A class of relations among multiple zeta values

0.3 MB·English

by  Gaku Kawashima

A class of relations among multiple zeta values
9
0
0
Gaku Kawashima
2
Graduate School of Mathematics,
n
a Nagoya University, Chikusa-ku, Nagoya464-8602, Japan
J
E-mail: [email protected]

8
2

] Abstract
T
Weproveanewclassofrelationsamongmultiplezetavalues(MZV’s)
N
whichcontainsOhno’srelation. Wealsogivetheformulaforthemaximal
.
h number of independent MZV’s of fixed weight, under our new relations.
t To derive our formula for MZV’s, we consider the Newton series whose
a
values at non-negativeintegers are finitemultiple harmonic sums.
m
keywords multiple zeta value, the Newton series, Ohno’s relation
[

3
v 1 Introduction
4
2
Letk ,...,k be positiveintegersandletk ≥2. Multiple zeta values (MZV’s)
8 1 p p
are defined by
2
1
0 ζ(k ,...,k )= .
07 1 p 0<n1X<···<np nk11···nkpp
/ Thesumk +···+k iscalledtheweightofthemultiplezetavalueζ(k ,...,k ).
h 1 p 1 p
t ThesenumberswereconsideredfirstbyEuler[2];hestudiedthecasep=2. The
a
general case was introduced in [4] and [19]. In recent years, many researchers
m
have studied these numbers in connection with Galois representations, arith-
v: metic geometry, quantum groups, invariants for knots, mathematical physics,
i etc. Many Q-linear relations are known among these values, e.g. Ohno’s rela-
X
tion,thecyclicsumformulaandthederivationrelation[7,8,15,16]. Thefamous
r
relation due to Ohno generalizes the duality, the sum formula and Hoffman’s
a
relation simultaneously. Ihara, Kaneko and Zagier investigated the regularized
doubleshufflerelationsin[8],wheretheyconjecturedthattheserelationsimply
all Q-linear relations. Recently in [10] Kaneko proposed a conjectural gener-
alization of the derivation relation for MZV’s and in [17] Tanaka proved this
conjecture by reducing it to the relationsstudied in this paper. The aimof this
paper is to give a new class of relations among MZV’s. We shall prove that
our new class of relations contains Ohno’s relation. We conjecture that it also
contains the cyclic sum formula. We have checked this up to weight 12.
We shall outline how our relations among MZV’s can be derived. Let µ =
(µ ,...,µ ) be a multi-index (i.e., an ordered set of positive integers) and n a
1 p
1
non-negative integer. We consider the finite multiple harmonic sums
1
s (n)=
µ (n1+1)µ1···(np+1)µp
0≤n1≤X···≤np=n
and
S (n)= s (k)
µ µ
0≤k<n
X
1
= .
(n1+1)µ1···(np+1)µp
0≤n1≤X···≤np<n
Inthe following,we denotethe setofnon-negativeintegersbyN={0,1,2,...}.
We define the operator ∇ on the space CN of complex-valued sequences by
n
n
(∇a)(n)= (−1)k a(k)
k
k=0 (cid:18) (cid:19)
X
fora∈CN andn∈N. InSection3,weshallproveourkeyformulaforthefinite
multiple harmonic sums s (n) (Theorem 3.8). As a corollary of Theorem 3.8,
µ
we shall prove that
∇sµ =sµ∗,
whereµ∗ issomemulti-indexdeterminedbyµ. Wenotethatthisassertionwas
proved independently by Hoffman in [6, Theorem 4.2].
In Section 5, in order to derive our relations among MZV’s, weconsider the
NewtonserieswhichinterpolatesthefinitemultipleharmonicsumsS (n). This
µ
Newton series F (z) is defined by
µ
∞
z z z(z−1)···(z−n+1)
F (z)= (−1)n(∇S )(n) , = ,
µ µ
n n n!
n=0 (cid:18) (cid:19) (cid:18) (cid:19)
X
where z is a complex number with Rez > −1. In fact, for any n ∈N, we have
F (n)=S (n). It is well known that S (n)S (n) is a Z-linear combination of
µ µ µ ν
S (n)’s:
λ
S (n)S (n)= c S (n), c ∈Z.
µ ν i λi i
i
X
The multi-indices λi and the coefficients c depend only on µ and ν, but not
i
on n. From this fact, we derive the functional equations
F (z)F (z)= c F (z), (1)
µ ν i λi
i
X
which are valid for Rez > −1. Since the derivatives F(m)(0) (m ≥ 1) can be
µ
expressedintermsofMZV’s,quadraticrelationsamongMZV’sarederivedfrom
(1) (Corollary 5.4).
