Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values

Shingo Saito and Tatsushi Tanaka Faculty of Mathematics Kyushu University 744, Motooka, Nishi-ku Fukuoka, 819–0395 Japan

              ssaito@math.kyushu-u.ac.jp
             t.tanaka@math.kyushu-u.ac.jp

Noriko Wakabayashi Faculty of Engineering Kyushu Sangyo University 3–1, Matsukadai 2-chome, Higashi-ku Fukuoka, 813–8503 Japan

               noriko@ip.kyusan-u.ac.jp

Abstract

The multiple zeta values are generalizations of the values of the Riemann zeta function at positive integers. They are known to satisfy a number of relations, among which are the cyclic sum formula. The cyclic sum formula can be stratified via linear operators defined by the second and third authors. We give the number of relations belonging to each stratum by combinatorial arguments.

1 Introduction

The Riemann zeta function

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}}

is one of the most important functions in mathematics, and its values ζ(k) at k ∈ ℤ_{≥2} are among the most interesting real numbers. The multiple zeta values (MZVs for short) are defined as generalizations of the values ζ(k):

Definition 1. For k_{1}, …, k_{l} ∈ ℤ_{≥1} with k_{1} ≥ 2, the multiple zeta value ζ(k_{1}, …, k_{l}) is a real number defined by

ζ(k_{1}, …, k_{l}) = ∑_{n_{1}>···>n_{l}≥1} \frac{1}{n_{1}^{k_{1}} ⋯ n_{l}^{k_{l}}} .

Little is known about the irrationality or transcendentality of MZVs; we however know that MZVs abound with relations among themselves, the simplest example being ζ(2, 1) = ζ(3).

The cyclic sum formula (CSF for short), described in Section 3, is a class of ℚ-linear relations among MZVs, established by Hoffman-Ohno [3]. Their proof appealed to partial fraction decomposition, whereas another proof given by the second and third authors [5] of the present article proceeded by showing that the CSF is included in Kawashima’s relation, which is believed to be rich enough to yield all relations among MZVs.

This present paper is aimed at providing combinatorial arguments to find the ranks of linear operators defined in [5]. In order to facilitate access for both algebraists and combinatorists, we strive to make the exposition as self-contained as possible.

Sets of multi-indices

The study of MZVs inevitably requires frequent use of multi-indices. We here summarize the sets of multi-indices used in this paper. For k, l ∈ ℤ_{≥1}, put

I_{k,l}^{1} = {(k_{1}, …, k_{l}) ∈ ℤ_{≥1}^{l} | k_{1} + ⋯ + k_{l} = k},
I_{k,l}^{0} = {(k_{1}, …, k_{l}) ∈ I_{k,l}^{1} | k_{1} ≥ 2},
Ǐ_{k,l}^{1} = {(k_{1}, …, k_{l}) ∈ I_{k,l}^{1} | not all of k_{1}, …, k_{l} are 1};

for k ∈ ℤ_{≥1}, put

I_{k}^{1} = ⋃_{l=1}^{∞} I_{k,l}^{1},  I_{k}^{0} = ⋃_{l=1}^{∞} I_{k,l}^{0},  Ǐ_{k}^{1} = ⋃_{l=1}^{∞} Ǐ_{k,l}^{1};

put

I^{1} = ⋃_{k,l=1}^{∞} I_{k,l}^{1},  I^{0} = ⋃_{k,l=1}^{∞} I_{k,l}^{0},  Ǐ^{1} = ⋃_{k,l=1}^{∞} Ǐ_{k,l}^{1}.

The elements of I_{k,l}^{1} are said to have weight k and depth l. For each k = (k_{1}, …, k_{l}) ∈ I^{0} the MZV ζ(k) = ζ(k_{1}, …, k_{l}) is defined.

2 Hoffman’s algebra

In the discussion of MZVs, it is convenient to use the algebra introduced by Hoffman. Let ℌ = ℚ⟨x, y⟩ be the noncommutative polynomial algebra over ℚ in two indeterminates x and y, and put ℌ^{0} = ℚ + xℌy, which is a subalgebra of ℌ. Write z_{k} = x^{k−1}y ∈ ℌ for k ∈ ℤ_{≥1}, so that

{1} ∪ {z_{k_{1}} ⋯ z_{k_{l}} | k = (k_{1}, …, k_{l}) ∈ I^{0}}

is a ℚ-vector space basis for ℌ^{0}. It follows that we may define a ℚ-linear map Z: ℌ^{0} → ℝ by setting Z(1) = 1 and Z(z_{k_{1}} ⋯ z_{k_{l}}) = ζ(k) for k = (k_{1}, …, k_{l}) ∈ I^{0}.

The MZVs are known to fulfill a number of ℚ-linear relations, each of which corresponds to an element of Ker Z ⊂ ℌ^{0}. Since Goncharov [1] conjectured that the MZVs ζ(k) with k having different weights are ℚ-linearly independent, we look at the ℚ-vector subspaces defined by

ℌ_{k}^{0} = span_{ℚ}{z_{k_{1}} ⋯ z_{k_{l}} | k = (k_{1}, …, k_{l}) ∈ I_{k}^{0}}
= {w ∈ ℌ^{0} | w is a homogeneous polynomial of degree k} ∪ {0} ⊂ ℌ^{0},
Z_{k} = Z(ℌ_{k}^{0}) = span_{ℚ}{ζ(k) | k ∈ I_{k}^{0}} ⊂ ℝ

for each k ∈ ℤ_{≥1}.

Let (d_{k})_{k≥1} be the Padovan sequence (\underline{A000931}) defined by d_{1} = 0, d_{2} = d_{3} = 1, and d_{k} = d_{k−2} + d_{k−3} for k ≥ 4. Zagier [7] conjectured the following:

Conjecture 2 (Zagier [7]). We have dim Z_{k} = d_{k} for all k ∈ ℤ_{≥1}.

Let k ∈ ℤ_{≥2}. In light of the fact that dim ℌ_{k}^{0} = #I_{k}^{0} = 2^{k−2}, Conjecture 2 means that the MZVs must satisfy plenty of ℚ-linear relations. Note that Conjecture 2 is equivalent to saying that the restriction Z|_{ℌ_{k}^{0}} : ℌ_{k}^{0} → ℝ of Z to ℌ_{k}^{0} has rank d_{k}, which is also equivalent to

dim Ker Z|_{ℌ_{k}^{0}} = dim(Ker Z ∩ ℌ_{k}^{0}) = 2^{k−2} − d_{k}.

Table 1: Dimensions concerning Z|_{ℌ_{k}^{0}}
k  2 3 4 5 6 7 8 9 10 Sequence Number
2^{k−2}(= dim ℌ_{k}^{0})  1 2 4 8 16 32 64 128 256  \underline{A000079}
d_{k}(=^{?} rank Z|_{ℌ_{k}^{0}})  1 1 1 2 2 3 4 5 7  \underline{A000931}
2^{k−2} − d_{k}(=^{?} dim Ker Z|_{ℌ_{k}^{0}}) 0 1 3 6 14 29 60 123 249  \underline{A038360}

Goncharov [2] and Terasoma [6] partially proved Conjecture 2:

Theorem 3 (Goncharov [2], Terasoma [6]). We have dim Z_{k} ≤ d_{k} for all k ∈ ℤ_{≥1}.

Since their proofs of Theorem 3 resort to algebraic geometry and fails to give concrete relations among MZVs, it still lies at the heart of research to find sufficiently many ℚ-linear relations. Also, the converse inequality is far from being solved.

