Euler’s
inversion formula between z = 0 and ∞ and Riemann zeta values at even
positive integers as the limit as z → 1.

2.7 Functional relations derived from z = 0, ∞ 51
2.7.1 The fundamental solution matrix on the neighborhood
of z = ∞
Let u = 1
z be a complex coordinate at z = ∞ and ρ∞ : X → M(2, C) be a
representation defined by
0
ρ∞(X) = ρ0(Y ) − ρ0(X) =
α
−β
0
, ρ∞(Y ) = ρ0(Y ) =
α + β
α
0
.
q
(2.73)
From d
d 1
u2
= −u2 and = −u− , one can rewrite Gauss hypergeometric
dz du 1−z 1−u
equation (2.33) as
d ρ∞(X)
G =
du u + ρ∞(Y
)
G, (2.74)
1 − u
and get the fundamental solution ρ∞(H0(u)). Comparing the asymptotic
properties, we have
Lemma 18.
1 1
Φ∞ = ρ∞(H0(u))
1 1
Proof. Put P =
. We show
Since ρ∞(X) = P
− α
β −1
α 0
0 β
− α
β
Φ∞P −1 u −ρ∞(X) → I (u → 0).
P −1 , we have
u −ρ∞(X) = P
Therefore by easy computation, we obtain
Φ∞P −1 u −ρ∞(X) = Φ∞P −1 P
=
− 1
= β
β − α
→ β
β − α
u −α 0
0 u −β
−α u 0
0 u−β
P −1 .
P −1 = Φ∞
uαF∞0 uβF∞1 β uα (αF∞0 + uF ′ ∞0) − 1
β uβ (βF∞1 + uF ′ ∞1)
where ϕ (∞)
0
− 1
α 1 − β
−1
. (2.75)
−α u 0
0 u−β
P −1
−α u 0
0 u−β
β
β − α P
F∞0 − α
β F∞1
F∞0 − F∞1
β (αF∞0 + uF ′ ∞0 − αF∞1 − uF ′ ∞1) − 1
β (αF∞0 + uF ′ ∞0 − βF∞1 − uF ′ ∞1)
0 − 1
β
= u α F∞0, ϕ (∞)
1
and F ′ stands for d
du
F .
0
= I,
(α − β)
= u β F∞1, F∞0, F∞1 are holomorphic at u = 0,

52 Chapter. 2
Then we obtain
Φ −1
∞ = β
1 1
β − α − α
β −1
ρ∞(H0(u)) −1 , (2.76)
ρ∞(H0(u)) −1 =
Li( ¯ S(w); u)ρ∞(W ).
w
In (2.76), it is difficult to calculate ρ∞(W ) concretely, but one can see
the limit of ρ∞(W ) and ρ∞(H0(u)) −1 for β → 0 as follows.
Lemma 19. For any word W = I in H, we have
lim
β→0 ρ∞(W
⎧
⎪⎨ α
) =
⎪⎩
|W |−d(w)
d(W )−1 0 0
(α + p)
α α + p
α |W |−d(w)
d(W )−1 0 0
(α + p)
α + p α + p
Proof. Under the limit as β tends to 0, we have
0
ρ∞(X) →
α
0
,
α
0
ρ∞(Y ) →
α
0
.
α + p
Therefore
ρ∞(X k1−1 Y · · · X kr−1 Y ) → α k1+···+kr−r
= α k1+···+kr−r (α + p) r−1
ρ∞(X k1−1 Y · · · X kr−1 Y X s ) → α k1+···+kr−r (α + p) r−1
= α k1+···+kr−r+s (α + p) r−1
hold. These results imply the lemma’s formulas.
r
(W ∈ HY )
(W ∈ HX).
(2.77)
0 0
α α + p
0
α
0
α
0
,
α + p
0
α
α + p
s−1
0
α
0 0
0
α
α + p α + p
Lemma 20. We denote by ˜g1(k, n) the sum of all words of h started with x
which consists of (k − n) x’s and n y’s. Then we have
˜g1(k, n) =
(−1) k−n−1−i (x k−n−i y n ) x x i . (2.78)
k−n−1
i=0

2.7 Functional relations derived from z = 0, ∞ 53
Proof. We prove the lemma by induction on n. Put k ′ = k−n. By definition,
we note that
(x a y) x x b b
a + s
=
x
s
a+s yx b−s
(2.79)
holds in general non-negative integer a, b. Thus, for n = 1, we have
s=0
k ′ −1
(−1)
i=0
k′ −1−i k
(x ′ k
−i i
y) x x =
′ −1
(−1)
i=0
k′ i
′
−1−i k − i + s
x
s
s=0
k′ −i+s i−s
yx
k
=
′ −1
k
s=0
′ −1
(−1)
j=s
j−s
j + 1
x
s
j+1 yx k′ −1−j
k
=
′ −1
(−1) j
j
(−1) s
j + 1
s
j=0
k
=
′ −1
j=0
s=0
x j+1 yx k′ −1−j = ˜g1(k ′ + 1, 1).
x j+1 yx k′ −1−j
For general, we suppose that the claim holds for ˜g1(k ′ + 1, 1), . . . , ˜g1(k ′ +
n, n). Therefore we obtain
k ′ −1
(−1) k′ −1−i k
(x ′ −i n+1 i
y ) x x
i=0
k
=
′ −1
(−1) k′ −1−i k
((x ′ k
−i n i
y ) x x )y +
′ −1
(−1) k′ −1−i k
((x ′ −i n+1 i−1
y ) x x )x
i=0
i=1
= ˜g1(k ′ k
+ n, n)y +
′ −2
(−1) k′ −2−i k
((x ′ −i n+1 i
y ) x x )x
i=0
= ˜g1(k ′ + n, n)y + ˜g1(k ′ k
− 1 + n, n)yx +
′ −3
= · · ·
(−1) k′ −3−i k
((x ′ −i n+1 i 2
y ) x x )x
= ˜g1(k ′ + n, n)y + ˜g1(k ′ − 1 + n, n)yx + ˜g1(k ′ − 2 + n, n)yx 2 + · · ·
i=0
+ ˜g1(2 + n, n)yx k′ −2 + (xy n+1 x x 0 )x k ′ −1
= ˜g1(k ′ + n, n)y + · · · + ˜g1(2 + n, n)yx k′ −2 + ˜g1(1 + n, n)yx k′ −1
= ˜g1(k ′ + n + 1, n + 1).

54 Chapter. 2
Proposition 21. The following formula holds:
1 0
where
H21 =
∞
k=1
+ p
lim
β→0 ρ∞(H0(u)) −1 =
(−1) k logk u
k! αk +
k>n≥1
H22 =
(−1) k
(−1)
k≥n≥0
k Li 1,...,1
n times
Proof. From Lemma 19,
−1
lim ρ∞(H0(u))
β→0
= I +
k≥n≥1 w∈hy
|w|=k,d(w)=n
+
= I +
(−1)
k≥n≥1
k Li 1,...,1
n times
H21 H22
k−n−1
(−1)
i=0
k−n−1−i Lik−n−i+1, 1,...,1
n−1 times
k>n≥0 w∈hx
|w|=k,d(w)=n
k≥n≥1 w∈yh
|w|=k,d(w)=n
+
= I +
k>n≥0 w∈xh
|w|=k,d(w)=n
k≥n≥1 w∈h
|w|=k,d(w)=n
+
k>n≥1 w∈xh
|w|=k,d(w)=n
+
k≥1
(−1) k Li(x k ; u)α k
, (2.80)
(u) logk−n u
(k − n)! αk−n+1 (α + p) n−1
(2.81)
(u) logi (u)
i! αk−n (α + p) n−1 ,
(u) logk−n u
(k − n)! αk−n (α + p) n . (2.82)
Li( ¯ S(w); u)α k−n (α + p) n−1
0 0
α α + p
Li( ¯ S(w); u)α k−n (α + p) n−1
0 0
α + p α + p
(−1) k Li(w; u)α k−n (α + p) n−1
0 0
α α + p
(−1) k Li(w; u)α k−n (α + p) n−1
0 0
α + p α + p
(−1) k Li(w; u)α k−n (α + p) n−1
(−1) k Li(w; u)α k−n (α + p) n−1
0 0
1 1
0 0
α α + p
0 0
p 0

