Weighted sum formula for multiple zeta values

Journal of Number Theory 129 (2009) 2747–2765

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Journal of Number Theory

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Weighted sum formula for multiple zeta values

Li Guo a,∗, Bingyong Xie b

a Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, USAb Department of Mathematics, Peking University, Beijing 100871, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 August 2008Revised 23 April 2009Available online 8 July 2009Communicated by David Goss

The sum formula is a basic identity of multiple zeta values thatexpresses a Riemann zeta value as a homogeneous sum of multiplezeta values of a given dimension. This formula was already knownto Euler in the dimension two case, conjectured in the early 1990sfor higher dimensions and then proved by Granville and Zagier.Recently a weighted form of Euler’s formula was obtained byOhno and Zudilin. We generalize it to a weighted sum formula formultiple zeta values of all dimensions.

© 2009 Elsevier Inc. All rights reserved.

1. Introduction

Multiple zeta values (MZVs) are special values of the multi-variable analytic function

ζ(s1, . . . , sk) =∑

n1>···>nk>0

1

ns11 · · ·nsk

k

(1)

at integers s1 � 2, si � 1, 1 � i � k. Their study in the two variable case went back to Goldbach andEuler [21]. The general concept was introduced in the early 1990s, with motivation from both math-ematics [15,33] and physics [5]. Since then the subject has been studied intensively with interactionsto a broad range of areas in mathematics and physics, including arithmetic geometry, combinatorics,number theory, knot theory, Hopf algebra, quantum field theory and mirror symmetry [2,3,9,14,16,18,20,23,30,32].

A principal goal in the theoretical study of MZVs is to determine all algebraic relations amongthem. Conjecturally all such relations come from the so-called extended double shuffle relations.

* Corresponding author.E-mail addresses: [email protected] (L. Guo), [email protected] (B. Xie).

0022-314X/$ – see front matter © 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.jnt.2009.04.018

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2748 L. Guo, B. Xie / Journal of Number Theory 129 (2009) 2747–2765

But there are no definite way to exhaust all of them and new identities of MZVs are being foundsteadily [3,6,13,19,28,35].

One of the first established and most well known among these identities is the striking sum for-mula, stating that, for given positive integers k and n � k + 1,

∑si�1, s1�2, s1+···+sk=n

ζ(s1, . . . , sk) = ζ(n), (2)

suggesting intriguing connection between Riemann zeta values and multiple zeta values. This formulawas obtained by Euler when k = 2, known as Euler’s sum formula [7],

n−1∑i=2

ζ(i,n − i) = ζ(n) (3)

which includes the basic example ζ(2,1) = ζ(3). Its general form was conjectured in [15] and provedby Granville [10] and Zagier [34]. Since then the sum formula has been generalized and extendedin various directions [4,6,19,22,24–27,29]. Ohno and Zudilin [28] recently proved a weighted form ofEuler’s sum formula (the weighted Euler sum formula)

n−1∑i=2

2iζ(i,n − i) = (n + 1)ζ(n), n � 2, (4)

and applied it to study multiple zeta star values.In this paper we generalize the weighted Euler sum formula of Ohno and Zudilin to higher dimen-

sions.

Theorem 1.1 (Weighted sum formula). For positive integers k � 2 and n � k + 1, we have

∑si�1, s1�2

s1+···+sk=n

[2s1−1 + (

2s1−1 − 1)((

k−1∑i=2

2Si−s1−(i−1)

)+ 2Sk−1−s1−(k−2)

)]ζ(s1, . . . , sk) = nζ(n),

where Si = s1 + · · · + si for i = 1, . . . ,k − 1.

See Theorem 2.2 for a more concise formulation. When k = 2, this formula becomes

n−1∑i=2

(2i − 1

)ζ(i,n − i) = nζ(n),

which gives Eq. (4) after applying Eq. (3).After introducing background notations and results, we prove our weighted sum formula in Sec-

tion 2 by combining sum formulas for the quasi-shuffle product, for the shuffle product, as well asfor MZVs in Eq. (2). The proofs of the sum formulas for the quasi-shuffle and shuffle product aregiven in Section 3 and Section 4, respectively. In the two-dimensional case [28], the sum formula forthe shuffle product was derived from Euler’s decomposition formula. We have generalized Euler’s de-composition formula to multiple zeta values [13]. Instead of applying this generalization directly, weobtain the sum formula for the shuffle product by induction.

L. Guo, B. Xie / Journal of Number Theory 129 (2009) 2747–2765 2749

2. Double shuffle relations and the weighted sum theorem

After introducing preliminary notations and results on MZVs in Section 2.1, we prove the weightedsum formula (Theorem 1.1) in Section 2.2 by applying several shuffle and quasi-shuffle (stuffle) rela-tions in Section 2.2. The proofs of these relations will be given in the next two sections.

2.1. Double shuffle relations of MZVs

As is well known, there are two ways to express a MZV:

ζ(s1, . . . , sk) : =∑

n1>···>nk�1

1

ns11 · · ·nsk

k

(5)

=1∫

0

t1∫0

· · ·t|�s|−1∫0

dt1

f1(t1)· · · dt|�s|

f |�s|(t|�s|)(6)

for integers si � 1 and s1 > 1. Here |�s| = s1 + · · · + sk and

f j(t) ={

1 − t j, if j = s1, s1 + s2, . . . , s1 + · · · + sk,

t j, otherwise.