3 Cyclic sum formula

Numerous concrete ℚ-linear relations among MZVs have been obtained so far, and our focus is on the following cyclic sum formula (CSF for short), first proved by Hoffman-Ohno [3]:

Theorem 4 (Cyclic sum formula). If (k_{1}, …, k_{l}) ∈ Ǐ^{1}, then

∑_{j=1}^{l} ∑_{i=1}^{k_{j}−1} ζ(k_{j} − i + 1, k_{j+1}, …, k_{l}, k_{1}, …, k_{j−1}, i) = ∑_{j=1}^{l} ζ(k_{j} + 1, k_{j+1}, …, k_{l}, k_{1}, …, k_{j−1}).

Example 5. The cyclic sum formula for l = 1 and k_{1} = 2 gives ζ(2, 1) = ζ(3).

In dealing with the CSF, it is convenient to extend the indices of k_{j} to all j ∈ ℤ by declaring k_{j} = k_{j′} whenever j ≡ j′ (mod l). Then the CSF can simply be written as

∑_{j=1}^{l} ∑_{i=1}^{k_{j}−1} ζ(k_{j} − i + 1, k_{j+1}, …, k_{j+l−1}, i) = ∑_{j=1}^{l} ζ(k_{j} + 1, k_{j+1}, …, k_{j+l−1}).

This convention will be used tacitly throughout the paper.

In order to describe the CSF in terms of Hoffman’s algebra, we write

ℌ̌^{1} = span_{ℚ}{z_{k_{1}} ⋯ z_{k_{l}} | k = (k_{1}, …, k_{l}) ∈ Ǐ^{1}}

= span_{ℚ}{w ∈ ℌ | w is a monomial ending with y but not a power of y} ⊂ ℌ

and define a ℚ-linear map ρ: ℌ̌^{1} → xℌy ⊂ ℌ^{0} by setting

ρ(z_{k_{1}} ⋯ z_{k_{l}}) = ∑_{j=1}^{l} ∑_{i=1}^{k_{j}−1} z_{k_{j}−i+1}z_{k_{j+1}} ⋯ z_{k_{j+l−1}}z_{i} − ∑_{j=1}^{l} z_{k_{j}+1}z_{k_{j+1}} ⋯ z_{k_{j+l−1}}

for k = (k_{1}, …, k_{l}) ∈ Ǐ^{1}. Then the CSF is equivalent to saying that Im ρ ⊂ Ker Z. For each k ∈ ℤ_{≥1}, if we put

ℌ̌_{k}^{1} = span_{ℚ}{z_{k_{1}} ⋯ z_{k_{l}} | (k_{1}, …, k_{l}) ∈ Ǐ_{k}^{1}}
= {w ∈ ℌ̌^{1} | w is a homogeneous polynomial of degree k} ∪ {0},

then ρ satisfies that ρ(ℌ̌_{k}^{1}) ⊂ ℌ_{k+1}^{0}. Therefore, it follows from Theorem 4 that

ρ(ℌ̌_{k−1}^{1}) ⊂ Ker Z ∩ ℌ_{k}^{0}

for all k ∈ ℤ_{≥2}.

In view of Conjecture 2, which is equivalent to dim(Ker Z ∩ ℌ_{k}^{0}) = 2^{k−2} − d_{k} for k ∈ ℤ_{≥2}, it is natural to ask for dim ρ(ℌ̌_{k−1}^{1}) because it can be regarded as the number of relations given by the CSF. The following theorem is known, though the authors have been unable to find a specific reference:

Theorem 6. For k ∈ ℤ_{≥2}, we have

dim ρ(ℌ̌_{k−1}^{1}) = \frac{1}{k − 1} ∑_{m|k−1} φ(\frac{k − 1}{m})2^{m} − 2,

where φ denotes Euler’s totient function.

Although this theorem can be proved rather easily, we omit its proof because it is a special case of our main theorem (Theorem 15).

Table 2: Dimensions concerning the CSF

k  2 3 4 5 6 7 8 9 10 Sequence Number
2^{k−2} − d_{k}(=^{?} dim(Ker Z ∩ ℌ_{k}^{0})) 0 1 3 6 14 29 60 123 249  \underline{A038360}
dim ρ(ℌ̌_{k−1}^{1})  0 1 2 4 6 12 18 34 58  \underline{A052823}

4 The operator ρ_{n} and the statement of our main theorem

4.1 The operator ρ_{n}

The second and third authors [5] of the present article defined linear maps ρ_{n}: ℌ → ℌ with the aim of giving an algebraic proof of the CSF by reducing it to Kawashima’s relation. We will not elaborate on their proof here, but focus on the stratification of the CSF provided by ρ_{n}. Note that our usage of indices is different from that in [5]: what we mean by ρ_{n} is denoted by ρ_{n+1} in [5].

Let n ∈ ℤ_{≥0} and consider the (n + 2)nd tensor power ℌ^{⊗(n+2)} of ℌ over ℚ. We first make ℌ^{⊗(n+2)} an ℌ-bimodule by setting

a ⋄ (w_{1} ⊗ ⋯ ⊗ w_{n+2}) ⋄ b = w_{1}b ⊗ w_{2} ⊗ ⋯ ⊗ w_{n+1} ⊗ aw_{n+2}

for a, b, w_{1}, …, w_{n+2} ∈ ℌ. Writing z = x + y, we define a ℚ-linear map C_{n}: ℌ → ℌ^{⊗(n+2)} by setting C_{n}(1) = 0, C_{n}(x) = x ⊗ z^{⊗n} ⊗ y, C_{n}(y) = −x ⊗ z^{⊗n} ⊗ y, and

C_{n}(ww′) = C_{n}(w) ⋄ w′ + w ⋄ C_{n}(w′).

We next define a ℚ-linear map M_{n}: ℌ^{⊗(n+2)} → ℌ by setting

M_{n}(w_{1} ⊗ ⋯ ⊗ w_{n+2}) = w_{1} ⋯ w_{n+2}.

Finally, set ρ_{n} = M_{n} ∘ C_{n}: ℌ → ℌ.

Remark 7. The recurrence relation in the definition of C_{n} shows that

C_{n}(w_{1} ⋯ w_{k}) = ∑_{j=1}^{k}(w_{1} ⋯ w_{j−1} ⋄ C_{n}(w_{j}) ⋄ w_{j+1} ⋯ w_{k})

for w_{1}, …, w_{k} ∈ ℌ. Proposition 8. We have ρ_{0}(w) = ρ(w) for all w ∈ ℌ̌^{1}. Proof. We may assume that w = z_{k_{1}} ⋯ z_{k_{l}} for some k = (k_{1}, …, k_{l}) ∈ Ǐ^{1}. For k ∈ ℤ_{≥1}, we have

C_{0}(z_{k}) = C_{0}(x^{k−1}y) = ∑_{i=1}^{k−1}(x^{i−1} ⋄ C_{0}(x) ⋄ x^{k−i−1}y) + x^{k−1} ⋄ C_{0}(y)

= ∑_{i=1}^{k−1}(x^{i−1} ⋄ (x ⊗ y) ⋄ x^{k−i−1}y) + x^{k−1} ⋄ (−x ⊗ y)

= ∑_{i=1}^{k−1}(x^{k−i}y ⊗ x^{i−1}y) − x ⊗ x^{k−1}y = ∑_{i=1}^{k−1}(z_{k−i+1} ⊗ z_{i}) − x ⊗ z_{k}.