2.7 Functional relations derived from z = 0, ∞ 55
= I +
(−1)
k≥n≥1
k Li(x k−n x y n ; u)α k−n (α + p) n−1
0 0
α α + p
+
(−1)
k>n≥1
k Li(˜g1(k, n); u)α k−n (α + p) n−1
0 0
p 0
+
(−1)
k≥1
k Li(x k ; u)α k
0 0
1 1
= I +
(−1)
k≥1
k logk u
k! αk
0 0
1 1
+
(−1)
k≥n≥1
k logk−n u
(k − n)! Li 1,...,1 (u)α
n times
k−n (α + p) n−1
0 0
α α + p
+
(−1) k Li(˜g1(k, n); u)α k−n (α + p) n−1
0 0
.
p 0
k>n≥1
By using Lemma 20, the third term can be calculated as
k>n≥1
−
k>n≥1
=
(−1) k Li(˜g1(k, n); u)α k−n (α + p) n−1
k>n≥1
(−1) k
(−1) k
Thus we have
k−n−1
i=0
−1
lim ρ∞(H0(u))
β→0
0 0
p 0
(−1) k−n−1−i Li((x k−n−i y n ) x x i ; u)α k−n (α + p) n−1
k−n−1
(−1)
i=0
k−n−1−i Lik−n−i+1, 1,...,1
n−1 times
11
H21 =
(−1) k logk u
k≥1
+
k≥n≥1
+ p
k>n≥1
k! αk
= 1,
(−1) k logk−n u
(−1) k
(k − n)! Li 1,...,1
(u) logi u
i!
−1
lim ρ∞(H0(u)) = 0,
β→0
12
n times
(u)α k−n+1 (α + p) n−1
k−n−1
(−1)
i=0
k−n−1−i Lik−n−i+1, 1,...,1
n−1 times
0 0
p 0
α k−n (α + p) n−1
0 0
.
p 0
(u) logi u
i! αk−n (α + p) n−1 ,

56 Chapter. 2
H22 = 1 +
(−1) k logk u
k! αk +
=
k≥n≥0
k≥1
(−1) k logk−n u
(k − n)! Li 1,...,1
n times
k≥n≥1
(−1) k logk−n u
(k − n)! Li 1,...,1
(u)α k−n (α + p) n .
n times
(u)α k−n (α + p) n
2.7.2 The functional relations derived from the (1, 1)element
of the connection formula between z = 0
and ∞
Multiplying both sides of the (1, 1)-elements of the connection formula (2.22)
(Φ −1
∞ )11(Φ0)11 + (Φ −1
∞ )12(Φ0)21 = (C 0∞ )11
(2.83)
by β−α
and then taking the limit as β → 0, we obtain the following functional
β
relations of multiple polylogarithms.
Proposition 22. The equation
1 + H21 + H22 lim
β→0
1 d
z F (α, β, γ; z)
β dz
−πiα Γ(1 − p)Γ(1 − α)
= e
Γ(1 − (α + p))
(2.84)
holds on |α|, |α + 1 − γ| are sufficiently small and yields functional relations
of multiple polylogarithms. Especially, as the coefficient of α m (α + 1 − γ) n
for any positive integers m, n and z ∈ U, we have
m 1 log z
m! −
m−1
i=0
m−1
(−1)
i=0
n+i+1 Lim−i, 1,...,1
n times
=
Lim−i(z) + (−1) m−i Lim−i( 1
z )
+
m−1
i=0
m1+m2+m3=m
n1+n2=n
n
j=0
( 1
z
log i 1
z
i!
(−2πi)
= Bm
m
, (2.85)
m!
1
z )logi
i! +
m−1
(−1)
i=0
n+i+1 Lim−i+1, 1,...,1
n−1 times
(−1) m+n−j−1 Li 1,...,1
m1 + n1
m1
(
1
z )
n−j times
×
s
( 1
z
)logi 1
z
i!
j
m − i − 1 + j − k
k=0
m − i − 1
1 logi z
G0(m − i + j, k + 1, s; z)
i!
(−1) m1 Pm1+n1(ζ)Pm2(ζ) (−πi)m3
m3! Pn2(−ζ),
(2.86)

2.7 Functional relations derived from z = 0, ∞ 57
where Bm are the Bernoulli numbers, namely the real numbers introduced
through the generating function tm
m
Bm m!
Proof. We try to compute the both sides of
= t
e t −1 .
β − α −1
lim (Φ∞ )11(Φ0)11 + (Φ
β→0 β
−1 β − α
∞ )12(Φ0)21 = lim
β→0 β (C0∞ )11. (2.87)
The right hand side is equal to
β − α
lim
β→0 β (C0∞ )11 = lim
β→0
β − α Γ(γ)Γ(β − α)
e−πiα
β Γ(β)Γ(γ − α)
−πiα Γ(γ)Γ(1 + β − α) Γ(γ)Γ(1 − α) Γ(1 − p)Γ(1 − α)
= lim e = e−πiα = e−πiα
β→0 Γ(1 + β)Γ(γ − α) Γ(1)Γ(γ − α) Γ(1 − (α + p))
∞ (−πiα)
=
k=0
k ∞
Pk(ζ)p
k!
k=0
k
∞
Pk(ζ)α
k=0
k
∞
Pk(−ζ)(α + p)
k=0
k
=
⎛
⎜
m1 + n1
⎝
(−1) m1Pm1+n1(ζ)Pm2(ζ) (−πi)m3
m3! Pn2(−ζ)
⎞
⎟
⎠
m,n
m1+m2+m3=m
n1+n2=n
m1
Especially if m = 0, we have
from eπiα Γ(1−p)Γ(1−α)
Γ(1−(α+p))
is
n1+n2=n
Pn1(ζ)Pn2(−ζ) =
0 (n ≥ 1)
1 (n = 0),
× α m (α + p) n .
= 1. Furthermore, since the coefficient of (α + p) α→0 0
−πiα Γ(1 + α − (α + p))Γ(1 − α)
e = e
Γ(1 − (α + p)) α+p→0
−πiα Γ(1 + α)Γ(1 − α)
= e −πiα πα
2πiα
= e−πiα
sin(πα) eπiα 2πiα
=
− e−πiα e2πiα − 1
= (2πi)
Bm
m
m! αm ,
m≥0
we have immediately
m1+m2+m3=m
(−1) m1Pm1(ζ)Pm2(ζ) (−πi)m3
m3!
= Bm
(2πi) m
m!