The product of two such sums is a Z-linear combination of other such sums and the product of twosuch integrals is a Z-linear combination of other such integrals. Thus the Z-linear span of the MZVsforms an algebra

MZV := Z{ζ(s1, . . . , sk)

∣∣ si � 1, s1 � 2}. (7)

The multiplication rules of the MZVs reflected by these two representations are encoded in twoalgebras, the quasi-shuffle (stuffle) algebra for the sum representation and the shuffle algebra for theintegral representation [16,20].

For the sum representation, let H∗ be the quasi-shuffle algebra whose underlying additive groupis that of the noncommutative polynomial algebra

Z〈zs | s � 1〉 = ZM(zs | s � 1),

where M(zs | i � 1) is the free monoid generated by the set {zs | s � 1}. So an element of M(zs | s � 1)

is either the unit 1, also called the empty word, or is of the form

zs1,...,sk := zs1 · · · zsk , s j � 1, 1 � j � k, k � 1.

The product on H∗ is the quasi-shuffle (also called harmonic shuffle or stuffle) product [2,16,17]defined recursively by

(zr1 u) ∗ (zs1 v) = zr1

(u ∗ (zs1 v)

) + zs1

((zr1 u) ∗ v

) + zr1+s1(u ∗ v), u, v ∈ M(zs | s � 1) (8)

with the convention that 1 ∗ u = u ∗ 1 = u for u ∈ M(zs | s � 1). In the quasi-shuffle algebra (H∗,∗)

there is a subalgebra

H∗0 := Z ⊕

(⊕s >1

Zzs1 · · · zsk

)⊆ H∗. (9)

1


Then the multiplication rule of MZVs according to their summation representations follows from thestatement that the linear map

ζ ∗ : H∗0 → MZV, zs1,...,sk → ζ(s1, . . . , sk) (10)

is an algebra homomorphism.For the integral representation, let HX be the shuffle algebra whose underlying additive group is

that of the noncommutative polynomial algebra

Z〈x0, x1〉 = ZM(x0, x1)

where M(x0, x1) is the free monoid generated by x0 and x1. The product on HX is the shuffle productdefined recursively by

(au) X (bv) = a(u X (bv)

) + b((au) X v

), a,b ∈ {x0, x1}, u, v ∈ M(x0, x1) (11)

with the convention that u X 1 = 1 X u = u for u ∈ M(x0, x1). In the shuffle algebra HX , there aresubalgebras

HX0 := Z ⊕ x0H

Xx1 ⊆ HX1 := Z ⊕ HXx1 ⊆ HX. (12)

Then the multiplication rule of MZVs according to their integral representations follows from thestatement that the linear map

ζX : HX0 → MZV, xs1−1

0 x1 · · · xsk−10 x1 → ζ(s1, . . . , sk) (13)

is an algebra homomorphism.There is a natural bijection of abelian groups (but not algebras)

η : HX1 → H∗, 1 ↔ 1, xs1−1

0 x1 · · · xsk−10 x1 ↔ zs1,...,sk . (14)

that restricts to a bijection of abelian groups

η : HX0 → H∗

0, 1 ↔ 1, xs1−10 x1 · · · xsk−1

0 x1 ↔ zs1,...,sk . (15)

Then the fact that MZVs can be multiplied in two ways is reflected by the commutative diagram

HX0

η

ζX

H∗0

ζ ∗

MZV

(16)

Through η, the shuffle product X on HX1 and HX

0 transports to a product X∗ on H∗ and H∗0 .

That is, for w1, w2 ∈ H∗0 , define

w1 X∗ w2 := η(η−1(w1) X η−1(w2)

). (17)


Then the double shuffle relation is simply the set{w1 X∗ w2 − w1 ∗ w2

∣∣ w1, w2 ∈ H∗0

}and the extended double shuffle relation [20,30] is the set

{w1 X∗ w2 − w1 ∗ w2, z1 X∗ w2 − z1 ∗ w2

∣∣ w1, w2 ∈ H∗0

}. (18)

Theorem 2.1. (See [16,20,30].) Let IEDS be the ideal of H∗0 generated by the extended double shuffle relation

in Eq. (18). Then IEDS is in the kernel of ζ ∗ .

It is conjectured that IEDS is in fact the kernel of ζ ∗ .

2.2. Proof of the weighted sum theorem

Let k be a positive integer � 2. For positive integers t1, . . . , tk−1, denote

C(t1, . . . , tk−1) : =(

k−1∑j=1

2t1+···+t j− j

)+ 2t1+···+tk−1−(k−1)

=(

k−2∑j=1

2t1+···+t j− j

)+ 2t1+···+tk−1−(k−1)+1. (19)

They satisfy the following simple relations that can be checked easily from the definition:

C(t1 + 1, t2, . . . , tk−1) = 2C(t1, t2, . . . , tk−1), (20)

C(1, t2, . . . , tk−1) = C(t2, . . . , tk−1) + 1, (21)

with the convention that C(t2, . . . , tk−1) = C(∅) = 1 when k = 2.Using the notation C(t1, . . . , tk−1) we can restate Theorem 1.1 as follows.

Theorem 2.2 (Second form of weighted sum formula). For positive integers k and n � k + 1, we have

∑ti�1, t1�2

t1+···+tk=n

[C(t1, . . . , tk−1) − C(t2, . . . , tk−1)

]ζ(t1, . . . , tk) = nζ(n), (22)

with the convention that C(t2, . . . , tk−1) = C(∅) = 1 when k = 2.

We will prove Theorem 2.2 and hence the weighted sum formula in Theorem 1.1 by using thefollowing auxiliary Theorem 2.4 and Theorem 2.6. They are a type of sum formulas on the products∗ and X∗ respectively and are interesting on their own right. Their proofs will be given in Section 3and Section 4, respectively.

We first display the sum formulas on the product ∗.