It follows that

C_{0}(w) = C_{0}(z_{k_{1}} ⋯ z_{k_{l}}) = ∑_{j=1}^{l}(z_{k_{1}} ⋯ z_{k_{j−1}} ⋄ C_{0}(z_{k_{j}}) ⋄ z_{k_{j+1}} ⋯ z_{k_{l}})

= ∑_{j=1}^{l} ∑_{i=1}^{k_{j}−1}(z_{k_{1}} ⋯ z_{k_{j−1}} ⋄ (z_{k_{j}−i+1} ⊗ z_{i}) ⋄ z_{k_{j+1}} ⋯ z_{k_{l}})

− ∑_{j=1}^{l}(z_{k_{1}} ⋯ z_{k_{j−1}} ⋄ (x ⊗ z_{k_{j}}) ⋄ z_{k_{j+1}} ⋯ z_{k_{l}})

= ∑_{j=1}^{l} ∑_{i=1}^{k_{j}−1}(z_{k_{j}−i+1}z_{k_{j+1}} ⋯ z_{k_{l}} ⊗ z_{k_{1}} ⋯ z_{k_{j−1}}z_{i}) − ∑_{j=1}^{l}(xz_{k_{j+1}} ⋯ z_{k_{l}} ⊗ z_{k_{1}} ⋯ z_{k_{j}}),

and so

ρ_{0}(w) = M_{0}(C_{0}(w))

= ∑_{j=1}^{l} ∑_{i=1}^{k_{j}−1} z_{k_{j}−i+1}z_{k_{j+1}} ⋯ z_{k_{l}}z_{k_{1}} ⋯ z_{k_{j−1}}z_{i} − ∑_{j=1}^{l} xz_{k_{j+1}} ⋯ z_{k_{l}}z_{k_{1}} ⋯ z_{k_{j}}

= ∑_{j=1}^{l} ∑_{i=1}^{k_{j}−1} z_{k_{j}−i+1}z_{k_{j+1}} ⋯ z_{k_{j+l−1}}z_{i} − ∑_{j=1}^{l} xz_{k_{j+1}} ⋯ z_{k_{j+l}},

which is equal to ρ(w) because

∑_{j=1}^{l} xz_{k_{j+1}} ⋯ z_{k_{j+l}} = ∑_{j=1}^{l} xz_{k_{j}} ⋯ z_{k_{j+l−1}} = ∑_{j=1}^{l} z_{k_{j}+1}z_{k_{j+1}} ⋯ z_{k_{j+l−1}}.

4.2 Properties of ρ_{n}

Definition 9. Define sgn: {x, y, z} → {1, −1, 0} by

sgn(u) = \begin{cases}1 & if u = x; \\ −1 & if u = y; \\ 0 & if u = z, \end{cases}

so that C_{n}(u) = sgn(u)(x ⊗ z^{⊗n} ⊗ y) for u ∈ {x, y, z}. Lemma 10. If w = u_{1} ⋯ u_{k}, where u_{1}, …, u_{k} ∈ {x, y, z}, then

ρ_{n}(w) = ∑_{j=1}^{k} sgn(u_{j})xu_{j+1} ⋯ u_{k}z^{n}u_{1} ⋯ u_{j−1}y.

Proof. We have

C_{n}(w) = C_{n}(u_{1} ⋯ u_{k}) = ∑_{j=1}^{k}(u_{1} ⋯ u_{j−1} ⋄ C_{n}(u_{j}) ⋄ u_{j+1} ⋯ u_{k})

= ∑_{j=1}^{k} sgn(u_{j})(u_{1} ⋯ u_{j−1} ⋄ (x ⊗ z^{⊗n} ⊗ y) ⋄ u_{j+1} ⋯ u_{k})

= ∑_{j=1}^{k} sgn(u_{j})(xu_{j+1} ⋯ u_{k} ⊗ z^{⊗n} ⊗ u_{1} ⋯ u_{j−1}y),

and so

ρ_{n}(w) = M_{n}(C_{n}(w)) = ∑_{j=1}^{k} sgn(u_{j})xu_{j+1} ⋯ u_{k}z^{n}u_{1} ⋯ u_{j−1}y.

Proposition 11. We have

ρ_{n+1}(w) = ρ_{n}(zw)

for all w ∈ ℌ. Proof. We may assume that w = u_{1} ⋯ u_{k} for some u_{1}, …, u_{k} ∈ {x, y}. Then Lemma 10 shows that

ρ_{n}(zw) = sgn(z)xu_{1} ⋯ u_{k}z^{n}y + ∑_{j=1}^{k} sgn(u_{j})xu_{j+1} ⋯ u_{k}z^{n}zu_{1} ⋯ u_{j−1}y

= ∑_{j=1}^{k} sgn(u_{j})xu_{j+1} ⋯ u_{k}z^{n+1}u_{1} ⋯ u_{j−1}y = ρ_{n+1}(w).

Corollary 12. We have

{0} = ρ_{k−2}(ℌ̌_{1}^{1}) ⊂ ρ_{k−3}(ℌ̌_{2}^{1}) ⊂ ⋯ ⊂ ρ_{0}(ℌ̌_{k−1}^{1}) = ρ(ℌ̌_{k−1}^{1}).

for k ∈ ℤ_{≥2}. Proof. For each n ∈ {0, …, k − 3}, we have ρ_{n+1}(ℌ̌_{k−n−2}^{1}) ⊂ ρ_{n}(ℌ̌_{k−n−1}^{1}) by Proposition 11 and by the fact that if w ∈ ℌ̌_{k−n−2}^{1}, then zw ∈ ℌ̌_{k−n−1}^{1}.

4.3 Statement of our main theorem

Corollary 12 can be interpreted as stratifying the ℚ-linear relations provided by the CSF. Since Theorem 6 tells us the dimension of the whole space, we may well wish to find the dimensions of the subspaces ρ_{n}(ℌ̌_{k}^{1}) in general. Our main theorem (Theorem 15) provides a complete solution to this problem, and it uses the following generalization of the Lucas sequence:

Definition 13. For n ∈ ℤ_{≥1}, the n-step Lucas sequence (L_{m}^{n})_{m≥1} is defined by

L_{m}^{n} = \begin{cases}2^{m} − 1, & for m = 1, …, n; \\ L_{m−1}^{n} + ⋯ + L_{m−n}^{n}, & for m ≥ n + 1. \end{cases}

We adopt the convention that L_{m}^{0} = 0 for all m ∈ ℤ_{≥1}.

Table 3: n-step Lucas sequences

n 1 2 3 4 5 6 7 Sequence Number
L_{m}^{1} 1 1 1 1 1 1 1  \underline{A000012}
L_{m}^{2} 1 3 4 7 11 18 29 \underline{A000032}, \underline{A000204}
L_{m}^{3} 1 3 7 11 21 39 71  \underline{A001644}
L_{m}^{4} 1 3 7 15 26 51 99 \underline{A073817}, \underline{A001648}
L_{m}^{5} 1 3 7 15 31 57 113 \underline{A074048}, \underline{A023424}

Lemma 14. Let n ∈ ℤ_{≥0}.

1. We have L_{n+1}^{n} = 2^{n+1} − n − 2.

2. We have L_{m}^{n} = 2L_{m−1}^{n} − L_{m−n−1}^{n} for m ≥ n + 2.

Proof. Since both equations are obvious for n = 0, we assume that n ≥ 1.

