# On a Reciprocity Law for Finite Multiple Zeta Values

Authors: Markus Kuba; Helmut Prodinger
Journal: International Journal of Combinatorics
 (2010)

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## Abstract

It was shown by Kirschenhofer and Prodinger (1998) and Kuba et al. (2008) that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from Kirschenhofer and Prodinger (1998) and Kuba et al. (2008)  can be generalized to finite variants of multiple zeta values, involving a finite variant of the shuffle identity for multiple zeta values. We present the generalized reciprocity relation and furthermore a combinatorial proof of the shuffle identity based on partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums.


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## Body

## 1. Introduction

LetHn=∑k=1n1/k denote the nth harmonic number and Hn(s)=∑k=1n1/ks the nth harmonic number of order s, with n,s∈ℕ and Hn=Hn(1). Kirschenhofer and Prodinger [1] analyzed the variance of the number of comparisons of the famous QUICKSELECT algorithm, also known as FIND [2] and derived a reciprocity relation for (first-order) harmonic numbers. Subsequently, the reciprocity relation of [1] was generalized [3], where the following identity was derived:(1.1)∑k=1jHN-k(a)kb+∑k=1N+1-jHN-k(b)ka=-1jb(N+1-j)a+Hj(b)HN+1-j(a)+RN(a,b),
where RN(a,b)=∑k=1NHN-k(a)/kb, which can be evaluated into a finite analogue of the so-called Euler identity for ζ(a)ζ(b) stated below,(1.2)RN(a,b)=∑i=1a(i+b-2b-1)ζN(i+b-1,a+1-i)+∑i=1b(i+a-2a-1)ζN(i+a-1,b+1-i),
where the multiple zeta values [4–9], and its finite counterpart are defined as follows:(1.3)ζ(a)=ζ(a1,…,ar):=∑n1>n2>⋯>nr≥11n1a1n2a2⋯nrar,ζN(a)=ζN(a1,…,ar):=∑N≥n1>n2>⋯>nr≥11n1a1n2a2⋯nrar.
Note that ζN(a)=HN(a). Finite multiple zeta values are also called truncated multiple zeta values. They are also of great importance in particle physics, see for example the works [10–12], and closely related to so-called harmonics sums. Let w=∑i=1rai denote the weight and d=r the depth of (finite) multiple zeta values. The aim of this note is to derive a generalization of the reciprocity relation (1.1), stated below in Theorem 2.1, by considering the more general sums(1.4)∑k=1jζk-1(b2,…,bs)ζN-k(a1,…,ar)kb1+∑k=1N+1-jζk-1(a2,…,ar)ζN-k(b1,…,bs)ka1,
instead of the previously considered sums ∑k=1jHN-k(a)/kb and ∑k=1N+1-jHN-k(b)/ka. The generalization involves a finite variant of the shuffle identity for multiple zeta values; see, for example, Hoffman [13] for a general algebraic framework for shuffle products. We will give an elementary proof of the shuffle identity using only partial fraction decomposition and the combinatorial properties of the shuffle product in Sections 3.1 and 3.2. Moreover, we discuss the close relation between this finite variant of the shuffle identity and the shuffle identity for generalized polylogarithm functions; it will turn out that the finite variant of the shuffle identity is equivalent to the shuffle identity for generalized polylogarithm functions.To simplify the presentation of this work, we will frequently use the shorthand notationsa=(a1,…,ar), a2=(a2,…,ar), and b=(b1,…,bs), b2=(b2,…,bs), respectively, with r,s∈ℕ and ai,bk∈ℕ for 1≤i≤r and 1≤k≤s.