58 Chapter. 2
for n = 0.
On the other hand, the left hand side of (2.87) is calculated as
β − α
lim
β→0 β
= lim
β→0
β − α
β
−1
(Φ∞ )11(Φ0)11 + (Φ −1
∞ )12(Φ0)21
β
β − α
(ρ∞(H0)) −1
11 + (ρ∞(H0)) −1
21 (Φ0)11
+ (ρ∞(H0)) −1
12 + (ρ∞(H0)) −1
22 (Φ0)21
= (1 + H21) lim
β→0 (Φ0)11 + H22 lim
β→0 (Φ0)21.
By Theorem 5, we have
lim
β→0 (Φ0)11 = 1,
lim
β→0 (Φ0)12 = lim
(
β→0 1 d
z F (α, β, γ; z))
β dz
= α
k>n≥1
G0(k − 1, n, ∗; z)p k−n−1 (α + p) n−1 ,
where G0(k, n, ∗; z) stands for
s G0(k, n, s; z) the sum of all multiple polylogarithms
which have weight k and depth n.
Therefore we obtain
(1 + H21) lim
β→0 (Φ0)11 + H22 lim
β→0 (Φ0)21
=
∞
(−1) m logm u
m! αm +
m=0
+ ((α + p) − α)
⎛
+ α ⎝
m,n≥0
m,n≥1
(−1) m+n Li 1,...,1
×
(−1)
m≥0,n≥1
m+n Li 1,...,1
n times
(−1) m+n
m−1
i=0
(−1) m−1−i Lim−i+1, 1,...,1
(u)
logm u
m! αm (α + p) n
⎞
⎠
n times
m,n≥1
(u) logm u
m! αm+1 (α + p) n−1
n−1 times
G0(m + n − 1, n, ∗; z)p m−1 (α + p) n−1
(u) logi (u)
i! αm (α + p) n−1

2.7 Functional relations derived from z = 0, ∞ 59
=
∞
(−1) m logm u
m=0
+
m! αm
(−1)
m≥1,n≥0
m+n Li 1,...,1
n+1 times
+
(u) logm−1 u
(m − 1)! αm (α + p) n
(−1)
m,n≥1
m+n
m−1
(−1)
i=0
m−1−i Lim−i+1, 1,...,1
n−1 times
+
m≥2,n≥0
+
m≥1,n≥0
(−1) m+n+1
m−2
m−1
i=0
n
j=0
×
i=0
(−1) m−2−i Lim−i, 1,...,1
n−1 times
(−1) m+n−i−j−1 Li 1,...,1
n−j times
k=0
Thus the coefficient of α m (α + p) n is
(u) logi (u)
i! αm (α + p) n
(u) logi (u)
i! αm (α + p) n−1
(u) logm−i−1 u
(m − i − 1)!
j
(−1) i
i + j + k
G0(i + j + 1, k + 1, ∗; z)
i
α m (α + p) n .
⎧
1 (m = n = 0)
0 (m = 0, n ≥ 1)
(−1)
⎪⎨
m logm m−1
u log
− (−1)m
m!
i=0
m−i−1 u
(m − i − 1)! Lii+1(z) (m ≥ 1, n = 0)
m−1
− (−1)
i=0
i Lim−i(u) logi u
i!
m−1
(−1) n+i+1 Lim−i, 1,...,1 (
1
1
z )logi (m, n ≥ 1)
z i!
⎪⎩
i=0
+
+
n times
m−1
(−1)
i=0
n+i+1 Lim−i+1, 1,...,1
n−1 times
m−1 n
(−1) m+n−j−1 Li 1,...,1
i=0
j=0
( 1
z
(
1
z )
n−j times
)logi 1
z
i!
j
m − i − 1 + j − k
k=0
Comparing these results, we complete the proof.
m − i − 1
×
G1(m − i + j, k + 1, s; z)
s
1 logi z .
i!

60 Chapter. 2
Especially, if m is an even positive integer, we have
(2πi)
−2ζ(m) = Bm
m
m!
(2.88)
when z tends to 1 in (2.85). Furthermore, taking the limit in (2.86) in the
case of n = 1 and 2, we obtain the following corollary.
Corollary 23. The following relations among MZVs holds.
(m + 2)ζ(m + 1) = 2
ζ(i + 1)ζ(2k) (m : odd),
and
i+2k=m,
i,k≥1
2ζ(m, 1) = mζ(m + 1) − 2
i+2k=m,
i,k≥1
(2.89)
ζ(i + 1)ζ(2k) (m : even),
(2.90)
(m + 2)ζ(m + 1, 1) = − π2 (m + 2)(m + 1)
ζ(m) + ζ(m + 2) (2.91)
2 2
−
(i + 1)ζ(i + 2)ζ(2k)
i+2k=m,
i,k≥1
−
i+j+2k=m,
i,j,k≥1
(i + 1)ζ(i + 1)ζ(j + 1)ζ(2k) (m : odd),
2ζ(m, 1, 1) = −mζ(m + 1, 1) + π2 (m + 2)(m + 1)
ζ(m) −
2 2
+
i+2k=m,
i,k≥1
+
i+j+2k=m,
i,j,k≥1
(i + 1)ζ(i + 2)ζ(2k)
ζ(m + 2)
(2.92)
(i + 1)ζ(i + 1)ζ(j + 1)ζ(2k) (m : even).
Proof. From the definition of MPLs and choice of the branches of them on the
interval I defined in Section 2.2, we have the following analytic continuation;
Li1,...,1
n times
( 1 1
) =
z n! (Li1(z) + log z + πi) n
(z ∈ I), (2.93)

2.7 Functional relations derived from z = 0, ∞ 61
1
lim Lik1,k2,...,kn( ) = lim
z→1 on I z z→1 on I Lik1,k2,...,kn(z) = ζ(k1, k2, . . . , kn) (2.94)
(k1 ≥ 2).
Here we note that Li(w, z) log z → 0 as z → 1 on I for any word w =
1 in h. By the analytic continuation of (2.86) and the algebraic relation
Li(w1; z)Li(w2; z)=Li(w1 x w2; z), we obtain
(the left hand side of (2.86) at n = 1)
→
z→1 (1 + (−1)mm)ζ(m + 1) + (1 + (−1) m )ζ(m + 1) + (−1) m ζ(m)πi,
(the left hand side of (2.86) at n = 2)
→
z→1 ((−1)m+1 − 1)ζ(m, 1, 1) + ((−1) m m π2
m − 1)ζ(m + 1, 1) + (−1)
m+1 m(m + 1)
+ (−1)
2
2 ζ(m)
ζ(m + 2) + (−1) m+1 ζ(m, 1)πi + (−1) m mζ(m + 1)πi.
On the other hand, differentiating the (1, 1)-element of C 0∞ by (α + 1 − γ),
taking the limit as (α + 1 − γ) → 0 and applying the equation (2.88), we
have
(the right hand side of (2.86) at n = 1)
= (−1) m ζ(m + 1) − 2
i+2k=m
i,k≥1
(the right hand side of (2.86) at n = 2)
(−1) i ζ(i + 1)ζ(2k) + (−1) m ζ(m)πi,
= (−1) m (m + 1)ζ(m + 2) + (−1) m+1 ζ(m + 1, 1)
−
i+2k=m
i,k≥1
(−1) i (i + 1)ζ(i + 2)ζ(2k) −
i=1
i+j+2k=m
i,j,k≥1
m 1
m−2
+ (−1) ζ(i + 1)ζ(m − i) + mζ(m + 1) πi.
2
(−1) i+j ζ(i + 1)ζ(j + 1)ζ(2k)
The real parts of these results yield the equations from (2.89) to (2.92). The
imaginary parts of them are nothing but the identity for n = 1 and the
relation (2.65) previously shown for n = 2.