Theorem 2.3. For positive integers k � 2 and n � k, we have

∑r,si�1

r+s1+···+sk−1=n

zr ∗ zs1,...,sk−1 = k∑ti�1

t1+···+tk=n

zt1,...,tk + (n − k + 1)∑ui�1

u1+···+uk−1=n

zu1,...,uk−1 . (23)


This theorem will be applied to MZVs through the following

Theorem 2.4. For positive integers k � 2 and n � k + 1, we have

∑r,si�1, s1�2

r+s1+···+sk−1=n

zr ∗ zs1,...,sk−1 =∑

ti�1, t1=1, t2�2t1+···+tk=n

zt1,t2,...,tk + (k − 1)∑

ti�1, t1�2, t2=1t1+···+tk=n

zt1,...,tk

+ k∑

ti�1, t1�2, t2�2t1+···+tk=n

zt1,...,tk + (n − k)∑

ui�1, u1�2u1+···+uk−1=n

zu1,...,uk−1 . (24)

The proofs of these two theorems will be given in Section 3. Similarly for the product X∗ , we havethe following sum formulas.

Theorem 2.5. For positive integers k � 2 and n � k, we have

∑r,si�1, r+s1+···+sk−1=n

zr X∗ zs1,...,sk−1 =∑

ti�1, t1+···+tk=n

C(t1, . . . , tk−1)zt1,...,tk . (25)

Theorem 2.6. For positive integers k � 2 and n � k + 1, we have

∑r,si�1, s1�2

r+s1+···+sk−1=n

zr X∗ zs1,...,sk−1

=∑ti�1

t1+···+tk=n

[C(t1, . . . , tk−1) − C(t2, . . . , tk−1)

]zt1,...,tk −

∑ti�1, t2=1

t1+···+tk=n

zt1,...,tk , (26)

with the convention that C(t2, . . . , tk−1) = C(∅) = 1 when k = 2.

The proofs of these two theorems will be given in Section 4. Now we derive Theorem 2.2 andhence Theorem 1.1 from Theorem 2.4 and Theorem 2.6.

Proof of Theorem 2.2. Regrouping the sums in Eq. (24) of Theorem 2.4 and applying the summationrelation ∑

t1=1, t2�2t1+···+tk=n

−∑

t1�2, t2=1t1+···+tk=n

=∑t1=1

t1+···+tk=n

−∑t2=1

t1+···+tk=n

we obtain

∑r,si�1, s1�2

r+s1+···+sk−1=n

zr ∗ zs1,...,sk−1

=∑

ti�1, t1=1t1+···+tk=n

zt1,...,tk + k∑

ti�1, t1�2t1+···+tk=n

zt1,...,tk −∑

ti�1, t2=1t1+···+tk=n

zt1,...,tk + (n − k)∑

ui�1, u1�2u1+···+uk−1=n

zu1,...,uk−1 .

From this equation and Theorem 2.6 we obtain


∑r,si�1, s1�2

r+s1+···+sk−1=n

(zr X∗ zs1,...,sk−1 − zr ∗ zs1,...,sk−1)

=∑

ti�1, t1=1t1+···+tk=n

[C(t1, . . . , tk−1) − C(t2, . . . , tk−1) − 1

]zt1,...,tk

+∑

ti�1, t1�2t1+···+tk=n

[C(t1, . . . , tk−1) − C(t2, . . . , tk−1) − k

]zt1,...,tk − (n − k)

∑ui�1, u1�2

u1+···+uk−1=n

zu1,...,uk−1

=∑

ti�1, t1�2t1+···+tk=n

[C(t1, . . . , tk−1) − C(t2, . . . , tk−1) − k

]zt1,...,tk − (n − k)

∑ui�1, u1�2

u1+···+uk−1=n

zu1,...,uk−1

by Eq. (21). Since ζ ∗(zr X∗ zs1,...,sk−1 − zr ∗ zs1,...,sk−1 ) = 0, by Theorem 2.1, this gives

∑ti�1, t1�2

t1+···+tk=n

[C(t1, . . . , tk−1) − C(t2, . . . , tk−1) − k

]ζ(t1, . . . , tk)

= (n − k)∑

ui�1, u1�2u1+···+uk−1=n

ζ(u1, . . . , uk−1).

The sum formula in Eq. (2) shows that

∑ti�1, t1�2

t1+···+tk=n

ζ(t1, . . . , tk) =∑

ui�1, u1�2u1+···+uk−1=n

ζ(u1, . . . , uk−1) = ζ(n).

Therefore

∑ti�1, t1�2

t1+···+tk=n

[C(t1, . . . , tk−1) − C(t2, . . . , tk−1)

]ζ(t1, . . . , tk) = nζ(n),

as desired. �3. Proofs of Theorem 2.3 and Theorem 2.4

In this section we prove the two sum relations of the quasi-shuffle product.