## 2. Results

We will state the main theorem and two corollaries below, and subsequently discuss their proofs and the precise definition of the shuffle relation for multiple zeta values.Theorem 2.1.
The finite multiple zeta valuesζN(a)=ζN(a1,…,ar), ζN(b)=ζN(b1,…,bs) satisfy the following reciprocity relation.
(2.1)∑k=1jζk-1(b2,…,bs)ζN-k(a1,…,ar)kb1+∑k=1N+1-jζk-1(a2,…,ar)ζN-k(b1,…,bs)ka1=ζN+1-j(a)ζj(b)-ζj-1(b2)ζN-j(a2)jb1(N+1-j)a1+RN(a;b).
The quantity RN(a;b)=∑k=1NζN-k(b)ζk-1(a2,…,ar)/ka1=RN(b;a) can be written as a sum of finite multiple zeta values, all of them having weight w=∑i=1rar+∑i=1sbi and depth d=r+s.Remark 2.2.
The quantityRN(a;b) satisfies a shuffle identity resembling the ordinary shuffle identity for multiple zeta values ζ(a)ζ(b)=ζ(a⧢b); see Sections 3.1, 3.2 and Proposition 3.4 for details.Corollary 2.3.
We obtain the complementary identity(2.2)∑k=1j-1ζk(b)ζN-k-1(a2)(N-k)a1+∑k=1N-jζk(a)ζN-k-1(b2)(N-k)b1=ζj-1(b)ζN-j(a2)(N+1-j)a1+ζN-j(a)ζj-1(b2)jb1-ζN+1-j(a)ζj(b)+ζj-1(b2)ζn-j(a2)jb1(N+1-j)a1+RN(a;b).Next we state an immediate asymptotic implication of the previous result.Corollary 2.4.
ForN=2n+1, j=n+1, with a1,b1∈ℕ∖{1} and n→∞, we obtain the following result:
(2.3)limn→∞(∑k=1jζk-1(b2)ζN-k(a)kb1+∑k=1N+1-jζk-1(a2)ζN-k(b)ka1)=2ζ(a)ζ(b).


## 3. The Proof of the Reciprocity Relation

In order to prove Theorem2.1, we proceed as follows (using the beforehand introduced shorthand notations).(3.1)∑k=1jζk-1(b2)ζN-k(a)kb1=∑k=1jζk-1(b2)kb1(ζN-j(a)+∑ℓ=N+1-jN-kζℓ-1(a2)ℓa1)=ζN-j(a)ζj(b)+∑k=1jζk-1(b2)kb1∑ℓ=N+1-jN-kζℓ-1(a2)ℓa1.
After changing summations, we obtain(3.2)∑k=1jζk-1(b2)ζN-k(a)kb1=ζN-j(a)ζj(b)+∑ℓ=N+1-jN-1ζℓ-1(a2)ℓa1∑k=1N-ℓζk-1(b2)kb1=ζN-j(a)ζj(b)+∑ℓ=N+1-jN-1ζℓ-1(a2)ζN-ℓ(b)ℓa1.
Using(3.3)ζN-j(a)ζj(b)+ζN-j(a2)ζj-1(b)(N+1-j)a1=ζN-j(a)ζj(b)+ζN-j(a2)(N+1-j)a1(ζj(b)-ζj-1(b2)jb1)=ζN+1-j(a)ζj(b)-ζN-j(a2)ζj-1(b2)(N+1-j)a1jb1,
and the fact that ζ0(b)=0 gives the intermediate result(3.4)∑k=1jζk-1(b2)ζN-k(a)kb1=ζN+1-j(a)ζj(b)-ζN-j(a2)ζj-1(b2)(N+1-j)a1jb1+∑ℓ=N+2-jNζℓ-1(a2)ζN-ℓ(b)ℓa1,
Add  the sum ∑k=1N+1-jζk-1(a2)ζN-k(b)/ka1 to both sides of the equation above. This proves the first part of Theorem 2.1 and(3.5)RN(a;b)=∑k=1NζN-k(b)ζk-1(a2,…,ar)ka1.
For the evaluation of RN(a;b), we note that R0(a;b)=0, and further(3.6)RN(a;b)=∑k=1N(Rk(a;b)-Rk-1(a;b)).
Since ζ0(b)=0, we have(3.7)RN(a;b)-RN-1(a;b)=∑k=1NζN-k(b)ζk-1(a2,…,ar)ka1-∑k=1N-1ζN-1-k(b)ζk-1(a2,…,ar)ka1=∑k=1N-1(ζN-k(b)-ζN-1-k(b)ζk-1(a2,…,ar)ka1=∑k=1N-1ζN-1-k(b2,…,bs)ζk-1(a2,…,ar)(N-k)b1ka1.
Now we use the following partial fraction decomposition (This identity has been rediscovered many times. For a fascinating historic account, see [14].), which appears already in [15],(3.8)1ka(N-k)b=∑i=1a(i+b-2b-1)Ni+b-1ka+1-i+∑i=1b(i+a-2a-1)Ni+a-1(N-k)b+1-i,
and obtain(3.9)∑k=1N-1ζk-1(a2,…,ar)ζN-1-k(b2,…,bs)(N-k)b1ka1=∑i=1a1∑k=1N-1(i+b1-2b1-1)ζk-1(a2,…,ar)ζN-1-k(b2,…,bs)Ni+b1-1ka1+1-i+∑i=1b1∑k=1N-1(i+a1-2a1-1)ζk-1(a2,…,ar)ζN-1-k(b2,…,bs)Ni+a1-1(N-k)b1+1-i.
Consequently, by summing up according to (3.6), we get the following recurrence relation for RN(a;b):(3.10)RN(a;b)=∑i=1a1∑n1=1N(i+b1-2b1-1)n1i+b1-1Rn1-1(a1+1-i,a2,…,ar;b2,…,bs)+∑i=1b1∑n1=1N(i+a1-2a1-1)n1i+a1-1Rn1-1(a2,…,ar;b1+1-i,b2,…,bs).
This recurrence relation suggests that there exists an evaluation of RN(a;b) into sums of finite multiple zeta values, all of them having weight w=∑i=1rar+∑i=1sbi and depth d=r+s. In order to specify this evaluation, we need to introduce the shuffle product for words over a noncommutative alphabet and to study the arising shuffle algebra, and its relation to (finite) multiple zeta values and RN(a;b). For a general algebraic framework for the shuffle product, we refer the reader to the work of Hoffman [13]. We remark that the recurrence relation above for RN(a;b) was already derived in the context of particle physics [11, 12]. Furthermore, weighted extensions including alternating sign versions have been treated there. An important algorithmic treatment of such sums is implemented in the package Summer for the computer algebra system Form.