Chapter 3
General representations and
many variable formal KZ
equations
In the previous chapter, we debated on the representation of the formal
1KZ equation with respect to Gauss hypergeometric equation. It is the most
fundamental case of non-trivial semisimple representations of the formal 1KZ
equation. In this chapter, we discuss general representations of the formal
1KZ equation and representations of the formal KZ equation of two variables.
The contents of this chapter are now under investigation, but are important
and interested targets to apply our method to make relations of MPLs and
MVZs introduced on Chapter 2.
3.1 Generalized hypergeometric equations and
multiple zeta values
In this section, we consider representations of the formal 1KZ equation (1.38)
with respect to the generalized hypergeometric equation. Let α1, . . . , αn+1,
β1, . . . , βn be complex parameters and λn : X → M(n + 1, C) be a representation
of X = C{X, Y } defined as
⎛
⎞
0 1 0 · · ·
⎜
⎜0
0 1 0 · · · ⎟
⎜
..
⎟
λn(X) = ⎜
. .
. ⎟ , (3.1)
⎜
0 0 0 · · · 1 0 ⎟
⎝0
0 0 · · · 0 1 ⎠
0 (−1) n+1 sn (−1) n sn−1 · · · −s2 s1
63

64 Chapter. 3
⎛
⎜
λn(Y ) = ⎜
⎝
0 0 0 · · · 0 0
. .
.
. .
0 0 0 · · · 0 0
tn+1 tn + (−1) n+1 sn tn−1 + (−1) n sn−1 · · · t2 − s2 t1 + s1
⎞
⎟
⎠ ,
(3.2)
where si stands for the i-th elementary symmetric functions of α1, . . . , αn+1
and ti the i-th elementary symmetric functions of p1 = (1 − β1), . . . , pn =
(1 − βn). One can show that the representation of the formal 1KZ equation
(1.38) by λn is equivalent to the equation
(ϑ(ϑ − p1) · · · (ϑ − pn) − z(ϑ + α1) · · · (ϑ + αn+1)) f = 0, (3.3)
where ϑ = z d is Euler operator. This equation is known as the generalized
dz
hypergeometric equation n+1En and has the solution holomorphic at z = 0
n+1Fn
α1, . . . , αn+1
β1, . . . , βn
; z :=
∞
k=0
(α1)k · · · (αn+1)k
(β1)k · · · (βn)kk! zk
(3.4)
the generalized hypergeometric function. For n = 1, 2E1 is nothing but
Gauss hypergeometric equation and 2F1( α1,α2 ; z) = F (α1, α2, β1; z) appeared
β1
in Section 1.3.1.
Thus the representation of the fundamental solution λn(H0(z)) is the fundamental
solution matrix of n+1En on a neighborhood of z = 0, applying our
method introduced in previous sections, we can an iterated integral expression
of the generalized hypergeometric equation and can expect to obtain
various relations of MPLs and MZVs.
On the other hand, Li([Li]) and Aoki-Kombu-Ohno([AKO]) extended the
result of Ohno-Zagier. On their works, the generalized hypergeometric function
with certain special parameters is correlated with MZVs.
For index k = (k1, . . . , kr), the i-height of k is defined as hi(k) = #{l|kl >
i}. 1-height is particular a height defined in Section 1.3.2 and 2.5. Let
G10(k, n, h1, . . . , hr; z) be a sum of MPLs whose weight is k, depth is n, iheight
is hi (i = 1, . . . , r) and the first index is greater than 1. Li said that
the generating function
Φ0 =
k≥n+ P r
j=1 hj
n≥h1≥...≥hr≥0
h1≥1
G10(k, n, h1, . . . , hr; z)x k−n−P r
1
j=1 hj
x n−h1
2 x h1−h2
3
· · · x hr−1−hr
r+1 x hr
r+2

3.1 Generalized hypergeometric equations 65
can be expressed by using the generalized hypergeometric function as
⎛
⎛
1 ⎜r−1
Φ0 =
⎜
xr+2 − x1xr+1
⎝ AjBjz
j=0
j ⎜ a1
× ⎜
+ j, a2 + j, . . . , ar+1 + j ⎟
r+1Fr ⎝
; z⎟
j + 1 − x1, j + 1, . . . , j + 1 ⎠
r−1 times
(3.5)
where a1 + · · · + ar+1 = x2 − x1,
1≤i1≤i2≤···≤ij≤r+1 ai1 · · · aij = xj+1 − x1xj
for j = 2, . . . , r + 1 and
r−1
i
r − 1
Aj = (xr+2−i − x1xr+1−i) + x1x2 ,
j
j
Bj =
i=j
1
(1 − x1)j(j!) r−1
r−1
i=1
(ai)j.
The notation { i
} stands for the Stirling number of the second kind.
j
Aoki-Kombu-Ohno([AKO]) defined multiple zeta star value (MVSV, for
short) as
ζ ∗ (k1, . . . , kr) =
m1≥m2≥···≥mr≥1
1
m k1
1 · · · m kr
r
and multiple star polylogarithm (MSPL, for short) as
Li ∗
k1,...,kr (z) =
m1≥m2≥···≥mr≥1
z m1
m k1
1 · · · m kr
r
⎞
(3.6)
. (3.7)
MVSVs can be expressed as Q-linear combinations of MZVs and vice versa.
Let G ∗ 10(z) be a sum of MSPLs defined as
G ∗ 10(k, n, s; z) = Li ∗ (g10(k, n, s); z)
and Φ ∗ 0(z) be the generating function of G ∗ 10(z), namely
Φ ∗ 0(z) =
k,n,s
G ∗ 10(k, n, s; z)u k−n−s v n−s w 2s−2 .
According to Aoki-Kombu-Ohno, Φ ∗ 0(z) is the unique solution to
z 2 (1 − z) d2f df
+ z((1 − z)(1 − u) − v)
dz2 dz + (uv − w2 )f = z (3.8)
vanishing at z = 0 and the value Φ∗ 0(1) can be written as
Φ ∗ 1
0(1) =
(1 − v)(1 − β) 3F2
1 − β, 1 − β + u, 1
; 1 , (3.9)
2 − v, 2 − β
− A0
⎞
⎟
⎠ ,

66 Chapter. 3
where β is a solution of the quadratic equation t 2 − (u + v)t + w 2 = 0.
For Li and Aoki-Kombu-Ohno’s results, we suggest the following conjecture.
Conjecture 24. Li and Aoki-Kombu-Ohno’s results can be interpreted through
the representation of the formal 1KZ equation by λn and can be extend as
relations of MPLs.
3.2 Representations of the formal generalized
1KZ equation and MZVs
In this section, we try to extend the formal 1KZ equation to the case with
many regular singular points on P 1 .
3.2.1 The formal generalized KZ equation of one variables
Let a1, . . . , am be non-zero mutually different complex numbers. We extend
the formal 1KZ equation as follows: Let X, Y1, . . . , Ym be non-commutative
formal elements. We call the equation
dG
dz =
X
z +
m
i=1
aiYi
1 − aiz
G (3.10)
the formal generalized 1KZ equation. For n = 1, a1 = 1, it is the ordinary
formal KZ equation of one variable (1.38). This equation is the universal
Fuchsian differential equation which has m + 2 regular singular points at
z = 0, 1 1 , . . . , , ∞ referred to as the formal Schlesinger system.
a1 am
Algebraic properties of formal generalized 1KZ equation and its fundamental
solutions can be constructed in the same way as the formal 1KZ
equation. For the formal generalized 1KZ equation, the infinitesimal pure
braid Lie algebra X = C{X, Y1, . . . , Yn} is the free Lie algebra generated by
X, Y1, . . . , Ym, and the dual Hopf algebra of U(X) is the free shuffle algebra
h = (C〈x, y1, . . . , ym〉, x, 1) generated by x = dz
z , y1 = a1dz
1−a1z , . . . , ym = amdz
1−amz .
We denote by h0 = C1 + hy1 + · · · + hym the x subalgebra of h generated by
words which is not ended with x and define the hyperlogarithm
L(x k1−1
yi1 · · · x kr−1
yir; z) = L( k1ai1 · · · krair; z) :=
z
0
x k1−1
yi1 · · · x kr−1
yir,
(3.11)

