Abstract
In this paper, we produce shuffle relations from multiple zeta values of the form ζ ({ 1 }m-1, n+1). Here { 1 }k is k repetitions of 1, and for a string of positive integers α1, α2, ...,αr with αr > 2 . ζ (α1, α1, ..., α1) = Σ n1-α1n2-α2... n r-αr 1 < n1 < n2 < ... < n r As applications of the sum formula and a newly developed weighted sum formula, we shall prove for even integers k, r > 0 that k r Σ Σ (-1)ℓ Σ ζ (α0, α1, ..., αj + βj, βj+1, ..., βk, βk+1 + 1) j = 0 ℓ = 0 |α| = j + r - ℓ + 1 |β| = k - j + ℓ + 2 + Σ Σ ζ (α0, α1, ..., αk r - ℓ + 3) = ζ (k + r + 4). 0 < ℓ < r |α| = k + ℓ + 1 ℓ : even Mathematical subject classification: Primary: 40A25, 40B05; Secondary: 11M99, 33E99.
multiple zeta values; sum formulae
Some alternating double sum formulae of multiple zeta values
Minking EieI; Fu-Yao YangII; Yao Lin OngIII
IDepartment of Mathematics, National Chung Cheng University, 168 University Rd. Minhsiung, Chiayi 62145, Taiwan
IIDepartment of Information Management, Toko University, 51 University Rd. Sec. 2, Pu-Tzu, Chiayi 613, Taiwan
IIIDepartment of Accounting and Information System, Chang Jung Christian University, Kway-Jen, Tainan 711, Taiwan. E-mails: minking@math.ccu.edu.tw / vipent@mail.toko.edu.tw / ylong@mail.cjcu.edu.tw
ABSTRACT
In this paper, we produce shuffle relations from multiple zeta values of the form ζ ({ 1 }m-1, n+1). Here { 1 }k is k repetitions of 1, and for a string of positive integers α1, α2, ...,αr with αr> 2 .
ζ (α1, α1, ..., α1) =
Σ
n1–α1n2–α2... nr–αr
1 < n1 < n2 < ... < nr
As applications of the sum formula and a newly developed weighted sum formula, we shall prove for even integers k, r > 0 that
j = 0
|α| = j + r ℓ + 1
+ Σ
0 < ℓ < r |α| = k + ℓ + 1
ℓ : even
Mathematical subject classification: Primary: 40A25, 40B05; Secondary: 11M99, 33E99.
Key words: multiple zeta values, sum formulae.
1 Introduction
For a pair of positive integers p and q with q > 2, the classical Euler sum Sp, q is defined as [2, 3, 8, 10]
The number p + q is the weight of Sp,q. When p = 1, or (p, q) = (2, 4), or (p, q) = (4, 2), or p = q, or p + q is odd, Sp,q can be expressed in terms of the special values of Riemann zeta function at positive integers. See [12, 13, 14] for the details of evaluations.
Multiple zeta values are multidimensional version of the Euler sums [1, 6, 9, 13, 14, 15]. For a string of positive integers α = (α1, α2, ..., αr) with αr> 2, the multiple zeta value or r-fold Euler sum ζ(α1, α2, ..., αr) is defined as
or equivalently as
Here the numbers r and |α| = α1 + α2+ ... + αr are the depth and the weight of ζ(α1, α2, ..., αr), respectively.
For convenience, we let {1}k be k repetitions of 1. For example,
There is an integral representation, due to Kontsevich [4, 5, 13], to expressmultiple zeta values in terms of iterated integrals (or Drinfeld integrals) over simplices of weight-dimension, namely,
where
Ωj = dtj/(1 - tj) if j ∈ {1, α1 + 1, α1 + α2 + 1, ... , α1 + ... + αr-1 + 1}
and Ωj = dtj/tj, otherwise. For our convenience, we rewrite the above integral representation as
An elementary consideration yields a depth-dimensional integral representation as
In particular, for positive integers m and n, we have
from which the so-called Drinfeld duality theorem
ζ({1}m-1, n + 1) = ζ({1}n-1, m + 1)
follows easily.