### 3.1. The Shuffle Algebra

Let𝒜 denote a finite noncommutative alphabet consisting of a set of letters. A word w on the alphabet 𝒜 consists of a sequence of letters from 𝒜. Let 𝒜* denote the set of all words on the alphabet 𝒜. A polynomial on 𝒜 over ℚ is a rational linear combination of words on 𝒜. The set of all such polynomials is denoted by ℚ〈𝒜〉. Let the shuffle product of two words w,v∈𝒜*, with w=x1⋯xn, v=xn+1⋯xn+m, xi∈𝒜 for 1≤i≤n+m, be defined as follows:(3.11)w⧢v:=∑xσ(1)xσ(2)⋯xσ(n+m),
where the sum runs over all (n+mn) permutations σ∈𝔖n+m which satisfy σ-1(j)<σ-1(k) for all 1≤j<k≤n and n+1≤j<k≤n+m. Note that the sum runs over all words of length n+m, counting multiplicities, in which the relative orders of the letters x1,…,xn and xn+1,…,xn+m are preserved. Equivalently, the shuffle product of two words w,v∈𝒜* can be defined in a recursive way:(3.12)∀w∈𝒜*,ɛ⧢w=w⧢ɛ=w,∀x,y∈𝒜,w,v∈𝒜*,xw⧢yv=x(w⧢yv)+y(xw⧢v).
The shuffle product extends to ℚ〈𝒜〉 by linearity. Note that the set ℚ〈𝒜〉, provided with the shuffle product ⧢, becomes a commutative and associative algebra. We remark that the term “shuffle” is used because such permutations arise in riffle shuffling a deck of n+m cards cut into one pile of n cards and a second pile of m cards [7].In the following, we will restrict ourselves to the non-commutative alphabet𝒜={ω0,ω1} and the arising shuffle algebra (ℚ〈𝒜〉,⧢). Hoang and Petitot [16] derived a shuffle identity for words A=ω0a-1ω1, B=ω0b-1ω1, which is stated below.Lemma 3.1.
Fora,b∈ℕ, let A=ω0a-1ω1 and B=ω0b-1ω1 be words on the non-commutative alphabet 𝒜={ω0,ω1}.
(3.13)A⧢B=∑i=0a-1(b-1+ib-1)ω0b-1+iω1ω0a-1-iω1+∑i=0b1-1(a-1+ia-1)ω0a-1+iω1ω0b-1-iω1.We will use a slight extension of this identity, which easily follows from the recursive definition of the shuffle product.Lemma 3.2.
Forr,s≥1 and ai,bj∈ℕ, 1≤i≤r, 1≤j≤s, let A:=ω0a1-1ω1⋯ω0ar-1ω1 and B:=ω0b1-1ω1⋯ω0bs-1ω1 be words on the non-commutative alphabet 𝒜={ω0,ω1}.
(3.14)A⧢B=∑i=1a1(i+b1-2b1-1)ω0i+b1-2ω1(Ai′⧢B2)+∑i=1b1(i+a1-2a1-1)ω0i+a1-2ω1(A2⧢Bi′),
with Ai′:=ω0a1-iω1ω0a2-1ω1⋯ω0ar-1ω1, Bi′:=ω0b1-iω1ω0b2-1ω1⋯ω0bs-1ω1 and further A2:=ω0a2ω1⋯ω0ar-1ω1, B2:=ω0b2-1ω1⋯ω0bs-1ω1.Note that the partial fraction decomposition (3.8) of 1/ka(N-k)b somewhat mimics the shuffle identity for words A=ω0a-1ω1, B=ω0b-1ω1, derived by Hoang and Petitot [16].