3.2 Rep. of the formal generalized 1KZ eq. 67
where i1, . . . , ir ∈ {1, . . . , m}. The hyperlogarithm is a many-valued analytic
function on P1 − {0, 1 1 , . . . , , ∞} and has a Taylor expansion
a1 am
L( k1 ai1 · · · kr air; z) =
n1>n2>···>nr>0
a n1−n2a
i1
n2−n3
i2
n k1
1 · · · n kr
r
· · · a nr
ir
z n1 (3.12)
on |z| < min{ 1
|ai | 1 , . . . , 1
|air |}. Moreover it converges as z tends to 1 − 0 for
k1 > 1 or k1 = 1, i1 = 1. If each ai’s are primitive m-th root of 1, the values
of the limit as z tends to 1 are multiple L values([ArK]).
We can also show that h = h0 [x] as x multiplication and can define
reg0 : h → h0 as taking the constant term of this decomposition. We define
an extended hyperlogarithm as
L(wx s ; z) =
s
i=0
L(reg 0 (wx s−i ); z) logi z
, (3.13)
i!
then the generating function of hyperlogarithms
H0(z) =
L(w; z)W (3.14)
w∈h word
is the fundamental solution to (3.10) normalized at z = 0.
3.2.2 Representations of the formal generalized 1KZ
equation
Let ρ : X → M(n, C) be a representation of X. The representation of the
formal generalized 1KZ equation by ρ
dG
dz =
ρ(X)
z +
m
aiρ(Yi)
G (3.15)
1 − aiz
is a Fuchsian differential equation which has regular singular points at z =
0, 1
a1
i=1
, . . . , 1
am , ∞ (exceptionally if ρ(X) = 0 (resp, ρ(Yi) = 0, ρ(−X − Yi) =
0), the equation is holomorphic at z = 0 (resp. z = 1
ai
, z = ∞)). Un-
fortunately, in general ρ, the representation of the formal generalized 1KZ
equation has accessory parameters (that is, parameters which are not determined
from asymptotic behaviors) and the connection problem is very
complicated.
Assume that the representation ρ is irreducible; namely if a subspace
V ⊂ C n satisfies ρ(X)(V ) ⊂ V, ρ(Yi)(V ) ⊂ V for all i, then V = {0} or C n .

68 Chapter. 3
Katz([Kat]) said that the equation (3.15) has no accessory parameters (in
this case, we say that the equation is rigid) if and only if the relation
(2 − (m + 1))n 2 +
i=0,...,m,∞
dim Z(Ai) = 2 (3.16)
holds, where A0 = exp(2πiρ(X)), Ai = exp(2πiρ(Yi)), A∞
= exp(2πiρ(−X −
Yi)), and Z(A) stands for the centralizer of A, namely the set of all matrices
in M(n, C) which commute to A.
If a representation ρ is irreducible and the equation (3.15) is rigid, the
connection coefficients of solutions to the equation are explicitly given due to
Oshima([Os]). Furthermore, Katz([Kat]), Yokoyama([Y]) and Oshima([Os])
gave the method for construct all irreducible rigid Fuchsian equations by two
manipulations called ”middle convolution” and ”addition”.
It is very interested problem to consider rigid and irreducible Fuchsian
equations through our viewpoint of a representations of the formal generalized
1KZ equation.
3.3 The formal KZ equation of two variables
In this section, we discuss the formal KZ equation of two variable (the formal
2KZ equation, for short) and its representations. We can define the formal
2KZ equation via the formal KZ equation on M0,5 in Section 1.2.2. However,
in this case, unfortunately the infinitesimal pure braid Lie algebra X is not
free, thus the dual Hopf algebra B has complicated structure. We introduce
an algebraic structure on the formal 2KZ equation and its solutions according
to [OU].
3.3.1 The formal KZ equation of two variables
The formal KZ equation on the moduli space M0,5 (1.31) can be written as
dG =
ζ1Z1 + ζ11Z11 + ζ2Z2 + ζ22Z22 + ζ12Z12 G (3.17)
on the cubic coordinate (z1, z2) defined by (1.30), where Z1 = X12 + X13 +
X23, Z2 = X23, Z11 = −X14, Z22 = −X12, Z12 = −X24 are formal elements
in the infinitesimal pure braid Lie algebra X and ζ1 = dz1
z1 , ζ11 = dz1
1−z1 , ζ2 =
dz2
z2 , ζ22 = dz2
1−z2 , ζ12 = d(z1z2)
1−z1z2
are differential 1-forms. This equation is equal to

3.3 The formal 2KZ equation 69
the following partial differential system;
∂
G =
∂z1
∂
G =
∂z2
Z1
z1
Z2
z2
+ Z11
+
1 − z1
z2Z12
1 − z1z2
+ Z22
+
1 − z2
z1Z12
1 − z1z2
G, (3.18)
G. (3.19)
The equation has the divisors D = {z1 = 0, 1, ∞} ∪ {z2 = 0, 1, ∞} ∪ {z1z2 =
1} on P 1 × P 1 and D are normal crossing at (z1, z2) = (0, 0), (1, 0), (0, 1).
The infinitesimal pure braid relation (1.32) for Z1, . . . , Z12 is written as
[Z1, Z2] = [Z11, Z2] = [Z1, Z22] = 0,
[Z11, Z22] = −[Z11, Z12] = [Z22, Z12] = −[Z1 − Z2, Z12], (3.20)
and the Arnold relation (1.33) for ζ1, . . . , ζ12 as
ζ1 ∧ ζ11 = ζ2 ∧ ζ22 = 0,
ζ22 ∧ ζ11 + ζ11 ∧ ζ12 + ζ12 ∧ (ζ22 + ζ2) = 0,
(ζ1 + ζ2) ∧ ζ12 = 0. (3.21)
The universal enveloping algebra U(X) has the following tensor product
decomposition.
Lemma 25 ([OU]). The decomposition
holds for {i1, i2} = {1, 2}, where X (ik)
i1⊗i2
U(X) = U(X (i1)
) ⊗ U(X(i2)
) (3.22)
i1⊗i2 i1⊗i2
stands for the Lie subalgebra
X (i1)
i1⊗i2 = C{Zi1, Zi1i1, Z12}, X (i2)
i1⊗i2 = C{Zi2, Zi2i2} ({i1, i2} = {1, 2}).
3.3.2 The reduced bar algebra and iterated integral of
two variables
Since X is a quotient of the free Lie algebra, the dual Hopf algebra B of U(X)
is a subalgebra of the free shuffle algebra generated by ζ1, . . . , ζ12. We call
the subalgebra B the reduced bar algebra and give a direct definition of B as
follows.
We denote by A = {ζ1, ζ11, ζ2, ζ22, ζ12} a set of letters and S(A) the free
shuffle algebra generated by A. We define that an element
ϕ =
cI ωi1 ◦ · · · ◦ ωis ∈ S(A)
I={i1,...,is}

70 Chapter. 3
(where each ωiα ∈ A, CI ∈ C) satisfies Chen’s integrability condition([C1])
if, for all l (1 ≤ l < s),
cI ωi1 ⊗ · · · ⊗ ωil ∧ ωil+1 ⊗ · · · ⊗ ωis = 0 (3.23)
I
holds as a multiple differential form and define B as a subspace of S(A)
spanned by elements satisfying Chen’s integrability condition. B has the
grading B = ∞
s=0 Bs, Bs = B ∩ Ss(A) as vector space with respect to the
length of words. We obtain clearly B0 = C1, B1 = Cζ1 ⊕ Cζ11 ⊕ Cζ2 ⊕
Cζ22 ⊕ Cζ12 and B2 is a 19 dimensional vector space given by
B2 =
Cω ◦ ω ⊕
Cζi ◦ ζii ⊕
ω∈A
⊕
ω1=ζ1,ζ11
ω2=ζ2,ζ22
i=1,2
i=1,2
Cζii ◦ ζi
C(ω1 ◦ ω2 + ω2 ◦ ω1) ⊕
ω∈A−{ζ12}
C(ω ◦ ζ12 + ζ12 ◦ ω)
⊕ C(ζ1 ◦ ζ12 + ζ2 ◦ ζ12) ⊕ C(ζ11 ◦ ζ12 + ζ22 ◦ ζ11 − ζ22 ◦ ζ12 − ζ2 ◦ ζ12)
(3.24)
by virtue of the AR (3.21). Furthermore we have
Bs =
s−1
j=1
Bj ◦ Bs−j =
s−2
j=0
B1 ◦ · · · ◦ B1
j times
◦B2 ◦ B1 ◦ · · · ◦ B1
s−j−2 times
(3.25)
for s > 2 due to Brown([B]). One can show that B is a Hopf subalgebra of
S(A) and is the dual Hopf algebra of U(X).
According to Chen([C2]), for any element ϕ ∈ B, the value of the iterated
integral of ϕ depends only on the homotopy class of the integral contour.
Thus we can define the iterated integral of two variable
(z1,z2)
ϕ as U(X)-
(z0 1 ,z0 2 )
valued many-valued analytic function on P1 × P1 − D for ϕ ∈ B. Especially
(z1,z2)
if ϕ ∈ B has no terms terminated by ζ1 and ζ2, the iterated integral
can be defined.
We denote by S 0 (A) the subspace of S(A) spanned by elements which
have no terms terminated by ζ1 and ζ2, and B 0 = B ∩ S 0 (A) the subspace of
B. Clearly S 0 (A) (resp. B 0 ) is a x-subalgebra of S(A) (resp. B).
In what follows, we assume that 0 < |z1|, |z2| < 1 and consider the following
two contours C1⊗2, C2⊗1 : (0, 0) → (z1, z2).
(0,0)
ϕ