3.1. Proof of Theorem 2.3

We first consider the case when k = 2. By the basic relation zr ∗ zs1 = zr,s1 + zs1,r + zr+s1 fromEq. (8), we have


∑r,s1�1, r+s1=n

(zr ∗ zs1) =∑

r,s1�1, r+s1=n

zr,s1 +∑

r,s1�1, r+s1=n

zs1,r +∑

r,s1�1, r+s1=n

zr+s1

= 2∑

ti�1, t1+t2=n

zt1,t2 + (n − 1)zn,

as needed.In the general case we prove Eq. (23) by induction on n. If n = 2, then k = 2 and we are done. For

a given integer m � 3, assume that Eq. (23) holds when n < m and consider Eq. (23) when n = m andk � 2. Since we have proved the case when k = 2, we may assume that k � 3. By Eq. (8), we have

∑r,si�1

r+s1+···+sk−1=n

(zr ∗ zs1,...,sk−1) =∑

r,si�1r+s1+···+sk−1=n

zr,s1,...,sk−1 +∑

r,si�1r+s1+···+sk−1=n

zs1(zr ∗ zs2,...,sk−1)

+∑

r,si�1r+s1+···+sk−1=n

zr+s1,s2,...,sk−1 . (27)

In the second sum on the right-hand side of Eq. (27), for any fixed s1 � 1, by the induction hypothesiswe have

∑r,si�1, r+s2+···+sk−1=n−s1

zr ∗ zs2,...,sk−1

= (k − 1)∑

ti�1, t1+···+tk−1=n−s1

zt1,...,tk−1 + (n − s1 − k + 2)∑

ui�1, u1+···+uk−2=n−s1

zu1,...,uk−2 .

So the second sum becomes

∑r,si�1

r+s1+s2+···+sk−1=n

zs1(zr ∗ zs2,...,sk−1)

= (k − 1)∑

s1�1, ti�1s1+t1+···+tk−1=n

zs1,t1,...,tk−1 + (n − s1 − k + 2)∑

s1�1, ui�1s1+u1+···+uk−2=n

zs1,u1,...,uk−2

= (k − 1)∑

ti�1, t1+···+tk=n

zt1,...,tk + (n − u1 − k + 2)∑

ui�1, u1+···+uk−1=n

zu1,...,uk−1 .

For the third sum on the right-hand side of Eq. (27), we have

∑r,si�1, r+s1+···+sk−1=n

zr+s1,s2,...,sk−1 =∑

ui�1, u1+···+uk−1=n

(u1 − 1)zu1,...,uk−1 .

Therefore Eq. (27) becomes

∑r,si�1

r+s1+···+sk−1=n

zr ∗ zs1,...,sk−1

=∑

t �1, t +···+t =n

zt1,...,tk +∑

t �1, t +···+t =n

(k − 1)zt1,...,tk

i 1 k i 1 k


+∑

ui�1, u1+···+uk−1=n

(n − u1 − k + 2)zu1,...,uk−1 +∑

ui�1, u1+···+uk−1=n

(u1 − 1)zu1,...,uk−1

=∑

ti�1, t1+···+tk=n

kzt1,...,tk +∑

ui�1, u1+···+uk−1=n

(n − k + 1)zu1,...,uk−1 .

This means that Eq. (23) holds when n = m, completing the inductive proof of Theorem 2.3.


We now prove Theorem 2.4 by applying Theorem 2.3. When k = 2 we can verify Eq. (24) directlyby Eq. (8) as in the case of Theorem 2.3. For k � 3, applying Eq. (8) and Eq. (23), we have

∑r,si�1

r+s2+···+sk−1=n−1

zr ∗ z1,s2,...,sk−1

=∑

r,si�1r+s2+···+sk−1=n−1

zr,1,s2,...,sk−1 +∑

r,si�1r+s2+···+sk−1=n−1

z1(zr ∗ zs2,...,sk−1)

+∑

r,si�1r+s2+···+sk−1=n−1

zr+1,s2,...,sk−1

=∑

ti�1, t2=1t1+···+tk=n

zt1,...,tk

+(

(k − 1)∑

t2+···+tk=n−1

z1,t2,...,tk + [(n − 1) − (k − 1) + 1

] ∑u2+···+uk−1=n−1

z1,u2,...,uk−1

)

+∑

ui�1, u1�2u1+···+uk−1=n

zu1,...,uk−1 ,

where k and n in Eq. (23) are replaced by k − 1 and n − 1. By regrouping, this expression can befurther simplified to

(k − 1)∑

ti�1, t1=1t1+···+tk=n

zt1,t2,...,tk +∑

ti�1, t2=1t1+···+tk=n

zt1,t2,...,tk + (n − k + 1)∑

ui�1, u1=1u1+···+uk−1=n

zu1,u2,...,uk−1

+∑

ui�1, u1�2u1+···+uk−1=n

zu1,u2,...,uk−1

= (k − 1)∑

ti�1, t1=1, t2�2t1+···+tk=n

zt1,t2,...,tk +∑

ti�1, t1�2, t2=1t1+···+tk=n

zt1,t2,...,tk + k∑

ti�1, t1=t2=1t1+···+tk=n

zt1,t2,...,tk

+ (n − k + 1)∑

ui�1, u1=1u1+···+uk−1=n

zu1,u2,...,uk−1 +∑

ui�1, u1�2u1+···+uk−1=n

zu1,u2,...,uk−1 .

Now Eq. (24) follows from this and Eq. (23).


4. Proofs of Theorem 2.5 and Theorem 2.6

In this section we give the proofs of the two sum relations of the shuffle product.

4.1. A preparational lemma

In this subsection we provide a lemma that is needed in the proofs of Theorem 2.5 and Theo-rem 2.6.

Let H∗+ be the subring of H∗ with a Z-basis consisting of words of the form z�s with �s ∈ Zk�1,

k � 1. Then

H∗ = Z ⊕ H∗+.

Likewise let HX+1 be the subring of HX

1 with a Z-basis consisting of words of the form ux1. Then

HX1 = Z ⊕ HX+

1 .

Define two operators

P : H∗+ → H∗, P (zs1,s2,...,sk ) = zs1+1,s2,...,sk ,

Q : H∗ → H∗, Q (w) ={

z1 w, w = 1,

z1, w = 1.(28)

These operators are simply the transports of the operators

I0 : HX+1 → HX

1 , I0(u) = x0u,

I1 : HX1 → HX

1 , I1(u) ={

x1u, u = 1,

x1, u = 1.