1. We have L_{n+1}^{n} = ∑_{m=1}^{n} L_{m}^{n} = ∑_{m=1}^{n}(2^{m} − 1) = 2^{n+1} − n − 2.

2. If m ≥ n + 2, then the recurrence relation shows that

L_{m}^{n} = L_{m−1}^{n} + ⋯ + L_{m−n}^{n} = L_{m−1}^{n} + (L_{m−2}^{n} + ⋯ + L_{m−n−1}^{n}) − L_{m−n−1}^{n}
= 2L_{m−1}^{n} − L_{m−n−1}^{n}.

Theorem 15 (Main Theorem). If n ∈ ℤ_{≥0} and k ∈ ℤ_{≥1}, then

dim ρ_{n}(ℌ̌_{k}^{1}) = \frac{1}{n + k} ∑_{m|n+k} φ(\frac{n + k}{m} )(2^{m} − L_{m}^{n}) − 2.

Remark 16. We may easily see that this theorem is a generalization of Theorem 6.

5 Proof of our main theorem

5.1 Cyclic equivalence and ρ_{0}

Definition 17. Let l ∈ ℤ_{≥1}. Set X_{l} = {y, z}^{l}, and let the cyclic group ℤ/lℤ of order l act on X_{l} by cyclic shifts, i.e.,

j(u_{1}, …, u_{l}) = (u_{j+1}, …, u_{j+l})

for j ∈ ℤ/lℤ and (u_{1}, …, u_{l}) ∈ X_{l}. The equivalence relation on X_{l} induced by the action is called the cyclic equivalence and denoted by ∼. Put Y_{l} = X_{l}/∼.

Proposition 18. For each l ∈ ℤ_{≥1}, we may define ρ̃_{0} : Y_{l} → ℌ by setting

ρ̃_{0}([(u_{1}, …, u_{l})]) = ρ_{0}(u_{1} ⋯ u_{l})

for (u_{1}, …, u_{l}) ∈ X_{l}.

Proof. Lemma 10 shows that if (u_{1}, …, u_{l}) ∈ X_{l}, then

ρ_{0}(u_{1} ⋯ u_{l}) = ∑_{j=1}^{l} sgn(u_{j})xu_{j+1} ⋯ u_{l}u_{1} ⋯ u_{j−1}y

= ∑_{\begin{matrix}j=1,...,l \\ (v_{1},...,v_{l})=j(u_{1},...,u_{l}) \end{matrix}} sgn(v_{1})xv_{2} ⋯ v_{l}y.

It follows that ρ_{0}(u_{1} ⋯ u_{l}) depends only on the equivalence class [(u_{1}, …, u_{l})] ∈ Y_{l}, as required.

Example 19. For l = 4, we have

ρ̃_{0}([(z, z, z, z)]) = 0,
ρ̃_{0}([(z, z, z, y)]) = −xzzzy,
ρ̃_{0}([(z, z, y, y)]) = −x(yzz + zzy)y,
ρ̃_{0}([(z, y, z, y)]) = −x(zyz + zyz)y,
ρ̃_{0}([(z, y, y, y)]) = −x(yyz + yzy + zyy)y,
ρ̃_{0}([(y, y, y, y)]) = −x(yyy + yyy + yyy + yyy)y.

Proposition 20. The family \{ρ̃_{0}(U) | U ∈ Y_{l} \ \{[(\underbrace{z, …, z}_{l})]\}\} is ℚ-linearly independent.

Proof. This is because the values ρ̃_{0}(U) for different equivalence classes U ∈ Y_{l}\{[(z, …, z)]} are nonzero and consist of different monomials.

5.2 Relationship between ρ_{n} and ρ_{0}

Definition 21. For l ∈ ℤ_{≥1} and n ∈ {0, …, l}, we write X_{l,n} for the subset of X_{l} consisting of all (u_{1}, …, u_{l}) ∈ X_{l} that contain at least n consecutive z’s when written cyclically.

Example 22. For l = 4, we have

X_{4,0} = {(z, z, z, z),

(z, z, z, y), (z, z, y, z), (z, y, z, z), (y, z, z, z), (z, z, y, y), (z, y, y, z), (y, y, z, z), (y, z, z, y), (z, y, z, y), (y, z, y, z), (z, y, y, y), (y, y, y, z), (y, y, z, y), (y, z, y, y), (y, y, y, y)}

= [(z, z, z, z)] ∪ [(z, z, z, y)] ∪ [(z, z, y, y)] ∪ [(z, y, z, y)] ∪ [(z, y, y, y)] ∪ [(y, y, y, y)] = X_{4},

X_{4,1} = {(z, z, z, z),

(z, z, z, y), (z, z, y, z), (z, y, z, z), (y, z, z, z), (z, z, y, y), (z, y, y, z), (y, y, z, z), (y, z, z, y), (z, y, z, y), (y, z, y, z), (z, y, y, y), (y, y, y, z), (y, y, z, y), (y, z, y, y)}

= [(z, z, z, z)] ∪ [(z, z, z, y)] ∪ [(z, z, y, y)] ∪ [(z, y, z, y)] ∪ [(z, y, y, y)],

X_{4,2} = {(z, z, z, z),

(z, z, z, y), (z, z, y, z), (z, y, z, z), (y, z, z, z), (z, z, y, y), (z, y, y, z), (y, y, z, z), (y, z, z, y)}

= [(z, z, z, z)] ∪ [(z, z, z, y)] ∪ [(z, z, y, y)],

X_{4,3} = {(z, z, z, z),

(z, z, z, y), (z, z, y, z), (z, y, z, z), (y, z, z, z)} = [(z, z, z, z)] ∪ [(z, z, z, y)],

X_{4,4} = {(z, z, z, z)}

= [(z, z, z, z)].

Remark 23. Each X_{l,n} is invariant under the action of ℤ/lℤ, which allows us to make the following definition.

Definition 24. For l ∈ ℤ_{≥1} and n ∈ {0, …, l}, we write

Y_{l,n} = X_{l,n}/∼ = \{[(\underbrace{z, …, z}_{n}, u_{n+1}, …, u_{l})] ∈ Y_{l} | u_{n+1}, …, u_{l} ∈ \{y, z\}\}.

Proposition 25. For n ∈ ℤ_{≥0} and k ∈ ℤ_{≥1}, we have

dim ρ_{n}(ℌ̌_{k}^{1}) = #Y_{n+k,n} − 2.

Proof. Since

ℌ̌_{k}^{1} = span_{ℚ}{z_{k_{1}} ⋯ z_{k_{l}} | (k_{1}, …, k_{l}) ∈ Ǐ_{k}^{1}}

= span_{ℚ}\{u_{1} ⋯ u_{k−1}y | (u_{1}, …, u_{k−1}) ∈ \{x, y\}^{k−1} \ \{(\underbrace{y, …, y}_{k−1})\}\}

= span_{ℚ}{u_{1} ⋯ u_{k−1}y − y^{k} | (u_{1}, …, u_{k−1}) ∈ {y, z}^{k−1}},

Proposition 11 shows that

ρ_{n}(ℌ̌_{k}^{1}) = span_{ℚ}{ρ_{n}(u_{1} ⋯ u_{k−1}y) − ρ_{n}(y^{k}) | (u_{1}, …, u_{k−1}) ∈ {y, z}^{k−1}}

= span_{ℚ}{ρ_{0}(z^{n}u_{1} ⋯ u_{k−1}y) − ρ_{0}(z^{n}y^{k}) | (u_{1}, …, u_{k−1}) ∈ {y, z}^{k−1}}

= span_{ℚ}\{ρ̃_{0}(U) − ρ̃_{0}([(\underbrace{z, …, z}_{n}, \underbrace{y, …, y}_{k})]) | U ∈ Y_{n+k,n} \ \{[(\underbrace{z, …, z}_{n+k})]\}\},

which implies that dim ρ_{n}(ℌ̌_{k}^{1}) = #Y_{n+k,n} − 2 because of Proposition 20.