2
Wemainlyconsiderthefirstderivativesonly. SinceF (0)vanishes,equation
µ
(1) implies a linear relation
0= c F′ (0).
i λi
i
X
In this way,we get a set of linear relations among MZV’s which can be written
explicitly(Corollary5.5). Weshallexplainthisformula. LetV betheQ-vector
k
spacewhosebasisisthesetofallmulti-indicesofweightkandletV = V .
k≥1 k
We define Q-linear mappings σ, u from V to V by
L
σ(µ ,...,µ )=(−1)p(µ ,...,µ ),
1 p 1 p
u(µ ,...,µ )= (ν ,...,ν ),
1 p 1 q
(ν1,...,νqX)≥(µ1,...,µp)
where(ν ,...,ν )runsoverallrefinementsof(µ ,...,µ )(see§2fortheprecise
1 q 1 p
definition). We define the Q-linear mapping ζ+ from V to R by
ζ+(µ ,...,µ )=ζ(µ ,...,µ ,µ +1).
1 p 1 p−1 p
Then Corollary 5.5 asserts that
ζ+(uσ(µ∗ν))=0
for any multi-indices µ and ν. (The multiplication ∗ on V is the harmonic
product.) For example, for µ=ν =(1), we have
µ∗ν =(2)+2(1,1),
σ(µ∗ν)=−(2)+2(1,1)
and
uσ(µ∗ν)=−{(2)+(1,1)}+2(1,1)=−(2)+(1,1).
Therefore we obtain
ζ+(2)=ζ+(1,1) i.e., ζ(3)=ζ(1,2) ,
which is due to Euler. (cid:0) (cid:1)
Under the harmonic product ∗, the Q-vector space Q⊕V becomes a com-
mutativeQ-algebra. Thisis isomorphicto the commutativeQ-algebraofquasi-
symmetricfunctions, whichis knownto be the polynomialalgebraonthe setof
Lyndonwords[12, Section 2],andwhose enumerationis wellknown. Fromthis
fact, it is easily seen that
d(k):=dim (uσ(V ∗V)∩V )
Q k
1 k
=2k−1− µ 2d (k ≥2), (2)
k d
Xd|k (cid:18) (cid:19)
3
whereµistheMo¨biusfunction. Theright-handsideof(2)appearedintherecent
research of Kaneko for the extended derivation relation, and this equation was
suggested to the author by Kaneko.
Weshallcomparethedimensionsofthreesubspacesofkerζ+∩V . Wedefine
k
the sequence {z }∞ by
k k=1
z =z =z =1, z =z +z (k ≥4)
1 2 3 k k−2 k−3
and put d (k) = 2k−1 −z . Then the dimension of the space kerζ+ ∩V is
Z k k
conjectured to be equal to d (k) by Zagier [19]. We denote the dimension of
Z
the space V ∩V by d (k), where V is the subspace of kerζ+ which
Ohno k O Ohno
corresponds to Ohno’s relation. The following table gives us the numerical
values of d (k), d(k) and d (k) for 2≤k ≤11.
Z O
k 2 3 4 5 6 7 8 9 10 11
d (k) 1 3 6 14 29 60 123 249 503 1012
Z
d(k) 1 2 5 10 23 46 98 200 413 838
d (k) 1 2 5 10 23 46 98 199 411 830
O
The product of two MZV’s is expressedas a Z-linear combinationof MZV’s
in two ways; the harmonic product relations and the shuffle product relations.
Weconjecturethatundertheshuffleproductrelations,Corollary5.4form=1,2
givesallQ-linearrelationsamongMZV’s. ThisconjectureisduetoTanaka. He
checkedthatthedimensionofthesubspaceofkerζ+ whichcorrespondstothese
linear relations is equal to d (k) for 2≤k ≤11.
Z
2 Multi-indices
A finite sequence of positive integers is called a multi-index and we denote the
set of all multi-indices by I. We denote by V the Q-vectorspace whose basis is
I. For µ = (µ ,...,µ ) ∈ I, we call l(µ) := p and |µ| := p µ the length
1 p i=1 i
andthe weightofµ,respectively. Fora positiveintegerm, wedenote the setof
P
all multi-indices of weight m by I . We define the mapping S for a positive
m m
integer m by
k
S : I →2{1,2,...,m−1}, (µ ,...,µ )7→ µ 1≤k <p .
m m 1 p i
( )
Xi=1 (cid:12)
(cid:12)
This is a bijection. For example, we obtain (cid:12)
S (2,2,1)={2,4} and S (1,1,3)={1,2}
5 5
by the following diagrams
↓ ↓ ↓ ↓
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) and (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) .