Thus the recursive definition of X in Eq. (11) gives the following Rota–Baxter relation [1,11,12,31]from [13].

Proposition 4.1. (See [13].) The multiplication X∗ on H∗ defined in Eq. (17) is the unique one such that

P (w1)X∗ P (w2) = P(

w1 X∗ P (w2)) + P

(P (w1)X∗ w2

), w1, w2 ∈ H∗+

,

Q (w1)X∗ Q (w2) = Q(

w1 X∗ Q (w2)) + Q

(Q (w1)X∗ w2

), w1, w2 ∈ H∗,

P (w1)X∗ Q (w2) = Q(

P (w1)X∗ w2) + P

(w1 X∗ Q (w2)

), w1 ∈ H∗+

, w2 ∈ H∗,

Q (w1)X∗ P (w2) = Q(

w1 X∗ P (w2)) + P

(Q (w1)X∗ w2

), w1 ∈ H∗, w2 ∈ H∗+

, (29)

with the initial condition that 1 X∗ w = w X∗ 1 = w for w ∈ H∗ .

Lemma 4.2. For positive integers k � 2 and n � k, we have

z1 X∗∑

si�1, s1�2s1+···+sk−1=n−1

zs1,...,sk−1

=∑

ti�1, t1=1, t2�2t +···+t =n

zt1,t2,...,tk + k∑

ti�1, t1�2, tk=1t +···+t =n

zt1,...,tk + (k − 1)∑

ti�1, t1,tk�2t +···+t =n

zt1,...,tk (30)

1 k 1 k 1 k


and

z1 X∗∑si�1

s1+···+sk−1=n−1

zs1,...,sk−1 = k∑

ti�1, tk=1t1+···+tk=n

zt1,...,tk + (k − 1)∑

ti�1, tk�2t1+···+tk=n

zt1,...,tk . (31)

Both sides of Eq. (30) are zero when n = k.

Proof. We prove Eq. (30) and (31) by induction on n. If n = 2,3, Eq. (30) and (31) can be verifieddirectly. Let m � 4 be an integer. Assume that Eq. (30) and (31) hold when n < m. Now assume thatn = m.

By Eq. (28), Eq. (29) and the induction hypothesis, the left-hand side of Eq. (30) becomes

z1 X∗∑

si�1, s1�2s1+···+sk−1=n−1

zs1,...,sk−1

= Q (1)X∗ P

( ∑si�1, s1+···+sk−1=n−2

zs1,...,sk−1

)

= Q

( ∑si�1, s1�2

s1+···+sk−1=n−1

zs1,...,sk−1

)+ P

(z1 X∗

∑si�1

s1+···+sk−1=n−2

zs1,...,sk−1

)

=∑

ti�1, t2�2t2+···+tk+1=n−1

z1,t2,...,tk+1 + P

(k

∑ti�1, tk=1

t1+···+tk=n−1

zt1,...,tk + (k − 1)∑

ti�1, tk�2t1+···+tk=n−1

zt1,...,tk

)

=∑

ti�1, t2�2t2+···+tk+1=n−1

z1,t2,...,tk+1 + k∑

ti�1, t1�2, tk=1t1+···+tk=n

zt1,...,tk + (k − 1)∑

ti�1, t1,tk�2t1+···+tk=n

zt1,...,tk ,

which shows that Eq. (30) holds when n = m.By a similar argument, we have

z1 X∗∑

s2+···+sk−1=n−2

z1,s2,...,sk−1

= Q

( ∑s2+···+sk−1=n−2

z1,s2,...,sk−1

)+ Q

(z1 X∗

∑s2+···+sk−1=n−2

zs2,...,sk−1

)

=∑

si�1, s2+···+sk−1=n−2

z1,1,s2,...,sk−1

+ Q

((k − 1)

∑ti�1, tk−1=1

t1+···+tk−1=n−1

zt1,...,tk−1 + (k − 2)∑

ti�1, tk−1�2t1+···+tk−1=n−1

zt1,...,tk−1

)

=∑

ti�1, t1=t2=1t +···+t =n

zt1,t2,...,tk + (k − 1)∑

ti�1, t1=tk=1t +···+t =n

zt1,t2,...,tk + (k − 2)∑

ti�1, t1=1, tk�2t +···+t =n

zt1,t2,...,tk .

1 k 1 k 1 k


Adding this with Eq. (30) we obtain

z1 X∗∑si�1

s1+···+sk−1=n−1

zs1,...,sk−1

=∑

ti�1, t1=1, t2�2t1+···+tk=n

zt1,t2,...,tk + k∑

ti�1, t1�2, tk=1t1+···+tk=n

zt1,...,tk + (k − 1)∑

ti�1, t1,tk�2t1+···+tk=n

zt1,...,tk

+∑

ti�1, t1=t2=1t1+···+tk=n

zt1,t2,...,tk + (k − 1)∑

ti�1, t1=tk=1t1+···+tk=n

zt1,t2,...,tk + (k − 2)∑

ti�1, t1=1, tk�2t1+···+tk=n

zt1,t2,...,tk

= k∑

ti�1, t1�2, tk=1t1+···+tk=n

zt1,...,tk + (k − 1)∑

ti�1, t1,tk�2t1+···+tk=n

zt1,...,tk +∑

ti�1, t1=1t1+···+tk=n

zt1,t2,...,tk

+ (k − 1)∑

ti�1, t1=tk=1t1+···+tk=n

zt1,t2,...,tk + (k − 2)∑

ti�1, t1=1, tk�2t1+···+tk=n

zt1,t2,...,tk

= k∑

ti�1, t1�2, tk=1t1+···+tk=n

zt1,...,tk + (k − 1)∑

ti�1, t1,tk�2t1+···+tk=n

zt1,...,tk + k∑

ti�1, t1=tk=1t1+···+tk=n

zt1,t2,...,tk

+ (k − 1)∑

ti�1, t1=1, tk�2t1+···+tk=n

zt1,t2,...,tk

= k∑

ti�1, tk=1t1+···+tk=n

zt1,...,tk + (k − 1)∑

ti�1, tk�2t1+···+tk=n

zt1,...,tk ,

which shows that Eq. (31) holds when n = m. This completes the induction. �4.2. Proof of Theorem 2.5