3.3 The formal 2KZ equation 71
z2
(0, 1) (1, 1)
C (1)
C1⊗2 = C
1⊗2
(1)
1⊗2 ◦ C (2)
1⊗2
✻
C (2)
1⊗2
C (1)
2⊗1
(z1, z2)
✲
✻
C (2)
2⊗1
C2⊗1 = C (2)
✲
(0, 0) (1, 0)
2⊗1 ◦ C (1)
2⊗1
C1⊗2 = C (1)
1⊗2 ◦ C (2)
1⊗2,
C (2)
1⊗2 : (0, 0) → (0, z2),
C (1)
1⊗2 : (0, z2) → (z1, z2).
C2⊗1 = C (2)
2⊗1 ◦ C (1)
2⊗1,
C (1)
2⊗1 : (0, 0) → (z1, 0),
C (2)
2⊗1 : (z1, 0) → (z1, z2).
3.3.3 The normalized fundamental solution to the formal
2KZ equation
Let L(z1, z2) be a solution to the formal 2KZ equation (3.17). We call
L(z1, z2) the fundamental solution normalized at (0, 0) if L(z1, z2) can be
written as
z1
L(z1, z2) = ˆ L(z1, z2)z Z1
1 z Z2
2 , (3.26)
where ˆ L(z1, z2) is a U(X)-valued many-valued analytic function holomorphic
on a neighborhood of (0, 0) and ˆ L(0, 0) = I.
Proposition 26 ([OU]). The fundamental solution normalized at (0, 0) to
the formal 2KZ equation (3.17) exists uniquely and expressed as
L(z1, z2) = ˆ L(z1, z2)z Z1
1 z Z2
2 , ˆ L(z1, z2) =
ˆLs(z1, z2) =
(z1,z2)
(0,0)
∞
s=0
ˆLs(z1, z2),
ad(Ω0) + µ(Ω ′ ) s (1 ⊗ I), (3.27)
where Ω0 = ζ1Z1 + ζ2Z2, Ω ′ = ζ11Z11 + ζ22Z22 + ζ12Z12, and the action ad
and µ on S(A) ⊗ U(X) stand for
ad(ω ⊗ X)(ϕ ⊗ F ) = (ω ◦ ϕ) ⊗ [X, F ],
µ(ω ⊗ X)(ϕ ⊗ F ) = (ω ◦ ϕ) ⊗ XF

72 Chapter. 3
for ϕ ⊗ F ∈ S(A) ⊗ U(X), ω ⊗ X ∈ B1 ⊗ X. Furthermore for any s ∈ Z≥0,
ad(Ω0) + µ(Ω ′ ) s (1 ⊗ I) belongs to B 0 ⊗ U(X).
The iterated integral on (3.27) can be calculate as follows. Let A (1)
1⊗2 =
{ζ1, ζ11, ζ (1)
12 }, A (2)
1⊗2 = {ζ2, ζ22}, A (2)
2⊗1 = {ζ2, ζ22, ζ (2)
12 } and A (1)
2⊗1 = {ζ1, ζ11} be
sets of letters and regard ζ (i)
12 as the 1-form
Proposition 27 ([OU]).
ˆL(z1, z2)
=
=
W ′ ∈W0 (X (1)
1⊗2 )
W ′′ ∈W0 (X (2)
1⊗2 )
W ′ ∈W 0 (X (2)
2⊗1 )
W ′′ ∈W 0 (X (1)
2⊗1 )
Here W 0 (X (ik)
i1⊗i2
ζ (1)
12 = z2dz1
, ζ
1 − z1z2
(2)
12 = z1dz2
. (3.28)
1 − z1z2
L(θ (1)
1⊗2(W ′ ); z1)L(θ (2)
1⊗2(W ′′ ); z2) α(W ′ )α(W ′′ )(I), (3.29)
L(θ (2)
2⊗1(W ′ ); z2)L(θ (1)
2⊗1(W ′′ ); z1) α(W ′ )α(W ′′ )(I). (3.30)
) is the set of all words of U(X(ik)
) which does not ended with
i1⊗i2
Z1 and Z2, α stands for the algebra homomorphism α : U(X) → End(U(X))
defined by
α : (Z1, Z11, Z2, Z22, Z12) ↦→ (ad(Z1), µ(Z11), ad(Z2), µ(Z22), µ(Z12))
and θ (ik)
i1⊗i2
: U(X(ik)
) → S(A(ik)
) is a map which gives the duality; it is
i1⊗i2 i1⊗i2
defined by the replacement of letters
θ (ik)
(Zik ) = ζik , θ(ik)
(Zikik ) = ζikik , θ(i1)
i1⊗i2 i1⊗i2 i1⊗i2 (Z12) = ζ (i1)
12 .
We note that L(θ (1)
1⊗2(W ′ ); z1) (resp. L(θ (2)
2⊗1(W ′ ); z2)) is a hyperlogarithm
defined by Section 3.2.1 for m = 2, z = z1, a1 = 1, a2 = z2, x = ζ1, ξ1 =
ζ11, ξ2 = ζ (1)
12 (resp. m = 2, z = z2, a1 = 1, a2 = z1, x = ζ2, ξ1 = ζ22, ξ2 =
ζ (2)
12 ) and L(θ (2)
1⊗2(W ′′ ); z2) (resp. L(θ (1)
2⊗1(W ′′ ); z2)) is a multiple polylogarithm
of one variable z2 (resp. z1).