### 3.2. The Shuffle Algebra and Finite Multiple Zeta Values

Leta denote an arbitrary r-tuple of positive integers a=(a1,…,ar) with ai∈ℕ for 1≤i≤r and r≥1. To any a, we will associate a unique word A=A(a) over the non-commutative alphabet 𝒜={ω0,ω1} as follows: A=A(a) such that A:=ω0a1-1ω1ω0a2-1ω1⋯ω0ar-1ω1. Let 𝒜* denote the set of all words over the alphabet 𝒜. Let (ZN)N≥1 denote a family of linear maps from the algebra ℚ〈𝒜〉 to the rational numbers, ZN:ℚ〈𝒜〉→ℚ, mapping words over the non-commutative alphabet 𝒜={ω0,ω1} to finite multiple zeta values in the following way. For words A:=ω0a1-1ω1ω0a2-1ω1⋯ω0ar-1ω1∈𝒜*, with r,N≥1, we define(3.15)ZN(A)=ZN(ω0a1-1ω1ω0a2-1ω1⋯ω0ar-1ω1)=ζN(a1,…,ar)=ζN(a).
Moreover, we additionally define Z0(A)=ζ0(a)=0 for all A∈𝒜*, and ZN(ɛ)=1 for all N≥1. The family of maps (ZN)N≥1 linearly extend to ℚ〈𝒜〉. By the recursive definition of the finite multiple zeta values, we can express the images of the maps ZN in a recursive way. Let A:=ω0a1-1ω1ω0a2-1ω1⋯ω0ar-1ω1∈𝒜*, with r≥1 and a1,…,ar≥1.(3.16)ZN(A)=ζN(a)=∑n1=1N1n1a1ζn1-1(a2,…,ar)=∑n1=1N1n1a1Zn1-1(ω0a2-1ω1⋯ω0ar-1ω1).
We need the following result.Lemma 3.3.
Forr,s≥1 and ai,bj∈ℕ, 1≤i≤r, 1≤j≤s, let A:=ω0a1-1ω1⋯ω0ar-1ω1 and B:=ω0b1-1ω1⋯ω0bs-1ω1 be words on the non-commutative alphabet 𝒜={ω0,ω1}. Then,
(3.17)ZN(A⧢B)=∑i=1a1∑n1=1N(i+b1-2b1-1)n1i+b1-1Zn1-1(Ai′⧢B2)+∑i=1b1∑n1=1N(i+a1-2a1-1)n1i+a1-1Zn1-1(A2⧢Bi′).
The depths d=r+s and the weights w=∑i=1rai+∑k=1sbk of the arising finite multiple zeta values are all the same.Proof.
By linearity of the mapsZN and Lemma 3.2, we get first
(3.18)ZN(A⧢B)=∑i=1a1(i+b1-2b1-1)ZN(ω0i+b1-2ω1(Ai′⧢B2))+∑i=1b1(i+a1-2a1-1)ZN(ω0i+a1-2ω1(A2⧢Bi′)),
using the notations of Lemma 3.2 for Ai′,Bi′,A2,B2. By definition of the shuffle product, Ai′⧢B2∈ℚ〈𝒜〉 and A2⧢Bi′∈ℚ〈𝒜〉 are rational linear combinations of words over 𝒜. Let {Ai′⧢B2} and {A2⧢Bi′} denote the sets of different words generated by the shuffles Ai′⧢B2 and A2⧢Bi′. Using the set notation, we write
(3.19)Ai′⧢B2=∑w∈{Ai′⧢B2}qww,A2⧢Bi′=∑w∈{A2⧢Bi′}qww,
with qw∈ℚ and w∈𝒜*, which helps to obtain a simple presentation of the subsequent calculations. We have
(3.20)ZN(A⧢B)=∑i=1a1(i+b1-2b1-1)ZN(ω0i+b1-2ω1∑w∈{Ai′⧢B2}qww)+∑i=1b1(i+a1-2a1-1)ZN(ω0i+a1-2ω1∑w∈{A2⧢Bi′}qww).
Using the linearity of the maps ZN and the fact that we can recursively describe their images, we get further
(3.21)ZN(A⧢B)=∑i=1a1(i+b1-2b1-1)∑w∈{Ai′⧢B2}qw∑n1=1N1n1i+b1-1Zn1-1(w)+∑i=1b1(i+a1-2a1-1)∑w∈{A2⧢Bi′}qw∑n1=1N1n1i+a1-1Zn1-1(w).
Interchanging the latter summations gives the stated result.
(3.22)ZN(A⧢B)=∑i=1a1(i+b1-2b1-1)∑n1=1N1n1i+b1-1∑w∈{Ai′⧢B2}qwZn1-1(w)+∑i=1b1(i+a1-2a1-1)∑n1=1N1n1i+a1-1∑w∈{A2⧢Bi′}qwZn1-1(w)=∑i=1a1∑n1=1N(i+b1-2b1-1)n1i+b1-1Zn1-1(Ai′⧢B2)+∑i=1b1∑n1=1N(i+a1-2a1-1)n1i+a1-1Zn1-1(A2⧢Bi′).
It can easily be checked that the finite multiple zeta values all have the same depth and weight.Now we are ready to provide the evaluation ofRN(a;b).Proposition 3.4.
For arbitraryr,s≥1, let a and b be given by a=(a1,…,ar) and b=(b1,…,bs), with ai,bj∈ℕ for 1≤i≤r, 1≤j≤s. Let A=A(a) and B=A(b) denote the words associated to a and b by A:=ω0a1-1ω1⋯ω0ar-1ω1 and B:=ω0b1-1ω1⋯ω0bs-1ω1. Then, for arbitrary N≥1,
(3.23)RN(a;b)=ZN(A⧢B).Proof.
We use induction with respect tod=r+s, corresponding to the depths of the arising finite multiple zeta values. The result clearly holds for depth d=2; see identity (1.2), as shown in [3]. Now assume that d≥3. Using the recurrence relation (3.10) for RN(a;b), we get
(3.24)RN(a;b)=∑i=1a1∑n1=1N(i+b1-2b1-1)n1i+b1-1Rn1-1(a1+1-i,a2,…,ar;b2,…,bs)+∑i=1b1∑n1=1N(i+a1-2a1-1)n1i+a1-1Rn1-1(a2,…,ar;b1+1-i,b2,…,bs).
The induction hypothesis states that RN(a;b)=ZN(A⧢B) for arbitrary r,s≥1 such that r+s<d and arbitrary N≥1. By the recurrence relation for RN(a;b), we can reduce RN(a;b) to values of the types Rn1-1(a1+1 – i,a2,…,ar;b2,…,bs) and Rn1-1(a2,…,ar;b1+1 – i,b2,…,bs), which are of depth smaller than d=r+s. Hence, we get by the induction hypothesis
(3.25)RN(a;b)=∑i=1a1∑n1=1N(i+b1-2b1-1)n1i+b1-1Zn1-1(Ai′⧢B2)+∑i=1b1∑n1=1N(i+a1-2a1-1)n1i+a1-1Zn1-1(A2⧢Bi′).
By Lemma 3.3, using the notations for Ai′,Bi′,A2,B2 of Lemma 3.2, we get
(3.26)∑i=1a1∑n1=1N(i+b1-2b1-1)n1i+b1-1Zn1-1(Ai′⧢B2)+∑i=1b1∑n1=1N(i+a1-2a1-1)n1i+a1-1Zn1-1(A2⧢Bi′)=ZN(A⧢B).
Consequently,
(3.27)RN(a;b)=ZN(A⧢B).
This proves the stated result for RN(a;b) and the corresponding statement of Theorem 2.1.Corollary2.3 can easily be deduced by noting that the sum of the left hand sides of Corollary 2.3 and Theorem 2.1 adds up to RN(a;b) plus the additional two extra terms. The proof of Corollary 2.4 will be given in the next section, which consists of several remarks.