For the proof of Theorem 2.5, we first consider the case of k = 2. In this case we have Euler’sdecomposition formula [8,13]

zr X∗ zs1 =∑

t1�1, t2�1, t1+t2=n

((t1 − 1

r − 1

)+

(t1 − 1

s1 − 1

))zt1,t2 .

So

∑r�1, s1�1, r+s1=n

zr X∗ zs1 =∑

t1�1, t2�1, t1+t2=n

( t1∑r=1

(t1 − 1

r − 1

)+

t1∑s1=1

(t1 − 1

s1 − 1

))zt1,t2 = 2t1 zt1,t2 ,

as required.For the general case we prove Eq. (25) by induction on n. If n = 2, then k = 2 and so we are done

by the above argument. Let m be a positive integer � 3 and assume that Eq. (25) holds for n < m.Now assume that n = m. Since we have dealt with the case of k = 2, we may assume that k � 3.

We decompose the left-hand side of Eq. (25) into three disjoint parts when r = 1, s1 � 1, whenr � 2, s1 � 2 and when r � 2, s1 = 1:


∑r,si�1

r+s1+···+sk−1=n

zr X∗ zs1,...,sk−1

=∑si�1

s1+···+sk−1=n−1

z1 X∗ zs1,...,sk−1 +∑

si�1, r,s1�2r+s1+···+sk−1=n

zr X∗ zs1,...,sk−1

+∑

r�2, si�1r+s2+···+sk−1=n−1

zr X∗ z1,s2,...,sk−1 . (32)

We denote the three sums by S1, S2 and S3 respectively with the given order.For the sum S2, by Eq. (28) and (29), we have

∑si�1, r,s1�2

r+s1+···+sk−1=n

zr X∗ zs1,...,sk−1

=∑

si�1, r,s1�2r+s1+···+sk−1=n

P (zr−1)X∗ P (zs1−1,...,sk−1)

= P

( ∑si�1, r,s1�2

r+s1+···+sk−1=n

zr−1 X∗ P (zs1−1,...,sk−1)

)+ P

( ∑si�1, r,s1�2

r+s1+···+sk−1=n

P (zr−1)X∗ zs1,...,sk−1

)

= P

( ∑si�1, r,s1�2

r+s1+···+sk−1=n

zr−1 X∗ zs1,...,sk−1

)+ P

( ∑si�1, r,s1�2

r+s1+···+sk−1=n

zr X∗ zs1−1,...,sk−1

).

We will denote S2,1 and S2,2 for the first and second sum respectively in the last expression. Simi-larly for the sum S3, we have

∑r�2, si�1

r+s2+···+sk−1=n−1

zr X∗ z1,s2,...,sk−1

=∑

r�2, si�1r+s2+···+sk=n−1

P (zr−1)X∗ Q (zs2,...,sk−1)

= P

( ∑r�2, si�1

r+s2+···+sk−1=n−1

zr−1 X∗ Q (zs2,...,sk−1)

)+ Q

( ∑r�2, si�1

r+s2+···+sk−1=n−1

P (zr−1)X∗ zs2,...,sk−1

)

= P

( ∑r�2, si�1

r+s2+···+sk−1=n−1

zr−1 X∗ z1,s2,...,sk−1

)+ Q

( ∑r�2, si�1

r+s2+···+sk−1=n−1

zr X∗ zs2,...,sk−1

).

Let S3,1 and S3,2 denote the first and the second sum respectively in the last expression.By Lemma 4.2 we have

S1 = k∑

t �1, t +···+t =n

zt1,...,tk −∑

t �1, t �2, t +···+t =n

zt1,...,tk . (33)
i 1 k i k 1 k


We also have

S2,1 + S3,1 = P

( ∑r�2, si�1

r+s1+···+sk−1=n

zr−1 X∗ zs1,s2,...,sk−1

)

= P

( ∑r,si�1

r+s1+···+sk−1=n−1

zr X∗ zs1,s2,...,sk−1

)

= P

( ∑ti�1

t1+···+tk=n−1

C(t1, . . . , tk−1)zt1,...,tk

)(by the induction hypothesis) (34)

=∑

ti�1, t1+···+tk=n−1

C(t1, . . . , tk−1)zt1+1,...,tk

=∑

ti�1, t1�2, t1+···+tk=n

C(t1 − 1, t2, . . . , tk−1)zt1,...,tk .