1 2 3 4 1 2 3 4
4
We define an orderon I by setting µ≥ν if |µ|=|ν|and S (µ)⊃S (ν). In
|µ| |ν|
other words, we write µ ≥ ν if µ is a refinement of ν. For example, the order
on I is given by the following Hasse diagram
4
(1,1,1,1)
(cid:16)(cid:16)PP
(cid:16) P
(cid:16) P
(cid:16) P
(2,1,1) (1,2,1) (1,1,2)
PP (cid:16)(cid:16)PP (cid:16)(cid:16)
P(cid:16) P(cid:16)
(cid:16) P (cid:16) P
(cid:16) P(cid:16) P
(3,1) (2,2) (1,3)
PP (cid:16)(cid:16)
P (cid:16)
P (cid:16)
P(cid:16)
(4) .
For µ=(µ ,...,µ )∈I with |µ|=m, we put
1 p
∗(µ)=µ∗ =S−1(S (µ)c),
m m
τ(µ)=(µ ,...,µ ),
p 1
σ(µ)=(−1)l(µ)µ,
u(µ)= ν
ν≥µ
X
and
d(µ)= ν.
ν≤µ
X
(S (µ)c denotes the complement of S (µ) in {1,2,...,m−1}.) We extend
m m
them as Q-linear mappings from V to V. All of these are bijections. We give
examples of µ∗. We have
(1,2,3)∗ =(2,2,1,1), (2,2,2)∗ =(1,2,2,1) and (4,1,1)∗ =(1,1,1,3)
by the diagrams
↓ ↓ ↓ ↓ ↓ ↓
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13), (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) and (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13),
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
wherethelowerarrowsareinthecomplementaryslotstotheupperarrows. We
can also calculate these correspondences by the diagrams
2 2 1 1 1 2 2 1 1 1 1 3
1(cid:13) 2(cid:13)(cid:13) 4(cid:13)(cid:13)(cid:13)(cid:13)
, and .
2(cid:13)(cid:13) 2 (cid:13)(cid:13) 1 (cid:13)
3 (cid:13)(cid:13)(cid:13) 2 (cid:13)(cid:13) 1 (cid:13)
It is easily seen that
5
l(µ)+l(µ∗)=|µ|+1 (3)
for any µ∈I. Let
µ=(µ ,...,µ ,1,...,1) (µ ≥2, r ≥0, l≥0)
1 r r
l
and µ∗ =(µ∗,...,µ∗). Then we|no{tze t}hat
1 q
µ∗ =l+1. (4)
q
Proposition 2.1. We have ∗τ =τ∗, στ =τσ, uτ =τu and dτ =τd.
Proof. These are easily seen.
Lemma 2.2. Let T ⊂T be finite sets. Then we have
1 2
(−1)#S = (−1)#T2 if T1 =T2
T1⊂XS⊂T2 (0 if T1 (T2.
Proof. The left-hand side equals
#(T2\T1) #(T \T )
(−1)#T1 (−1)#S =(−1)#T1 (−1)k 2 1
k
S⊂(XT2\T1) Xk=0 (cid:18) (cid:19)
=(−1)#T1(1−1)#(T2\T1).
Thus we proved the lemma.
Proposition 2.3. We have
(i) ∗d∗=u,
(ii) dσdσ =id ,
V
(iii) uσuσ =id ,
V
(iv) d∗d−1 =−uσ,
(v) u−1∗u=−σd.
Proof. Let µ be any element of I and let |µ|=m.
(i) We have
∗d∗(µ)= ν∗ = ν∗ =u(µ).
ν≤µ∗ ν∗≥µ
X X
(ii)We have
dσdσ(µ)=(−1)l(µ) (−1)l(ν) λ=(−1)l(µ) (−1)l(ν) λ.
 
νX≤µ λX≤ν λX≤µλ≤Xν≤µ 
By Lemma 2.2, we have  
(−1)l(µ) if λ=µ
(−1)l(ν) = (−1)#Sm(ν)+1 =
(0 if λ<µ.
λ≤Xν≤µ Sm(λ)⊂SXm(ν)⊂Sm(µ)
6
Therefore we obtain dσdσ(µ)=µ.