## 3.1. The Shuffle Algebra

Let𝒜 denote a finite noncommutative alphabet consisting of a set of letters. A word w on the alphabet 𝒜 consists of a sequence of letters from 𝒜. Let 𝒜* denote the set of all words on the alphabet 𝒜. A polynomial on 𝒜 over ℚ is a rational linear combination of words on 𝒜. The set of all such polynomials is denoted by ℚ〈𝒜〉. Let the shuffle product of two words w,v∈𝒜*, with w=x1⋯xn, v=xn+1⋯xn+m, xi∈𝒜 for 1≤i≤n+m, be defined as follows:(3.11)w⧢v:=∑xσ(1)xσ(2)⋯xσ(n+m),
where the sum runs over all (n+mn) permutations σ∈𝔖n+m which satisfy σ-1(j)<σ-1(k) for all 1≤j<k≤n and n+1≤j<k≤n+m. Note that the sum runs over all words of length n+m, counting multiplicities, in which the relative orders of the letters x1,…,xn and xn+1,…,xn+m are preserved. Equivalently, the shuffle product of two words w,v∈𝒜* can be defined in a recursive way:(3.12)∀w∈𝒜*,ɛ⧢w=w⧢ɛ=w,∀x,y∈𝒜,w,v∈𝒜*,xw⧢yv=x(w⧢yv)+y(xw⧢v).
The shuffle product extends to ℚ〈𝒜〉 by linearity. Note that the set ℚ〈𝒜〉, provided with the shuffle product ⧢, becomes a commutative and associative algebra. We remark that the term “shuffle” is used because such permutations arise in riffle shuffling a deck of n+m cards cut into one pile of n cards and a second pile of m cards [7].In the following, we will restrict ourselves to the non-commutative alphabet𝒜={ω0,ω1} and the arising shuffle algebra (ℚ〈𝒜〉,⧢). Hoang and Petitot [16] derived a shuffle identity for words A=ω0a-1ω1, B=ω0b-1ω1, which is stated below.Lemma 3.1.
Fora,b∈ℕ, let A=ω0a-1ω1 and B=ω0b-1ω1 be words on the non-commutative alphabet 𝒜={ω0,ω1}.
(3.13)A⧢B=∑i=0a-1(b-1+ib-1)ω0b-1+iω1ω0a-1-iω1+∑i=0b1-1(a-1+ia-1)ω0a-1+iω1ω0b-1-iω1.We will use a slight extension of this identity, which easily follows from the recursive definition of the shuffle product.Lemma 3.2.
Forr,s≥1 and ai,bj∈ℕ, 1≤i≤r, 1≤j≤s, let A:=ω0a1-1ω1⋯ω0ar-1ω1 and B:=ω0b1-1ω1⋯ω0bs-1ω1 be words on the non-commutative alphabet 𝒜={ω0,ω1}.
(3.14)A⧢B=∑i=1a1(i+b1-2b1-1)ω0i+b1-2ω1(Ai′⧢B2)+∑i=1b1(i+a1-2a1-1)ω0i+a1-2ω1(A2⧢Bi′),
with Ai′:=ω0a1-iω1ω0a2-1ω1⋯ω0ar-1ω1, Bi′:=ω0b1-iω1ω0b2-1ω1⋯ω0bs-1ω1 and further A2:=ω0a2ω1⋯ω0ar-1ω1, B2:=ω0b2-1ω1⋯ω0bs-1ω1.Note that the partial fraction decomposition (3.8) of 1/ka(N-k)b somewhat mimics the shuffle identity for words A=ω0a-1ω1, B=ω0b-1ω1, derived by Hoang and Petitot [16].