The above Drinfeld integral representation for multiple zeta values also enables us to express the product of two multiple zeta values as a linear combination of multiple zeta values through the shuffle product formula of two multiple zeta values. The shuffle product formula of two multiple zeta values is defined as
where the sum is taken over all permutations σ of the set { 1, 2, ..., m+n }, which preserve the orders of strings of differential forms Ω1Ω2...Ωm and Ωm+1Ωm+2 ...Ωm+n. More precisely, the permutation σ satisfies the condition
σ-1(i) < σ-1(j)
for all 1 < i < j < m and m+1 < i < j < m+n.
We restrict our attention to shuffle product formulae obtained from a group of multiple zeta values of the form ζ ({ 1 }m-1, n+1) , which can be further expressed as integrals in one variable or double integrals in two variables. The following propositions are main tools for our exploration.
Proposition 1. [7] For a pair of positive integers m and n, we have
Proposition 2. [7] For an integer p > 0 and positive integers q, m and n with m > q, we have
In particular, for integers k, r > 0, one has
Proposition 3. [13] For positive integers a1, b1, a2, b2, ..., ar, br, let
and p' be the dual of p,
Then for any integer ℓ> 0, we have
In this paper, we shall consider integrals of the form
which can be expressed as a finite sum of products of multiple zeta values of the form ζ({ 1 }m-1, n+1). All possible interlacing of the variables t 1,t 2 and u1,u2 then produce 6 simplices. Integrations over each simplex give another expression in terms of 6 sums of multiple zeta values. Our alternating double sum of multiple zeta values is just one among them.
Theorem 4. For a pair of even integers k,r > 0, we have
Theorem 5. For a pair of even integers k,r > 0, we have
Some extensions of these theorems would be discussed in section 4.
2 Shuffle relations and the sum formula
In 1997, A. Granville proved the sum formula
which was originally conjectured independently by C. Moen and M. Schmidt around 1990 [11, 12, 13]. Also he mentioned that it was proved independently by Zagier in one of his unpublished papers. Here we show that the sum formula is equivalent to the evaluations of multiple zeta values of the form ζ({ 1 }m-1, n+1).
Proposition 6. For a pair of integers k,r > 0, we have
In particular, when k+r is even
Proof. We begin with the integral
Rewriting the integrand of the above integral as
we see immediately the integral is separable, and by Proposition 1, its value is equal to
As a replacement of shuffle process, we decompose the region of integrationinto 6 simplices produced from all possible interlacing of variables t1,t2 and u1,u2. They are
(1) D1: (t1, t2, u1, u2) ∈ [0, 1]4, 0 < t1 < t2 < u1 < u2 < 1,
(2) D2: (t1, t2, u1, u2) ∈ [0, 1]4, 0 < u1 < u2 < t1 < t2 < 1,
(3) D3: (t1, t2, u1, u2) ∈ [0, 1]4, 0 < t1 < u1 < t2 < u2 < 1,
(4) D4: (t1, t2, u1, u2) ∈ [0, 1]4, 0 < t1 < u1 < u2 < t2 < 1,
(5) D5: (t1, t2, u1, u2) ∈ [0, 1]4, 0 < u1 < t1 < t2 < u2 < 1 and
(6) D6: (t1, t2, u1, u2) ∈ [0, 1]4, 0 < u1 < t1 < u2 < t2 < 1.
The integration over D3, D4, D5 and D6 are easily to get. For the simplex D3:0 < t1 < u1 < t2 < u2 < 1, we rewrite the integral as
It comes from the Drinfeld integral
and hence its value is ζ ({ 1 }k+1,r+3). A similar consideration leads to the values of the integrations over D4, D5 and D6 are
(-1)rζ({ 1 }k+1,r+3), (-1)kζ ({ 1 }k+1,r+3) and (-1)k+rζ ({ 1 }k+1,r+3),
respectively.