(iii) The proof of (iii) is similar to the proof of (ii).
(iv) We have
d∗σd(µ)= (−1)l(ν) λ=  (−1)l(ν)λ.
νX≤µ λX≤ν∗ λX∈ImννX≤≤λµ∗ 
By Lemma 2.2, we have  
−1 if S (µ)⊂S (λ)
(−1)l(ν) = (−1)#Sm(ν)+1 = m m
ννX≤≤λµ∗ Sm(ν)⊂SXm(µ)∩Sm(λ)c (0 if Sm(µ)6⊂Sm(λ).
Hence we get
d∗σd(µ)=− λ=−u(µ).
λ≥µ
X
This implies the assertion.
(v) The proof of (v) is similar to the proof of (iv).
We define the multiplication ∗ on V known as the harmonic product. We
shall give a combinatorial description of the harmonic product. For µ, ν ∈ I,
we consider 2×l (l≥1) matrices M =(m ) with the following properties:
ij
• Each entry of M is a non-negative integer.
• If we exclude zeros from the first row of M, we get µ.
• If we exclude zeros from the second row of M, we get ν.
• For 1≤j ≤l, we have m +m >0.
1j 2j
(Then it must hold that max{l(µ),l(ν)}≤l ≤l(µ)+l(ν).) We denote the set
of all such matrices by H . For example, for µ=(1) and ν =(2,3), we have
µ,ν
1 0 0 0 1 0 0 0 1 1 0 0 1
H = , , , , . (5)
µ,ν 0 2 3 2 0 3 2 3 0 2 3 2 3
(cid:26)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:27)
Definition 2.4. For µ, ν ∈I, we define
µ∗ν = (m +m ,...,m +m )∈V
11 21 1l 2l
M=(mXij)∈Hµ,ν
and
µ∗ν = (−1)l(µ)+l(ν)−l(m +m ,...,m +m )∈V,
11 21 1l 2l
M=(mXij)∈Hµ,ν
where l is the number of columns of M. We extend ∗, ∗: I ×I →V bilinearly
on V ×V.
7
For example, we have
(1)∗(2,3)=(1,2,3)+(2,1,3)+(2,3,1)+(3,3)+(2,4)
and
(1)∗(2,3)=(1,2,3)+(2,1,3)+(2,3,1)−(3,3)−(2,4)
from (5).
Proposition 2.5. For any v, w ∈V, we have d(v∗w)=d(v)∗d(w).
Proof. For a proof of this proposition, see [14, Proof of Proposition 2.4].
For µ=(µ ,...,µ ), ν =(ν ,...,ν )∈I, we define
1 p 1 q
µ#ν =(µ ,...,µ ,ν ,...,ν ),
1 p 1 q
µ#˙ ν =(µ ,...,µ ,µ +ν ,ν ,...,ν )
1 p−1 p 1 2 q
and
µ+ =(µ ,...,µ ,µ +1).
1 p−1 p
In addition, if |µ|>1, we define
(µ ,...,µ ,µ −1) if µ >1
µ− = 1 p−1 p p
((µ1,...,µp−1) if µp =1.
We extend #, #˙ : I ×I → V bilinearly on V ×V and µ 7→ µ+ linearly on V.
The following Propositions 2.6 and 2.7 are easily proved.
Proposition 2.6. For any µ, ν ∈I, we have (µ#ν)∗ =µ∗#˙ ν∗.
Proposition 2.7. For any µ, ν ∈I, we have
(i) u(µ#ν)=u(µ)#u(ν),
(ii) u(µ#˙ ν)=u(µ)#u(ν)+u(µ)#˙ u(ν),
(iii) d(µ#˙ ν)=d(µ)#˙ d(ν),
(iv) d(µ#ν)=d(µ)#d(ν)+d(µ)#˙ d(ν).
Here, we introduce the empty multi-index φ. We denote {φ}∪I by I˜and
Qφ⊕V by V˜. It is natural to define
φ+ =(1) and (1)− =φ.
We define ∗(φ), τ(φ), σ(φ), u(φ) and d(φ) to be φ. Moreover,we define for any
µ∈I˜
φ∗µ=µ∗φ=µ, φ∗µ=µ∗φ=µ
and
φ#µ=µ#φ=µ, φ#˙ µ=µ#˙ φ=µ.
Proposition 2.8 follows from the definition of the harmonic product ∗.