5.3 Calculation of #Y_{l,n}

Proposition 25 reduces our main theorem (Theorem 15) to the following proposition: Proposition 26. For l ∈ ℤ_{≥1} and n ∈ {0, …, l}, we have

#Y_{l,n} = \frac{1}{l} ∑_{m|l} φ(\frac{l}{m})(2^{m} − L_{m}^{n}).

We first invoke the Cauchy-Frobenius lemma: Proposition 27 (Cauchy-Frobenius lemma). If a finite group G acts on a finite set X, then we have

#(X/G) = \frac{1}{#G} ∑_{g∈G} #\{x ∈ X | gx = x\}.

Lemma 28. For l ∈ ℤ_{≥1} and n ∈ {0, …, l}, we have

#Y_{l,n} = \frac{1}{l} ∑_{m|l} φ(\frac{l}{m})#\{u ∈ X_{l,n} | mu = u\}.

Proof. Applying the Cauchy-Frobenius lemma with G = ℤ/lℤ and X = X_{l,n} gives

#Y_{l,n} = \frac{1}{l} ∑_{j∈ℤ/lℤ}#\{u ∈ X_{l,n} | ju = u\}

= \frac{1}{l} ∑_{j∈ℤ/lℤ}#\{u ∈ X_{l,n} | gcd(j, l)u = u\}

= \frac{1}{l} ∑_{m|l}(#\{j ∈ ℤ/lℤ | gcd(j, l) = m\} · #\{u ∈ X_{l,n} | mu = u\})

= \frac{1}{l} ∑_{m|l} φ(\frac{l}{m})#\{u ∈ X_{l,n} | mu = u\}.

Lemma 29. If l ∈ ℤ_{≥1}, n ∈ {0, …, l}, and m is a positive divisor of l, then

#{u ∈ X_{l,n} | mu = u} = 2^{m} − L_{m}^{n}.

Proof. The map f_{m,l}: X_{m} → X_{l} defined by

f_{m,l}(v) = (\underbrace{v, …, v}_{l/m})

is injective and has image {u ∈ X_{l} | mu = u}. Therefore it suffices to show that

#f_{m,l}^{−1}(X_{l,n}) = 2^{m} − L_{m}^{n}.

Observe that for each (m, n) ∈ ℤ_{≥1} × ℤ_{≥0}, the set f_{m,l}^{−1}(X_{l,n}) is the same for all multiples l of m with l ≥ n. Therefore, we may put Z_{m,n} = X_{m} \ f_{m,l}^{−1}(X_{l,n}) = f_{m,l}^{−1}(X_{l,n}^{c}), aiming to show that #Z_{m,n} = L_{m}^{n} for all m ∈ ℤ_{≥1} and n ∈ ℤ_{≥0}.

If m ≤ n, then

#Z_{m,n} = #{v ∈ X_{m} | at least one component of v is y} = 2^{m} − 1 = L_{m}^{n}.

If m = n + 1, then

#Z_{m,n} = #{v ∈ X_{m} | at least two components of v are y}
= 2^{m} − m − 1 = 2^{n+1} − n − 2 = L_{m}^{n}

by Lemma 14 (1). Suppose that m ≥ n + 2. If we put

Z_{m,n,i} = {(v_{1}, …, v_{m}) ∈ Z_{m,n} | v_{1} = ⋯ = v_{i−1} = z, v_{i} = y}

for i = 1, …, n, then {Z_{m,n,i} | i = 1, …, n} is a partition of Z_{m,n}. Set

Z_{m,n,i}^{y} = {(v_{1}, …, v_{m}) ∈ Z_{m,n,i} | v_{i+1} = y},
Z_{m,n,i}^{z} = {(v_{1}, …, v_{m}) ∈ Z_{m,n,i} | v_{i+1} = z}.

Then removing the (i + 1)st component gives a bijection from Z_{m,n,i}^{y} to Z_{m−1,n,i} and an injection from Z_{m,n,i}^{z} to Z_{m−1,n,i} with image Z_{m−1,n,i} \ Z_{m−1,n,i}^{z···zy}, where

Z_{m−1,n,i}^{z···zy} = {(v_{1}, …, v_{m−1}) ∈ Z_{m−1,n,i} | v_{i+1} = ⋯ = v_{i+n−1} = z, v_{i+n} = y}.

Moreover removing components from the (i + 1)st to the (i + n)th gives a bijection from Z_{m−1,n,i}^{z···zy} to Z_{m−n−1,n,i}. It follows that

#Z_{m,n,i} = #Z_{m,n,i}^{y} + #Z_{m,n,i}^{z} = #Z_{m−1,n,i} + #(Z_{m−1,n,i} \ Z_{m−1,n,i}^{z···zy})
= 2#Z_{m−1,n,i} − #Z_{m−n−1,n,i}.

Summing up for i = 1, …, n gives #Z_{m,n} = 2#Z_{m−1,n} − #Z_{m−n−1,n}. Hence the lemma follows from Lemma 14 (2).

Lemmas 28 and 29 imply Proposition 26, thereby establishing our main theorem.

6 Multiple zeta-star values

6.1 Multiple zeta-star values and Hoffman’s algebra

The multiple zeta-star values (MZSVs for short) are defined as the MZVs with equality allowed in the index of summation:

Definition 30. For k = (k_{1}, …, k_{l}) ∈ I^{0}, the multiple zeta-star value ζ^{⋆}(k_{1}, …, k_{l}) is a real number defined by

ζ^{⋆}(k) = ζ^{⋆}(k_{1}, …, k_{l}) = ∑_{n_{1}≥···≥n_{l}≥1} \frac{1}{n_{1}^{k_{1}} ⋯ n_{l}^{k_{l}}} .

Example 31. We have

ζ^{⋆}(4, 2, 1) = ∑_{n_{1}≥n_{2}≥n_{3}≥1} \frac{1}{n_{1}^{4}n_{2}^{2}n_{3}}

= ∑_{n_{1}>n_{2}>n_{3}≥1} \frac{1}{n_{1}^{4}n_{2}^{2}n_{3}} + ∑_{n_{1}=n_{2}>n_{3}≥1} \frac{1}{n_{1}^{4}n_{2}^{2}n_{3}} + ∑_{n_{1}>n_{2}=n_{3}≥1} \frac{1}{n_{1}^{4}n_{2}^{2}n_{3}} + ∑_{n_{1}=n_{2}=n_{3}≥1} \frac{1}{n_{1}^{4}n_{2}^{2}n_{3}}

= ∑_{n_{1}>n_{2}>n_{3}≥1} \frac{1}{n_{1}^{4}n_{2}^{2}n_{3}} + ∑_{n_{1}>n_{3}≥1} \frac{1}{n_{1}^{6}n_{3}} + ∑_{n_{1}>n_{2}≥1} \frac{1}{n_{1}^{4}n_{2}^{3}} + ∑_{n_{1}≥1} \frac{1}{n_{1}^{7}}

= ζ(4, 2, 1) + ζ(6, 1) + ζ(4, 3) + ζ(7).