3.3 The formal 2KZ equation 73
3.3.4 Decomposition theorem of the normalized fundamental
solution and the generalized harmonic
product relation
We consider the following four formal (generalized) 1KZ equation.
dz1G(z1, z2) = Ω (1)
1⊗2G(z1, z2), Ω (1)
1⊗2 = ζ1Z1 + ζ11Z11 + ζ (1)
12 Z12, (3.31)
dz2G(z2) = Ω (2)
1⊗2G(z2), Ω (2)
1⊗2 = ζ2Z2 + ζ22Z22, (3.32)
dz2G(z1, z2) = Ω (2)
2⊗1G(z1, z2), Ω (2)
2⊗1 = ζ2Z2 + ζ22Z22 + ζ (2)
12 Z12, (3.33)
dz1G(z1) = Ω (1)
2⊗1G(z1), Ω (1)
2⊗1 = ζ1Z1 + ζ11Z11, (3.34)
where dz1 (resp. dz2) stands for the exterior differentiation by the variable
z1 (resp. z2). The fundamental solution normalized at the origin to each
equations L (1)
1⊗2, L (2)
1⊗2, L (2)
2⊗1, L (1)
2⊗1 satisfies the conditions
ˆL (ik)
i1⊗i2 =
∞
L (ik)
i1⊗i2 = ˆ L (ik)
i1⊗i2 zZi k
ik ,
ˆL
s=0
(ik)
i1⊗i2,s , L ˆ(ik) i1⊗i2,s
zi =0
k
= 0 (s > 0), L ˆ(ik) i1⊗i2,0 = I.
Proposition 28. The fundamental solution L(z1, z2) to (3.17) normalized
at the origin decomposes to product of the normalized fundamental solutions
of the (generalized) formal 1KZ equations as follows:
L(z1, z2) = L (1)
1⊗2L (2)
1⊗2 = ˆ L (1)
1⊗2 ˆ L (2)
1⊗2z Z1
1 z Z2
2
= L (2)
2⊗1L (1)
2⊗1 = ˆ L (2)
2⊗1 ˆ L (1)
2⊗1z Z1
1 z Z2
2 .
The expressions (3.29) and (3.30) mean that each decomposition in Proposition
28 corresponds to the choice of the integral contours C1⊗2, C2⊗1. Furthermore,
for the connection problem of the formal 2KZ equation, we obtain
Proposition 29. The connection formula between the fundamental solutions
of the formal 2KZ equation (3.17) normalized at (z1, z2) = (0, 0) and
(1, 0), (0, 1) is given as
L (0,0) (z1, z2) = L (1,0) (z1, z2)ΦKZ(Z1, Z11), (3.35)
L (0,0) (z1, z2) = L (0,1) (z1, z2)ΦKZ(Z2, Z22). (3.36)
In addition, [OU] claimed the followings:

74 Chapter. 3
Proposition 30 ([OU]). We define linear maps ι1⊗2 : B → S(A (1)
1⊗2) ⊗
S(A (2)
1⊗2) by
ιi1⊗i2 =
Pr (1)
1⊗2
⊗ Pr
B
(2)
1⊗2
◦
B
¯ ∆, (3.37)
where ¯ ∆ is the coproduct of the Hopf algebra S(A) (1.9) and Pr (1)
1⊗2 : S(A) →
S(A (1)
1⊗2) (resp. Pr (1)
1⊗2 : S(A) → S(A (2)
1⊗2)) stands for a projection defined by
(ζ1, ζ11, ζ2, ζ22, ζ12) ↦→ (ζ1, ζ11, 0, 0, ζ (1)
12 ) (resp. (0, 0, ζ2, ζ22, 0)). Then
and
ι1⊗2 : B → S(A (1)
1⊗2) ⊗ S(A (2)
1⊗2)
ι1⊗2| B 0 : B 0 → S 0 (A (1)
1⊗2) ⊗ S 0 (A (2)
1⊗2)
are both x algebra isomorphism. We can also define the x isomorphism
ι2⊗1 : B → A (2)
2⊗1 ⊗ A (1)
2⊗1 in the same way. Furthermore, the decomposition
holds as x multiplication.
B = B 0 [ζ1, ζ2] (3.38)
For ψ1⊗ψ2 ∈ S0 (A (i1)
i1⊗i2 )⊗S0 (A (i2)
), we define the integral
i1⊗i2
by
Ci 1 ⊗i 2
ψ1 ⊗ ψ2 :=
zi1 zi2 ψ1
zi =0 1
zi =0 2
ψ2.
Ci 1 ⊗i 2
ψ1⊗ψ2
The map ι1⊗2 (resp. ι2⊗1) picks up the terms of B 0 whose iterated integral
along C1⊗2 (resp. C2⊗1) does not vanish, namely
for ϕ ∈ B 0 .
(z1,z2)
(0,0)
ϕ =
C1⊗2
Proposition 31 ([OU]). Putting
ι1⊗2(ϕ) =
C2⊗1
ϕ(W ′ , W ′′ ) = ι −1
1⊗2(θ (1)
1⊗2(W ′ ) ⊗ θ (2)
1⊗2(W ′′ )) ∈ B 0
for W ′ ∈ W 0 (X (1)
1⊗2), W ′′ ∈ W 0 (X (2)
1⊗2), we have
ι2⊗1(ϕ) (3.39)
ι1⊗2(ϕ(W
C1⊗2
′ , W ′′ )) = L(θ (1)
1⊗2(W ′ ); z1)L(θ (2)
1⊗2(W ′′ ); z2), (3.40)

3.3 The formal 2KZ equation 75
and
ˆL(z1, z2) =
Furthermore the equation
W ′ ∈W0 (X (1)
1⊗2 )
W ′′ ∈W0 (X (2)
1⊗2 )
(z1,z2)
ϕ(W
(0,0)
′ , W ′′ ) α(W ′ )α(W ′′ )(I). (3.41)
L(θ (1)
1⊗2(W ′ ); z1)L(θ (2)
1⊗2(W ′′ ); z2) =
for each W ′ ∈ W0 (X (1)
1⊗2), W ′′ ∈ W0 (X (2)
C2⊗1
ι2⊗1(ϕ(W ′ , W ′′ )) (3.42)
1⊗2) yields the functional relation
among hyperlogarithms referred to as the generalized harmonic product of
hyperlogarithm. The relations are equivalent to the relations derived from
comparing coefficients of each element of U(X) on (3.29) and (3.30).
These relations contains the harmonic product of MPLs which is the
relation expressing a product of MPLs as sum of MPLs of two variables by
using series expression, for instance,
Lik1(z1) Lil1(z2) = z
m>0
m 1
mk1 z
n>0
n
2
= + l1 n
m>n>0
+
m=n>0
z
n>m>0
m 1 zn 2
mk1nl1 = Lik1,l1(1, 1; z1, z2) + Lik1+l1(z1z2) + Lik1,l1(1, 1; z2, z1),
where MPLs of two variables Lik1,...,ki+j (i, j; z1, z2) is a special case of hyperlogarithm
defined by
Lik1,...,ki+j (i, j; z1, z2) := L( k1 1 · · · ki 1 ki+1 z2 · · · ki+j z2; z1) (3.43)
=
n1>n2>···>nr>0
z n1
1 z ni+1
2
n k1
1 · · · n kr
r
Taking limits of them as z tends to 1, we obtain the harmonic product
of MZVs (1.25). Thus the harmonic product of MZVs can be interpreted as
the connection problem of the formal 2KZ equation.
3.3.5 Representations of the formal 2KZ equation
Now we can consider a representation of the formal 2KZ equation and its
fundamental solution. Let ρ : X → M(n, C) be a representation of the
infinitesimal pure braid Lie algebra X. We call Pfaffian system
dG =
ρ(Z1) dz1
z1
+ ρ(Z11) dz1
+ ρ(Z2)
1 − z1
dz2
+ ρ(Z22)
z2
dz2
+ ρ(Z12)
1 − z2
d(z1z2)
.
1 − z1z2
(3.44)