For the simplex D1:0 < t1 < t2 < u1 < u2 < 1, we substitute the factor
by
Consequently, in terms of multiple zeta values, the value of the integration over D1 is
For 1 < j < k, we have
so that the sum in (2.1) is equal to
Identifying r-l+1 as a new dummy variable, the above sum is
Exchanging the roles of t1,t2 and u1,u2, the value of the integration over D2 is
□
Remark 7. When k+r is even, the sum formula
is equivalent to the evaluation
To obtain the relation when the weight is odd, we consider the integral
instead. Finally, we get the following relation
In particular, when k+r is even, we have
Note that the second sum of multiple zeta values is equal to
by Ohno's generalization of the sum formula and the duality theorem.
The following proposition plays an important role in our proof of Theorem 4 and 5.
Proposition 8. For a pair of integers k, r > 0 with k even, we have
Proof. Consider the integral
The above integral is separable and its value is given by
Let Dj(j = 1,2,3,4,5,6) be simplices obtained from all possible interlacing of variables t1,t2 and u1,u2. Note that the integrand of the integral is invariant if we exchange the roles of t1,t2 and u1,u2. Therefore, it suffices to evaluate the integration over D1: 0 < t1 < t2 < u1 < u2 < 1, D3:0 < t1 < u1 < t2 < u2 < 1 and D4: 0 < t1 < u1 < u2 < t2 < 1.
For the simplex D1, we rewrite the integral as
In terms of multiple zeta values, it is equal to
Sum over k leads to
Identifying r - l + 1 as a new variable αk+1, it is
Both the integrations over D3 and D4 have the same value
(2r+1-1) ζ({ 1 }k+1, r+3).
Including the integrations over D2, D5 and D6, we get the identity
3 The proof of Theorem 4
In our previous considerations, the integrands are so simple that it is easy to evaluate the integrations over all the simplices Dj (j = 1,2,3,4,5,6). It will be a different story for our next consideration. For our convenience, we shall use the notation
ζ(α,m) = ζ(α0,α1,...,αk,m).
Now we are ready to prove Theorem 4.
Proof of Theorem 4. For a pair of integers k, r > 0, we consider the integral
Rewrite the second factor in the integrand as
Also we have
and
Consequently, the integral is separable and its value is given by
When both k and r are even, the above sum is equal to
The sum of the second summation, by Proposition 6, is equal to
and hence the total is
by Proposition 8.
Next we evaluate the integrations over Dj(j = 1,2,3,4,5,6). For the simplex D1: 0 < t1 < t2 < u1 < u2 < 1, we rewrite the integral as
so it can be expressed as
or
For fixed l, by counting the number of ζ (α0,α1, ..., αk,r - l +2) appeared in the summation, we conclude that the sum is equal to
Extending the above sum to l = -1 and then set l+1 as a new dummy variable, the sum is
The integration over D2 is just the alternating double sum
which would be denoted by G in the after.
The integrations over D3, D4, D5 and D6 yield the following multiple zeta values
Adding together all the values obtained from the integrations over the simplices Dj (j = 1,2,3,4,5,6), we get the identity
Our assertion follows from
On the other hand, if we consider the integral
it has the value
The assertion in Theorem 5 follows after a similar procedure.
4 A final remark
Through the double generating function
we are able to express ζ ({ 1 }m,n+2) in terms of the special values of Riemann zeta function at positive integers. The shuffle relations in Proposition 6 provide us to evaluate ζ({ 1 }m,n+2) recursively in terms of special values of Riemann zeta function.
Another way to prove Theorem 4 is to count the number of appearances of each ζ(α0,α1,...,αk,2) in the complicated double alternating sum. Therefore the identity is just a problem of counting and can be extended. For example, for a pair of even integers k,r > 0, we have
or for any positive integer m > 2,
or for a string of nonnegative integers p0,p1,...,pk
All these identities are difficult to be proved otherwise.
Received: 26/III/09.
Accepted: 16/XII/09.
The third author was partially supported by Grant NSC99-2115-M-309-001 from the National Science Council of Taiwan, Republic of China.
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Publication Dates
-
Publication in this collection
22 Nov 2010 -
Date of issue
2010
History
-
Accepted
16 Dec 2009 -
Received
26 Mar 2009