As the above example indicates, each MZSV can be expressed as a ℤ-linear combination of MZVs. Hoffman’s algebra is useful for describing this relationship between MZVs and MZSVs.

Definition 32. Define a ℚ-linear map Z̄: ℌ^{0} → ℝ by setting Z̄(1) = 1 and Z̄(z_{k_{1}} ⋯ z_{k_{l}}) = ζ^{⋆}(k) for k = (k_{1}, …, k_{l}) ∈ I^{0}.

Definition 33. Let γ denote the algebra automorphism on ℌ satisfying γ(x) = x and γ(y) = z. Define a ℚ-linear transformation d on ℚ + ℌy by setting d(1) = 1 and d(wy) = γ(w)y for w ∈ ℌ.

Proposition 34. We have Z̄ = Z ∘ d: ℌ^{0} → ℝ.

Proof. Easy and well known.

6.2 Cyclic sum formula for multiple zeta-star values

Ohno and the third author [4] proved the following analog of the CSF for MZSVs:

Theorem 35 (Cyclic sum formula for multiple zeta-star values). If (k_{1}, …, k_{l}) ∈ Ǐ^{1}, then

∑_{j=1}^{l} ∑_{i=1}^{k_{j}−1} ζ^{⋆}(k_{j} − i + 1, k_{j+1}, …, k_{j+l−1}, i) = (k_{1} + ⋯ + k_{l})ζ(k_{1} + ⋯ + k_{l} + 1).

The second and third authors [5] defined operators ρ̄_{n} for MZSVs as well as ρ_{n} for MZVs: Definition 36. Let n ∈ ℤ_{≥0}. Define a ℚ-linear map C̄_{n}: ℌ → ℌ^{⊗(n+2)} by setting C̄_{n}(1) = 0, C̄_{n}(x) = x ⊗ y^{⊗(n+1)}, C̄_{n}(y) = −x ⊗ y^{⊗(n+1)}, and

C̄_{n}(ww′) = C̄_{n}(w) ⋄ γ^{−1}(w′) + γ^{−1}(w) ⋄ C̄_{n}(w′).

Write ρ̄_{n} = M_{n} ∘ C̄_{n}. Proposition 37. We have the following commutative diagram:

\xymatrix{ & xℌ ⊗ ℌ^{⊗n} ⊗ ℌy \ar[dd]^{γ⊗γ^{⊗n}⊗d}_{≅} \ar[r]_{M_{n}} & ℌ^{0} \ar[dd]_{d}^{≅} \ar[dr]_{Z̄} \\ ℌ \ar[ur]_{C̄_{n}} \ar[dr]^{C_{n}} \ar[urr]^{ρ̄_{n}} \ar[drr]_{ρ_{n}} &  &  & ℝ. \\  & xℌ ⊗ ℌ^{⊗n} ⊗ ℌy \ar[r]^{M_{n}} & ℌ^{0} \ar[ur]^{Z}}

Proof. Straightforward. Proposition 38. If k_{1}, …, k_{l} ∈ ℤ_{≥1}, then

∑_{j=1}^{l}(ρ̄_{0} ∘ α ∘ d)(z_{k_{j}} ⋯ z_{k_{j+l−1}}) = ∑_{j=1}^{l} ∑_{i=1}^{k_{j}−1} z_{k_{j}−i+1}z_{k_{j+1}} ⋯ z_{k_{j+l−1}}z_{i} − (k_{1} + ⋯ + k_{l})z_{k_{1}+···+k_{l}+1}.

Here α: ℌy → ℌy denotes the ℚ-linear map representing the division by depth, i.e.,

α(z_{k′_{1}} ⋯ z_{k′_{l′}}) = \frac{1}{l′}z_{k′_{1}} ⋯ z_{k′_{l′}}

for all monomials z_{k′_{1}} ⋯ z_{k′_{l′}} ∈ ℌy. Proof. For simplicity, let A and B respectively denote the left- and right-hand sides of the desired identity. By the injectivity of d, it suffices to show that d(A) = d(B); setting A′ = d(A) and B′ = d(B), we have

A′ = ∑_{j=1}^{l}(d ∘ ρ̄_{0} ∘ α ∘ d)(z_{k_{j}} ⋯ z_{k_{j+l−1}}) = ∑_{j=1}^{l}(ρ_{0} ∘ α ∘ d)(z_{k_{j}} ⋯ z_{k_{j+l−1}}),
B′ = ∑_{j=1}^{l} ∑_{i=1}^{k_{j}−1} d(z_{k_{j}−i+1}z_{k_{j+1}} ⋯ z_{k_{j+l−1}}z_{i}) − (k_{1} + ⋯ + k_{l})z_{k_{1}+···+k_{l}+1}.

For a ∈ ℤ_{≥1}, let δ_{a}: ℌy → ℌy be the ℚ-linear map that extracts the depth-a part, i.e.,

δ_{a}(z_{k′_{1}} ⋯ z_{k′_{l′}}) = \begin{cases}z_{k′_{1}} ⋯ z_{k′_{l′}} & if l′ = a; \\ 0 & otherwise \end{cases}

for all monomials z_{k′_{1}} ⋯ z_{k′_{l′}} ∈ ℌy. Then it is enough to prove that δ_{a}(A′) = δ_{a}(B′) for a = 1, …, l + 1. Note that

(δ_{a} ∘ ρ_{0})(w) = \begin{cases}(δ_{a} ∘ ρ_{0} ∘ δ_{a})(w) + (δ_{a} ∘ ρ_{0} ∘ δ_{a−1})(w) & if a ≥ 2; \\ (δ_{a} ∘ ρ_{0} ∘ δ_{a})(w) & if a = 1, \end{cases}
(δ_{a} ∘ α)(w) = \frac{1}{a}δ_{a}(w)

for all w ∈ ℌy.

For a = 1, we have

δ_{1}(A′) = ∑_{j=1}^{l}(δ_{1} ∘ ρ_{0} ∘ α ∘ d)(z_{k_{j}} ⋯ z_{k_{j+l−1}}) = ∑_{j=1}^{l}(δ_{1} ∘ ρ_{0} ∘ δ_{1} ∘ α ∘ d)(z_{k_{j}} ⋯ z_{k_{j+l−1}})

= ∑_{j=1}^{l}(δ_{1} ∘ ρ_{0} ∘ δ_{1} ∘ d)(z_{k_{j}} ⋯ z_{k_{j+l−1}}) = ∑_{j=1}^{l}(δ_{1} ∘ ρ_{0})(z_{k_{j}+···+k_{j+l−1}})

= ∑_{j=1}^{l}(δ_{1} ∘ ρ_{0})(z_{k_{1}+···+k_{l}}) = ∑_{j=1}^{l}(−z_{k_{1}+···+k_{l}+1}) = −lz_{k_{1}+···+k_{l}+1},

δ_{1}(B′) = ∑_{j=1}^{l} ∑_{i=1}^{k_{j}−1}(δ_{1} ∘ d)(z_{k_{j}−i+1}z_{k_{j+1}} ⋯ z_{k_{j+l−1}}z_{i}) − (k_{1} + ⋯ + k_{l})z_{k_{1}+···+k_{l}+1}

= ∑_{j=1}^{l} ∑_{i=1}^{k_{j}−1} z_{(k_{j}−i+1)+k_{j+1}+···+k_{j+l−1}+i} − (k_{1} + ⋯ + k_{l})z_{k_{1}+···+k_{l}+1}

= ∑_{j=1}^{l}(k_{j} − 1)z_{k_{1}+···+k_{l}+1} − (k_{1} + ⋯ + k_{l})z_{k_{1}+···+k_{l}+1}

= −lz_{k_{1}+···+k_{l}+1},

as required.