For the sum S2,2, we have

S2,2 = P

( ∑r�2, si�1, r+s1+···+sk−1=n−1

zr X∗ zs1,...,sk−1

)

= P

( ∑r,si�1, r+s1+···+sk−1=n−1

zr X∗ zs1,...,sk−1 −∑

si�1, s1+···+sk−1=n−2

z1 X∗ zs1,...,sk−1

)

= P

( ∑ti�1, t1+···+tk=n−1

C(t1, . . . , tk−1)zt1,...,tk − k∑

ti�1, tk=1, t1+···+tk=n−1

zt1,...,tk

−(k − 1)∑

ti�1, tk�2t1+···+tk=n−1

zt1,...,tk

) (by the induction hypothesis and Eq. (31)

)

=∑ti�1

t1+···+tk=n−1

C(t1, . . . , tk−1)zt1+1,...,tk − k∑ti�1

t1+···+tk=n−1

zt1+1,...,tk +∑

ti�1, tk�2t1+···+tk=n−1

zt1+1,...,tk

=∑

ti�1, t1�2t1+···+tk=n

(C(t1 − 1, . . . , tk−1) − k

)zt1,...,tk +

∑ti�1, t1�2, tk�2

t1+···+tk=n

zt1,...,tk . (35)

For the sum S3,2, we have

S3,2 = Q

( ∑r,s �1, r+s +···+s =n−1

zr X∗ zs2,...,sk−1 −∑

s +···+s =n−2

z1 X∗ zs2,...,sk−1

)
i 2 k−1 2 k−1


= Q

( ∑ti�1, t2+···+tk=n−1

C(t2, . . . , tk−1)zt2,...,tk −[ ∑

ti�1, tk=1,t2+···+tk=n−1

(k − 1)zt2,...,tk

+∑

ti�1, tk�2t2+···+tk=n−1

(k − 2)zt2,...,tk

]) (by the induction hypothesis and Eq. (31)

)

= Q

( ∑ti�1, t2+···+tk=n−1

[C(t2, . . . , tk−1) − (k − 1)

]zt2,...,tk +

∑ti�1, tk�2

t2+···+tk=n−1

zt2,...,tk

)

=∑

ti�1, t1=1t1+···+tk=n

[C(t1, . . . , tk−1) − k

]zt1,t2,...,tk +

∑ti�1, t1=1, tk�2

t1+···+tk=n

zt1,t2,...,tk (36)

by Eq. (21).Adding Eq. (34) and (35) we obtain

S2,1 + S2,2 + S3,1 =∑

ti�1, t1�2t1+···+tk=n

(2C(t1−1, . . . , tk−1) − k

)zt1,...,tk +

∑ti�1, t1,tk�2t1+···+tk=n

zt1,...,tk

=∑

ti�1, t1�2t1+···+tk=n

(C(t1, . . . , tk−1) − k

)zt1,...,tk +

∑ti�1, t1,tk�2t1+···+tk=n

zt1,...,tk (37)

by Eq. (20). Next adding Eq. (36) and (37) together, we get

S2,1 + S2,2 + S3,1 + S3,2 =∑ti�1

t1+···+tk=n

(C(t1, . . . , tk−1) − k

)zt1,...,tk +

∑ti�1, tk�2

t1+···+tk=n

zt1,...,tk . (38)

Finally adding Eq. (33) and (38), we obtain Eq. (25). This completes the inductive proof of Theo-rem 2.5.


The proof of Theorem 2.6 is similar to that of Theorem 2.5.First we prove that Eq. (26) holds when k = 2. This follows from

∑r�1, s1�2

r+s1=n

zr X∗ zs1 =∑

r�1, s1�2r+s1=n

∑t1,t2�1t1+t2=n

((t1 − 1

r − 1

)+

(t1 − 1

s1 − 1

))zt1 zt2

=∑

t1,t2�1, t1+t2=n

( min(t1,n−2)∑r=1

(t1 − 1

r − 1

)+

t1∑s1=2

(t1 − 1

s1 − 1

))zt1 zt2

=(

n−2∑t1=1

(2t1 − 1

)zt1,n−t1

)+ (

2n−1 − 2)zn−1,1

=(

n−1∑t =1

(2t1 − 1

)zt1,n−t1

)− zn−1,1.

1


For the general case we prove Eq. (26) by induction on n. If n = 3, then k = 2 and we are done.Let m � 4 be an integer. Assume that Eq. (26) holds when n � m − 1. We will prove that it holdswhen n = m. Since we have dealt with the case of k = 2, we may assume that k � 3 without loss ofgenerality.

By Eq. (29) the left-hand side of Eq. (26) is equal to

∑r,si�1, s1�2

r+s1+···+sk−1=n

zr X∗ zs1,...,sk−1

= z1 X∗( ∑

si�1, s1�2s1+···+sk−1=n−1

zs1,...,sk−1

)+

∑r�2, si�1, s1�2r+s1+···+sk−1=n

P (zr−1)X∗ P (zs1−1,s2,...,sk−1)

= z1 X∗( ∑

si�1, s1�2s1+···+sk−1=n−1

zs1,...,sk−1

)+ P

( ∑r�2, si�1, s1�2r+s1+···+sk−1=n

zr−1 X∗ zs1,s2,...,sk−1

)

+ P

( ∑r�2, si�1, s1�2r+s1+···+sk−1=n

zr X∗ zs1−1,s2,...,sk−1

)

= z1 X∗( ∑

si�1, s1�2s1+···+sk−1=n−1

zs1,...,sk−1

)+ P

( ∑r�1, si�1, s1�2

r+s1+···+sk−1=n−1

zr X∗ zs1,s2,...,sk−1

)

+ P

( ∑r�2, si�1

r+s1+···+sk−1=n−1

zr X∗ zs1,s2,...,sk−1

).