## 3.2. The Shuffle Algebra and Finite Multiple Zeta Values

Leta denote an arbitrary r-tuple of positive integers a=(a1,…,ar) with ai∈ℕ for 1≤i≤r and r≥1. To any a, we will associate a unique word A=A(a) over the non-commutative alphabet 𝒜={ω0,ω1} as follows: A=A(a) such that A:=ω0a1-1ω1ω0a2-1ω1⋯ω0ar-1ω1. Let 𝒜* denote the set of all words over the alphabet 𝒜. Let (ZN)N≥1 denote a family of linear maps from the algebra ℚ〈𝒜〉 to the rational numbers, ZN:ℚ〈𝒜〉→ℚ, mapping words over the non-commutative alphabet 𝒜={ω0,ω1} to finite multiple zeta values in the following way. For words A:=ω0a1-1ω1ω0a2-1ω1⋯ω0ar-1ω1∈𝒜*, with r,N≥1, we define(3.15)ZN(A)=ZN(ω0a1-1ω1ω0a2-1ω1⋯ω0ar-1ω1)=ζN(a1,…,ar)=ζN(a).
Moreover, we additionally define Z0(A)=ζ0(a)=0 for all A∈𝒜*, and ZN(ɛ)=1 for all N≥1. The family of maps (ZN)N≥1 linearly extend to ℚ〈𝒜〉. By the recursive definition of the finite multiple zeta values, we can express the images of the maps ZN in a recursive way. Let A:=ω0a1-1ω1ω0a2-1ω1⋯ω0ar-1ω1∈𝒜*, with r≥1 and a1,…,ar≥1.(3.16)ZN(A)=ζN(a)=∑n1=1N1n1a1ζn1-1(a2,…,ar)=∑n1=1N1n1a1Zn1-1(ω0a2-1ω1⋯ω0ar-1ω1).
We need the following result.Lemma 3.3.
Forr,s≥1 and ai,bj∈ℕ, 1≤i≤r, 1≤j≤s, let A:=ω0a1-1ω1⋯ω0ar-1ω1 and B:=ω0b1-1ω1⋯ω0bs-1ω1 be words on the non-commutative alphabet 𝒜={ω0,ω1}. Then,
(3.17)ZN(A⧢B)=∑i=1a1∑n1=1N(i+b1-2b1-1)n1i+b1-1Zn1-1(Ai′⧢B2)+∑i=1b1∑n1=1N(i+a1-2a1-1)n1i+a1-1Zn1-1(A2⧢Bi′).
The depths d=r+s and the weights w=∑i=1rai+∑k=1sbk of the arising finite multiple zeta values are all the same.Proof.
By linearity of the mapsZN and Lemma 3.2, we get first
(3.18)ZN(A⧢B)=∑i=1a1(i+b1-2b1-1)ZN(ω0i+b1-2ω1(Ai′⧢B2))+∑i=1b1(i+a1-2a1-1)ZN(ω0i+a1-2ω1(A2⧢Bi′)),
using the notations of Lemma 3.2 for Ai′,Bi′,A2,B2. By definition of the shuffle product, Ai′⧢B2∈ℚ〈𝒜〉 and A2⧢Bi′∈ℚ〈𝒜〉 are rational linear combinations of words over 𝒜. Let {Ai′⧢B2} and {A2⧢Bi′} denote the sets of different words generated by the shuffles Ai′⧢B2 and A2⧢Bi′. Using the set notation, we write
(3.19)Ai′⧢B2=∑w∈{Ai′⧢B2}qww,A2⧢Bi′=∑w∈{A2⧢Bi′}qww,
with qw∈ℚ and w∈𝒜*, which helps to obtain a simple presentation of the subsequent calculations. We have
(3.20)ZN(A⧢B)=∑i=1a1(i+b1-2b1-1)ZN(ω0i+b1-2ω1∑w∈{Ai′⧢B2}qww)+∑i=1b1(i+a1-2a1-1)ZN(ω0i+a1-2ω1∑w∈{A2⧢Bi′}qww).
Using the linearity of the maps ZN and the fact that we can recursively describe their images, we get further