For a = l + 1, we have

δ_{l+1}(A′) = ∑_{j=1}^{l}(δ_{l+1} ∘ ρ_{0} ∘ α ∘ d)(z_{k_{j}} ⋯ z_{k_{j+l−1}})

= ∑_{j=1}^{l}(δ_{l+1} ∘ ρ_{0} ∘ δ_{l+1} ∘ α ∘ d)(z_{k_{j}} ⋯ z_{k_{j+l−1}}) + ∑_{j=1}^{l}(δ_{l+1} ∘ ρ_{0} ∘ δ_{l} ∘ α ∘ d)(z_{k_{j}} ⋯ z_{k_{j+l−1}})

= \frac{1}{l + 1} ∑_{j=1}^{l}(δ_{l+1} ∘ ρ_{0} ∘ δ_{l+1} ∘ d)(z_{k_{j}} ⋯ z_{k_{j+l−1}}) + \frac{1}{l} ∑_{j=1}^{l}(δ_{l+1} ∘ ρ_{0} ∘ δ_{l} ∘ d)(z_{k_{j}} ⋯ z_{k_{j+l−1}})

= \frac{1}{l} ∑_{j=1}^{l}(δ_{l+1} ∘ ρ_{0})(z_{k_{j}} ⋯ z_{k_{j+l−1}}) = \frac{1}{l} ∑_{j=1}^{l} ∑_{j′=1}^{l} ∑_{i=1}^{k_{j+j′−1}−1} z_{k_{j+j′−1}−i+1}z_{k_{j+j′}} ⋯ z_{k_{j+j′+l−2}}z_{i}

= ∑_{j=1}^{l} ∑_{i=1}^{k_{j}−1} z_{k_{j}−i+1}z_{k_{j+1}} ⋯ z_{k_{j+l−1}}z_{i} = δ_{l+1}(B′),

as required.

Now let 2 ≤ a ≤ l. We first compute δ_{a}(A′). It is easy to see that δ_{a}(A′) = P + Q, where

P = \frac{1}{a} ∑_{j=1}^{l}(δ_{a} ∘ ρ_{0} ∘ δ_{a} ∘ d)(z_{k_{j}} ⋯ z_{k_{j+l−1}}),

Q = \frac{1}{a − 1} ∑_{j=1}^{l}(δ_{a} ∘ ρ_{0} ∘ δ_{a−1} ∘ d)(z_{k_{j}} ⋯ z_{k_{j+l−1}}).

For b ∈ ℤ_{≥1}, set

M_{b} = I_{l,b}^{1} = {m = (m_{1}, …, m_{b}) ∈ ℤ_{≥1}^{b} | m_{1} + ⋯ + m_{b} = l}.

For each m = (m_{1}, …, m_{b}) ∈ M_{b}, we extend the indices of m_{p} to all p ∈ ℤ by declaring m_{p} = m_{p′} whenever p ≡ p′ (mod b). For j = 1, …, l, m = (m_{1}, …, m_{b}) ∈ M_{b}, and p = 1, …, b, write

k_{j,m,p} = ∑_{i=j+m_{1}+···+m_{p−1}}^{j+m_{1}+···+m_{p}−1} k_{i} = k_{j+m_{1}+···+m_{p−1}} + ⋯ + k_{j+m_{1}+···+m_{p}−1}.

Then we have

P = \frac{1}{a} ∑_{j=1}^{l}(δ_{a} ∘ ρ_{0})( ∑_{m∈M_{a}} z_{k_{j,m,1}} ⋯ z_{k_{j,m,a}})

= −\frac{1}{a} ∑_{\begin{matrix}j=1,...,l \\ m∈M_{a} \end{matrix}} ∑_{p=1}^{a} z_{k_{j,m,p}+1}z_{k_{j,m,p+1}} ⋯ z_{k_{j,m,p+a−1}}.

Note here that

∑_{\begin{matrix}j=1,...,l \\ m∈M_{a} \end{matrix}} z_{k_{j,m,p}+1}z_{k_{j,m,p+1}} ⋯ z_{k_{j,m,p+a−1}}

does not depend on p = 1, …, a, because the bijection from {1, …, l} × M_{a} to itself defined by

(j, m) = (j, (m_{1}, …, m_{a})) ↦ (j′, m′) = (j + m_{1} + ⋯ + m_{p−1}, (m_{p}, …, m_{p+a−1}))

has the property that

z_{k_{j,m,p}+1}z_{k_{j,m,p+1}} ⋯ z_{k_{j,m,p+a−1}} = z_{k_{j′,m′,1}+1}z_{k_{j′,m′,2}} ⋯ z_{k_{j′,m′,a}}.

It follows that

P = − ∑_{\begin{matrix}j=1,...,l \\ m∈M_{a} \end{matrix}} z_{k_{j,m,a}+1}z_{k_{j,m,1}} ⋯ z_{k_{j,m,a−1}}.

Similar reasoning shows that

Q = \frac{1}{a − 1} ∑_{j=1}^{l}(δ_{a} ∘ ρ_{0})( ∑_{m∈M_{a−1}} z_{k_{j,m,1}} ⋯ z_{k_{j,m,a−1}})

= \frac{1}{a − 1} ∑_{\begin{matrix}j=1,...,l \\ m∈M_{a−1} \end{matrix}} ∑_{p=1}^{a−1} ∑_{i=1}^{k_{j,m,p}−1} z_{k_{j,m,p}−i+1}z_{k_{j,m,p+1}} ⋯ z_{k_{j,m,p+a−2}}z_{i}

= ∑_{\begin{matrix}j=1,...,l \\ m∈M_{a−1} \end{matrix}} ∑_{i=1}^{k_{j,m,a−1}−1} z_{k_{j,m,a−1}−i+1}z_{k_{j,m,1}} ⋯ z_{k_{j,m,a−2}}z_{i}.

We next compute δ_{a}(B′). Observe that δ_{a}(B′) = R + S, where

R = ∑_{\begin{matrix}j=1,...,l \\ m∈M_{a−1} \end{matrix}} ∑_{i=1}^{k_{j}−1} z_{k_{j,m,1}−i+1}z_{k_{j,m,2}} ⋯ z_{k_{j,m,a−1}}z_{i},

S = ∑_{\begin{matrix}j=1,...,l \\ m∈M_{a} \end{matrix}} ∑_{i=1}^{k_{j}−1} z_{k_{j,m,1}−i+1}z_{k_{j,m,2}} ⋯ z_{k_{j,m,a−1}}z_{k_{j,m,a}+i}.