We denote the three sums in the last expression by S1, S2 and S3 respectively with the given order.By Eq. (30) and (21), we have

S1 =∑

ti�1, t1=1, t2�2t1+···+tk=n

zt1,t2,...,tk + k∑

ti�1, t1�2, tk=1t1+···+tk=n

zt1,...,tk + (k − 1)∑

ti�1, t1,tk�2t1+···+tk=n

zt1,...,tk

=∑

ti�1, t1=1t1+···+tk=n

[C(t1, . . . , tk) − C(t2, . . . , tk−1)

]zt1,...,tk −

∑ti�1, t1=t2=1

t1+···+tk=n

zt1,...,tk

+k∑

ti�1, t1�2, tk=1t1+···+tk=n

zt1,...,tk + (k − 1)∑

ti�1, t1�2, tk�2t1+···+tk=n

zt1,...,tk . (39)

By the induction hypothesis, we have

S2 = P

( ∑r�1, si�1, s1�2

r+s1+···+sk−1=n−1

zr X∗ zs1,s2,...,sk−1

)

= P

( ∑ti�1

t +···+t =n−1

[C(t1, . . . , tk−1) − C(t2, . . . , tk−1)

]zt1,...,tk −

∑ti�1, t2=1

t +···+t =n−1

zt1,...,tk

)
1 k 1 k


=∑ti�1

t1+···+tk=n−1

[C(t1, . . . , tk−1) − C(t2, . . . , tk−1)

]zt1+1,...,tk −

∑ti�1, t2=1t1+···+tk=n−1

zt1+1,...,tk

=∑

ti�1, t1�2t1+···+tk=n

[C(t1 − 1, . . . , tk−1) − C(t2, . . . , tk−1)

]zt1,...,tk −

∑ti�1, t1�2, t2=1

t1+···+tk=n

zt1,...,tk .

By Theorem 2.5 and Eq. (31), we have

S3 = P

( ∑r�2, si�1

r+s1+···+sk−1=n−1

zr X∗ zs1,s2,...,sk−1

)

= P

( ∑r�1, si�1

r+s1+···+sk−1=n−1

zr X∗ zs1,s2,...,sk−1 −∑si�1

s1+···+sk−1=n−2

z1 X∗ zs1,s2,...,sk−1

)

= P

( ∑ti�1

t1+···+tk=n−1

C(t1, . . . , tk−1)zt1,...,tk

−k∑

ti�1, tk=1t1+···+tk=n−1

zt1,...,tk − (k − 1)∑

ti�1, tk�2t1+···+tk=n−1

zt1,...,tk

)

= P

( ∑ti�1, t1+···+tk=n−1

[C(t1, . . . , tk−1) − k

]zt1,...,tk +

∑ti�1, tk�2

t1+···+tk=n−1

zt1,...,tk

)

=∑

ti�1, t1+···+tk=n−1

[C(t1, . . . , tk−1) − k

]zt1+1,...,tk +

∑ti�1, tk�2

t1+···+tk=n−1

zt1+1,...,tk

=∑

ti�1, t1�2, t1+···+tk=n

[C(t1 − 1, t2, . . . , tk−1) − k

]zt1,...,tk +

∑ti�1, t1,tk�2t1+···+tk=n

zt1,...,tk .

Hence, using Eq. (20) we obtain

S2 + S3 =∑

ti�1, t1�2t1+···+tk=n

(2C(t1 − 1, t2, . . . , tk−1) − C(t2, . . . , tk−1) − k

)zt1,...,tk

−∑

ti�1, t1�2, t2=1t1+···+tk=n

zt1,...,tk +∑

ti�1, t1,tk�2t1+···+tk=n

zt1,...,tk (40)

=∑

ti�1, t1�2t1+···+tk=n

(C(t1, t2, . . . , tk) − C(t2, . . . , tk−1) − k

)zt1,...,tk

−∑

ti�1, t1�2, t2=1t +···+t =n

zt1,...,tk +∑

ti�1, t1,tk�2t +···+t =n

zt1,...,tk .

1 k 1 k


Adding Eq. (39) and (40) together we obtain

S1 + S2 + S3 =∑

ti�1, t1=1t1+···+tk=n

[C(t1, . . . , tk) − C(t2, . . . , tk−1)

]zt1,...,tk −

∑ti�1, t1=t2=1

t1+···+tk=n

zt1,...,tk

+ k∑

ti�1, t1�2, tk=1t1+···+tk=n

zt1,...,tk + (k − 1)∑

ti�1, t1�2, tk�2t1+···+tk=n

zt1,...,tk

+∑

ti�1, t1�2t1+···+tk=n

[C(t1, t2, . . . , tk) − C(t2, . . . , tk−1) − k

]zt1,...,tk

−∑

ti�1, t1�2, t2=1t1+···+tk=n

zt1,...,tk +∑

ti�1, t1,tk�2t1+···+tk=n

zt1,...,tk

=∑

ti�1, t1=1t1+···+tk=n

[C(t1, . . . , tk) − C(t2, . . . , tk−1)

]zt1,...,tk −

∑ti�1, t1=t2=1

t1+···+tk=n

zt1,...,tk

+∑

ti�1, t1�2t1+···+tk=n

[C(t1, t2, . . . , tk) − C(t2, . . . , tk−1)

]zt1,...,tk −

∑ti�1, t1�2, t2=1

t1+···+tk=n

zt1,...,tk

=∑ti�1

t1+···+tk=n

[C(t1, t2, . . . , tk) − C(t2, . . . , tk−1)

]zt1,...,tk −

∑ti�1, t2=1

t1+···+tk=n

zt1,...,tk ,

as desired. This completes the inductive proof of Theorem 2.6.

Acknowledgments

Both authors thank the Max Planck Institute for Mathematics in Bonn for providing them with theenvironment to carry out this research. They also thank Don Zagier and Wadim Zudilin for helpfuldiscussions. The first author acknowledges the support from NSF grant DMS-0505643.

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Weighted sum formula for multiple zeta values