8
Proposition 2.8. For any µ=(µ ,...,µ ), ν =(ν ,...,ν )∈I, we have
1 p 1 q
µ∗ν =(µ−∗ν)#(µ )+(µ∗ν−)#(ν )+(µ−∗ν−)#(µ +ν ).
p q p q
In the last of this section, we define bilinear mappings ⊛, ⊛: V ×V → V,
which are used in the following sections, and prove a proposition (Proposition
2.11). Lemma 2.9 is used in the proof of Proposition 2.11. We note that
p
d(µ)= d(µ ,...,µ )#(µ +···+µ ) (6)
1 i−1 i p
i=1
X
for any µ=(µ ,...,µ )∈I, which is immediate from the definition of d.
1 p
Lemma 2.9. For any µ=(µ ,...,µ ), ν =(ν ,...,ν )∈I, we have
1 p 1 q
p
d(µ)∗d(ν)= d(µ ,...,µ )∗d(ν) #(µ +···+µ )
1 i−1 i p
Xi=1(cid:16) (cid:17)
q
+ d(µ)∗d(ν ,...,ν ) #(ν +···+ν )
1 j−1 j q
Xj=1(cid:16) (cid:17)
p q
+ d(µ ,...,µ )∗d(ν ,...,ν ) #(µ +···+µ +ν +···+ν ).
1 i−1 1 j−1 i p j q
Xi=1Xj=1(cid:16) (cid:17)
Proof. It follows from (6) and Proposition 2.8.
Definition 2.10. For µ=(µ ,...,µ ), ν =(ν ,...,ν )∈I, we define
1 p 1 q
µ⊛ν =(µ−∗ν−)#(µ +ν )
p q
and
µ⊛ν =(µ−∗ν−)#(µ +ν ).
p q
We extend ⊛, ⊛: I ×I →V bilinearly on V ×V.
Proposition 2.11. For any v, w ∈V, we have d(v⊛w)=d(v)⊛d(w).
Proof. It suffices to show that the equality d(µ⊛ν) = d(µ)⊛d(ν) holds for
any µ, ν ∈I. Let µ=(µ ,...,µ ), ν =(ν ,...,ν )∈I. Then we have
1 p 1 q
d(µ⊛ν)= d(µ−)∗d(ν−) #(µ +ν )+ d(µ−)∗d(ν−) #˙ (µ +ν ) (7)
p q p q
(cid:16) (cid:17) (cid:16) (cid:17)
9
by the definition of ⊛, Propositions 2.7 and 2.5. By Lemma 2.9, we have
d(µ−)∗d(ν−) #˙ (µ +ν )
p q
(cid:16) p−1 (cid:17)
= d(µ ,...,µ )∗d(ν−) #(µ +···+µ +ν )
1 i−1 i p q
Xi=1(cid:16) (cid:17)
q−1
+ d(µ−)∗d(ν ,...,ν ) #(µ +ν +···+ν )
1 j−1 p j q
Xj=1(cid:16) (cid:17)
p−1q−1
+ d(µ ,...,µ )∗d(ν ,...,ν ) #(µ +···+µ +ν +···+ν ).
1 i−1 1 j−1 i p j q
Xi=1Xj=1(cid:16) (cid:17)
Hence the right-hand side of (7) equals
p q
d(µ ,...,µ )∗d(ν ,...,ν ) #(µ +···+µ +ν +···+ν ),
1 i−1 1 j−1 i p j q
Xi=1Xj=1(cid:16) (cid:17)
which is equal to d(µ)⊛d(ν) by (6).
3 A formula for finite multiple harmonic sums
In this section, we prove a formula for finite multiple harmonic sums (Theorem
3.8). We note that N denotes the set of non-negative integers.
Definition 3.1. (i) We define the difference ∆: CN →CN by putting
(∆a)(n)=a(n)−a(n+1)
for a ∈ CN and n ∈ N. We denote the composition of ∆ with itself n times by
∆n.
(ii) We define the inversion ∇: CN →CN by putting
(∇a)(n)=(∆na)(0)
for a∈CN and n∈N.
Let C[[x,y]] be the ring of formalpower series over the complex numbers in
two indeterminates x and y. We put ∂ =∂/∂x and ∂ =∂/∂y. For
x y
∞ xnyk
f(x,y)= a(n,k) ∈C[[x,y]],
n!k!
n,k=0
X
we have
∞ xnyk
(∂ +∂ −1)f(x,y)= {a(n+1,k)+a(n,k+1)−a(n,k)} . (8)
x y
n!k!
n,k=0
X
10