Since the bijection from {1, …, l} × M_{a−1} to itself defined by

(j, m) = (j, (m_{1}, …, m_{a−1})) ↦ (j′, m′) = (j + m_{1}, (m_{2}, …, m_{a−1}, m_{1}))

has the property that

∑_{i=1}^{k_{j}−1} z_{k_{j,m,1}−i+1}z_{k_{j,m,2}} ⋯ z_{k_{j,m,a−1}}z_{i} = ∑_{i=1}^{k_{j′+m′_{1}+···+m{′ \atop a−2}}−1} z_{k_{j′,m′,a−1}−i+1}z_{k_{j′,m′,1}} ⋯ z_{k_{j′,m′,a−2}}z_{i},

we have

R = ∑_{\begin{matrix}j=1,...,l \\ m=(m_{1},...,m_{a−1})∈M_{a−1} \end{matrix}} ∑_{i=1}^{k_{j+m_{1}+···+m_{a−2}}−1} z_{k_{j,m,a−1}−i+1}z_{k_{j,m,1}} ⋯ z_{k_{j,m,a−2}}z_{i}.

Similar reasoning shows that

S = ∑_{\begin{matrix}j=1,...,l \\ m=(m_{1},...,m_{a})∈M_{a} \end{matrix}} ∑_{i=1}^{k_{j+m_{1}+···+m_{a−1}}−1} z_{k_{j,m,a}−i+1}z_{k_{j,m,1}} ⋯ z_{k_{j,m,a−2}}z_{k_{j,m,a−1}+i}.

What needs to be shown is that P + Q = R + S. Note that

S − P = ∑_{\begin{matrix}j=1,...,l \\ m=(m_{1},...,m_{a})∈M_{a} \end{matrix}} ∑_{i=0}^{k_{j+m_{1}+···+m_{a−1}}−1} z_{k_{j,m,a}−i+1}z_{k_{j,m,1}} ⋯ z_{k_{j,m,a−2}}z_{k_{j,m,a−1}+i}

= ∑_{\begin{matrix}j=1,...,l \\ m=(m_{1},...,m_{a})∈M_{a} \end{matrix}} ∑_{i=k_{j,m,a−1}}^{k_{j,m,a−1}+k_{j+m_{1}+···+m_{a−1}}−1} z_{k_{j,m,a−1}+k_{j,m,a}−i+1}z_{k_{j,m,1}} ⋯ z_{k_{j,m,a−2}}z_{i}.

Fix j = 1, …, l and consider the map ψ: M_{a} → M_{a−1} defined by

ψ(m_{1}, …, m_{a}) = (m_{1}, …, m_{a−2}, m_{a−1} + m_{a}).

If ψ(m) = m′, then k_{j,m,a−1} + k_{j,m,a} = k_{j,m′,a−1} and k_{j,m,p} = k_{j,m′,p} for p = 1, …, a − 2. Moreover, for each m′ = (m′_{1}, …, m′_{a−1}) ∈ M_{a−1}, the sets

{i ∈ ℤ | k_{j,m,a−1} ≤ i ≤ k_{j,m,a−1} + k_{j+m_{1}+···+m_{a−1}} − 1}

for m = (m_{1}, …, m_{a}) ∈ ψ^{−1}(m′) are disjoint with union

{i ∈ ℤ | k_{j+m′_{1}+···+m′_{a−2}} ≤ i ≤ k_{j,m′,a−1} − 1}.

It follows that

S − P = ∑_{\begin{matrix}j=1,...,l \\ m′=(m′_{1},...,m′_{a−1})∈M_{a−1} \end{matrix}} ∑_{i=k_{j+m′_{1}+···+m{′ \atop a−2}}}^{k_{j,m′,a−1}−1} z_{k_{j,m′,a−1}−i+1}z_{k_{j,m′,1}} ⋯ z_{k_{j,m′,a−2}}z_{i}.

We therefore conclude that

S − P + R = ∑_{\begin{matrix}j=1,...,l \\ m∈M_{a−1} \end{matrix}} ∑_{i=1}^{k_{j,m,a−1}−1} z_{k_{j,m,a−1}−i+1}z_{k_{j,m,1}} ⋯ z_{k_{j,m,a−2}}z_{i} = Q,

as required.

Proposition 39. We have ρ̄_{0}(ℌ̌^{1}) ⊂ Ker Z̄ and ρ̄_{0}(ℌ̌_{k−1}^{1}) ⊂ Ker Z̄ ∩ ℌ_{k}^{0} for k ∈ ℤ_{≥2}.

Proof. If w ∈ ℌ̌^{1}, then Proposition 37, Proposition 8, and Theorem 4 show that

Z̄(ρ̄_{0}(w)) = Z(ρ_{0}(w)) = Z(ρ(w)) = 0,

from which the first assertion follows. The second assertion is now obvious.

Proof of Theorem 35. Immediate from Propositions 38 and 39.

Proposition 40. We have

{0} = ρ̄_{k−2}(ℌ̌_{1}^{1}) ⊂ ρ̄_{k−3}(ℌ̌_{2}^{1}) ⊂ ⋯ ⊂ ρ̄_{0}(ℌ̌_{k−1}^{1}) ⊂ Ker Z̄ ∩ ℌ_{k}^{0}

for all k ∈ ℤ_{≥2}.

Proof. Observe that ρ̄_{n+1}(w) = ρ̄_{n}(zw) for all n ∈ ℤ_{≥0} and w ∈ ℌ; indeed, we have

ρ̄_{n+1}(w) = d^{−1}(ρ_{n+1}(w)) = d^{−1}(ρ_{n}(zw)) = ρ̄_{n}(zw).

This finishes the proof.

Theorem 41. If n ∈ ℤ_{≥0} and k ∈ ℤ_{≥1}, then

dim ρ̄_{n}(ℌ̌_{k}^{1}) = dim ρ_{n}(ℌ̌_{k}^{1}) = \frac{1}{n + k} ∑_{m|n+k} φ(\frac{n + k}{m} )(2^{m} − L_{m}^{n}) − 2.

Proof. The ℚ-vector spaces ρ̄_{n}(ℌ̌_{k}^{1}) and ρ_{n}(ℌ̌_{k}^{1}) are isomorphic as given by d, and the theorem follows from Theorem 15.

7 Acknowledgments

The authors are grateful to Professor Yasuo Ohno for valuable comments that inspired this work, and to Mr. Hiroki Kondo for carefully reading the manuscript.

References

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[2] A. B. Goncharov, Periods and mixed motives, http://arxiv.org/abs/math/0202154, 2002.

[3] M. E. Hoffman and Y. Ohno, Relations of multiple zeta values and their algebraic expression, J. Algebra 262 (2003), 332–347.

[4] Y. Ohno and N. Wakabayashi, Cyclic sum of multiple zeta values, Acta Arith. 123 (2006), 289–295.

[5] T. Tanaka and N. Wakabayashi, An algebraic proof of the cyclic sum formula for multiple zeta values, J. Algebra 323 (2010), 766–778.

[6] T. Terasoma, Mixed Tate motives and multiple zeta values, Invent. Math. 149 (2002), 339–369.

[7] D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992), 497–512, Progr. Math., 120 (1994).

2010 Mathematics Subject Classification: Primary 11M32; Secondary 05A05. Keywords: multiple zeta values, multiple zeta-star values, cyclic sum formula, Lucas numbers.

(Concerned with sequences \underline{A000012}, \underline{A000032}, \underline{A000079}, \underline{A000204}, \underline{A000931}, \underline{A001644}, \underline{A001648}, \underline{A023424}, \underline{A038360}, \underline{A052823}, \underline{A073817}, and \underline{A074048}.)

Received June 20 2010; revised version received February 2 2011. Published in Journal of Integer Sequences, February 20 2011.

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