76 Chapter. 3
a representation of the formal 2KZ equation and the formal sum
ρ(L(z1, z2)) (3.45)
a representation of the fundamental solution normalized at origin. A representation
of the fundamental solution can be expressed as a formal sum whose
coefficients are product of hyperlogarithms and multiple polylogarithms. If
it converges absolutely, it is a fundamental solution matrix of (3.44).
Representations of the formal 2KZ equation contain the differential equations
satisfied by Appell hypergeometric function F1, F2 and F4 ([Ko],[Ka]).
Indeed, let µ1 be a representation of X defined as
⎛
⎞
⎛
⎞
0 1 1
0 0 0
µ1(Z1) = ⎝0
1 − γ 0 ⎠ , µ1(Z11) = ⎝αβ
α + β + 1 − γ β⎠
,
0 0 1 − γ
0 0 0
⎛
⎞
⎛
⎞
0 0 1
0 0 0
µ1(Z2) = ⎝0
0 −β ⎠ , µ1(Z22) = ⎝0
β
0 0 β + 1 − γ
′ −β
0 −β ′ ⎠ ,
⎛
⎞
β
µ1(Z12) = ⎝
0 0 0
0 0 0
αβ ′ β ′ α + β + 1 − γ
⎠ .
Then the Appell hypergeometric function
F1(α, β, β ′ ∞
, γ; u, v) :=
m,n=0
(α)m+n(β)m(β ′ )n
u
(γ)m+nm!n!
m v n
(3.46)
satisfies the representation of the formal 2KZ equation (3.17) by µ1 through
the blowing up (u, v) = (z1, z1z2).
For other Appell hypergeometric function
F2(α, β, β ′ , γ, γ ′ ; u, v) :=
F4(α, β, γ, γ ′ ; u, v) :=
∞
m,n=0
∞
m,n=0
(α)m+n(β)m(β ′ )n
(γ)m(γ ′ )nm!n! um v n
(α)m+n(β)m+n
(γ)m(γ ′ )nm!n! um v n
the correspondence representations µ2, µ4 are given as follows.
⎛
−β − β
⎜
µ2(Z1) = ⎜
⎝
′ ⎞
0 1 0
⎟
⎠ ,
−βδ −β ′ + δ 0 0
−β ′ δ ′ 0 −β + δ ′ 0
0 −β ′ δ ′ −βδ −α + 1
(u, v) = (z, zw) ,
(3.47)
(u, v) = (z 2 1w2, (1 − z1)(1 − z1z2)) ,
(3.48)

3.3 The formal 2KZ equation 77
⎛
0 1 0 0
⎜
µ2(Z11) = ⎜0
−β
⎝
′ ⎞
+ α + δ − 1 0 0 ⎟
0 0 0 0⎠
, µ2(Z2)
⎛
−β
⎜
= ⎜
⎝
′ 0 1 0
0 −β ′ ⎞
0 1 ⎟
⎠ ,
0 −β ′ δ ′ 0 0
−β ′ δ ′ 0 δ ′ 0
0 −β ′ δ ′ 0 δ ′
⎛
0
⎜
µ2(Z22) = ⎜0
⎝0
0
0
0
0
0
0
0
1
1
0 0 0 α + δ + δ ′ ⎞
⎟
⎠
− 1
, µ2(Z12)
⎛
0
⎜
= ⎜0
⎝0
0
0
0
1
0
−β + α + δ
0
0
′ − 1
⎞
⎟
0⎠
0 0 −βδ 0
,
⎛
0
⎜
µ4(Z1) = ⎜0
⎝0
1 1
1 − γ + ε 0
0 1 − γ + ε
0
1
1
⎞
⎟
⎠
0 0 0 2(1 − γ)
,
⎛
0
⎜
µ4(Z11) = ⎜αβ
⎝
0
γ
0 0
′ 0 0
ε
0
0
0
0 0 (α + ε)(β + ε) γ ′
⎞
⎟
⎠ , µ4(Z2)
⎛
0
⎜
= ⎜0
⎝0
0
0
0
1
ε
1 − γ
0
1
0
⎞
⎟
⎠
0 0 0 1 − γ
,
⎛
0 0
⎜
µ4(Z22) = ⎜0
−ε
⎝0
ε
⎞
0 0
ε 0 ⎟
−ε 0⎠
0 0 0 0
, µ4(Z12)
⎛
0
⎜
= ⎜ 0
⎝αβ
0
0
−ε
0 0
0 0
−γ ′ 0
0 (α + ε)(β + ε) 0 γ ′
⎞
⎟
⎠ ,
where δ = β + 1 − γ, δ ′ = β ′ + 1 − γ ′ and ε = γ + γ ′ − α − β − 1. We remark
that the equation satisfied by Appell hypergeometric function
F3(α, α ′ , β, β ′ , γ; u, v) :=
∞
m,n=0
(α)m(α ′ )n(β)m(β ′ )n
u
(γ)m+nm!n!
m v n
(3.49)
also can be expressed as a representation of the formal 2KZ equation, however
the expression is too complicated to write concretely.
We call the representation of the formal 2KZ equation (3.17) by µi (i =
1, 2, 4) Appell hypergeometric equation of type i. Proposition 27 and 29 give
us the way to calculate the iterated integral expressions of Appell hypergeometric
functions and we can discuss connection problems and relationship
to MZVs. Calculating the iterated integral expressions and considering correspondence
relations of hyperlogarithms are issues in the future.

82 Bibliography
[Oh] Y. Ohno, A generalization of the duality and sum formulas on
the multiple zeta values, J. Number Theory 74 (1999), no. 1,
39–43.
[Ok] J. Okuda, Duality formulas of the special values of multiple
polylogarithms, Bull. London Math. Soc. 37 (2005), no. 2, 230–
242.
[OkU] J. Okuda and K. Ueno, The sum formula for multiple zeta values
and connection problem of the formal Knizhnik-Zamolodchikov
equation, Zeta Functions, Topology and Quantum Physics, 145–
170, Dev. Math., 14, Springer, New York, 2005.
[OkU2] J. Okuda and K. Ueno, Relations for multiple zeta values and
Mellin transforms of multiple polylogarithms, Publ. Res. Inst.
Math. Sci. 40 (2004), no. 2, 537–564.
[Os] T. Oshima, Classification of Fuchsian systems and their connection
problem, preprint (2008), arXiv:math.CA/0811.2916.
[OS] P. Orlik and L. Solomon, Combinatorics and topology of complements
of hyperplanes, Invent. Math. 56 (1980), no. 2, 167–189.
[OT] P. Orlik and H. Terao, Arrangements of Hyperplanes,
Grundlehren der Mathematischen Wissenschaften, 300,
Springer-Verlag, Berlin, 1992.
[OU] S. Oi and K. Ueno, The formal KZ equation on the moduli space
M0,5 and the harmonic product of multiple zeta values, preprint
(2009), arXiv:math.QA/0910.0718.
[OZ] Y. Ohno and D. Zagier, Multiple zeta values of fixed weight,
depth, and height Indag. Math. (N.S.) 12 (2001), no. 4, 483–
487.
[R] G. Racinet, Doubles mélanges des polylogarithmes multiples aux
racines de l’unité, Publ. Math. Inst. Hautes Études Sci. No. 95
(2002), 185–231.
[Re] C. Reutenauer, Free Lie Algebras, London Mathematical Society
Monographs. New Series, 7. Oxford Science Publications. The
Clarendon Press, Oxford University Press, New York, 1993.
[T1] T. Terasoma, Selberg integrals and multiple zeta values, Compositio
Math. 133 (2002), no. 1, 1–24.

[T2] T. Terasoma, Mixed Tate motives and multiple zeta values, Invent.
Math. 149 (2002), no. 2, 339–369.
[W] Z. Wojtkowiak, Monodromy of iterated integrals and nonabelian
unipotent periods, Geometric Galois actions, 2, 219–
289, London Math. Soc. Lecture Note Ser., 243, Cambridge
Univ. Press, Cambridge, 1997.
[WW] E.T. Whittaker and G.N. Watson, A course of modern analysis,
Reprint of the fourth (1927) edition, Cambridge Mathematical
Library, Cambridge Univ. Press, Cambridge, 1996.
[Y] T. Yokoyama, Construction of systems of differential equations
of Okubo normal form with rigid monodromy, Math. Nachr.,
279 (2006), 327–348.
[Z1] D. Zagier, Values of zeta functions and their applications, First
European Congress of Mathematics, Vol. II (Paris, 1992), 497–
512, Progr. Math., 120, Birkhäuser, Basel, 1994.
[Z2] D. Zagier, Multiple zeta values, preprint.
83