(3.21)ZN(A⧢B)=∑i=1a1(i+b1-2b1-1)∑w∈{Ai′⧢B2}qw∑n1=1N1n1i+b1-1Zn1-1(w)+∑i=1b1(i+a1-2a1-1)∑w∈{A2⧢Bi′}qw∑n1=1N1n1i+a1-1Zn1-1(w).
Interchanging the latter summations gives the stated result.
(3.22)ZN(A⧢B)=∑i=1a1(i+b1-2b1-1)∑n1=1N1n1i+b1-1∑w∈{Ai′⧢B2}qwZn1-1(w)+∑i=1b1(i+a1-2a1-1)∑n1=1N1n1i+a1-1∑w∈{A2⧢Bi′}qwZn1-1(w)=∑i=1a1∑n1=1N(i+b1-2b1-1)n1i+b1-1Zn1-1(Ai′⧢B2)+∑i=1b1∑n1=1N(i+a1-2a1-1)n1i+a1-1Zn1-1(A2⧢Bi′).
It can easily be checked that the finite multiple zeta values all have the same depth and weight.Now we are ready to provide the evaluation ofRN(a;b).Proposition 3.4.
For arbitraryr,s≥1, let a and b be given by a=(a1,…,ar) and b=(b1,…,bs), with ai,bj∈ℕ for 1≤i≤r, 1≤j≤s. Let A=A(a) and B=A(b) denote the words associated to a and b by A:=ω0a1-1ω1⋯ω0ar-1ω1 and B:=ω0b1-1ω1⋯ω0bs-1ω1. Then, for arbitrary N≥1,
(3.23)RN(a;b)=ZN(A⧢B).Proof.
We use induction with respect tod=r+s, corresponding to the depths of the arising finite multiple zeta values. The result clearly holds for depth d=2; see identity (1.2), as shown in [3]. Now assume that d≥3. Using the recurrence relation (3.10) for RN(a;b), we get
(3.24)RN(a;b)=∑i=1a1∑n1=1N(i+b1-2b1-1)n1i+b1-1Rn1-1(a1+1-i,a2,…,ar;b2,…,bs)+∑i=1b1∑n1=1N(i+a1-2a1-1)n1i+a1-1Rn1-1(a2,…,ar;b1+1-i,b2,…,bs).
The induction hypothesis states that RN(a;b)=ZN(A⧢B) for arbitrary r,s≥1 such that r+s<d and arbitrary N≥1. By the recurrence relation for RN(a;b), we can reduce RN(a;b) to values of the types Rn1-1(a1+1 – i,a2,…,ar;b2,…,bs) and Rn1-1(a2,…,ar;b1+1 – i,b2,…,bs), which are of depth smaller than d=r+s. Hence, we get by the induction hypothesis
(3.25)RN(a;b)=∑i=1a1∑n1=1N(i+b1-2b1-1)n1i+b1-1Zn1-1(Ai′⧢B2)+∑i=1b1∑n1=1N(i+a1-2a1-1)n1i+a1-1Zn1-1(A2⧢Bi′).
By Lemma 3.3, using the notations for Ai′,Bi′,A2,B2 of Lemma 3.2, we get
(3.26)∑i=1a1∑n1=1N(i+b1-2b1-1)n1i+b1-1Zn1-1(Ai′⧢B2)+∑i=1b1∑n1=1N(i+a1-2a1-1)n1i+a1-1Zn1-1(A2⧢Bi′)=ZN(A⧢B).
Consequently,
(3.27)RN(a;b)=ZN(A⧢B).
This proves the stated result for RN(a;b) and the corresponding statement of Theorem 2.1.Corollary2.3 can easily be deduced by noting that the sum of the left hand sides of Corollary 2.3 and Theorem 2.1 adds up to RN(a;b) plus the additional two extra terms. The proof of Corollary 2.4 will be given in the next section, which consists of several remarks.


## 4. Remarks on Polylogarithms and the Finite Shuffle Identity

For givena=(a1,…,ar) and b=(b1,…,bs), one may define the shuffle product ζN(a⧢b) in terms of the images of the maps ZN using the words A=A(a) and B=A(b) associated to a and b by A:=ω0a1-1ω1⋯ω0ar-1ω1 and B:=ω0b1-1ω1⋯ω0bs-1ω1,(4.1)ζN(a⧢b):=ZN(A⧢B).
It turns out that this definition coincides with the usual definition of the shuffle product for multiple zeta values; for an excellent overview concerning the shuffle product for multiple zeta values, we refer the reader to [5, 16, 17].LetLia(z) denote the (multiple) polylogarithm function with parameters a1,…,ar, defined by(4.2)Lia(z)=Lia1,…,ar(z)=∑n1>n2>⋯>nr≥1zn1n1a1n2a2⋯nrar.
The value RN(a;b) can be obtained by coefficient extraction in the following way:(4.3)RN(a;b)=∑k=1Nζk-1(a2,…,ar)ζN-k(b1,…,bs)ka1=[zN]Lia(z)Lib(z)1-z.
On the other hand, by the finite shuffle identity (3.23) for RN(a;b), one can show the following representation:(4.4)RN(a;b)=[zN]Lia⧢b(z)1-z.
Here the shuffle product for polylogarithm functions Lia⧢b(z) is defined in the usual way. We do not want to go into the proof details concerning the equation above since we would have to state and use the precise definition of the shuffle product for multiple zeta values and polylogarithm functions; avoiding repetition, we skip the details and only refer the interested reader to [17], and Theorem 5.1. We want to remark that the result of Proposition 3.4 for RN(a;b) implies that the shuffle identity for polylogarithm functions, and consequently also for multiple zeta values, can be developed entirely from finite sums using only basic partial fraction decomposition and the combinatorics behind the shuffle product and the shuffle algebra; see Hoffman [13] for an important discussion of the shuffle product. Note that by evaluating at z=1, the shuffle identity for polylogarithm functions implies the shuffle identity for multiple zeta values. The identity above is well known; see for example the article [5]. The shuffle identity for polylogarithm functions is due to the iterated Drinfeld integral representation of polylogarithm functions and multiple zeta values due to Kontsevich  [9]. As remarked in [5], the shuffle identity for polylogarithm functions can be deduced from the fact that the product of two simplex integrals consists of a sum of simplex integrals over all possible interlacings of the respective variables of integration.Finally, we turn to the proof of Corollary2.4. For N=2n+1 and j=n+1 and n→∞, we have(4.5)limn→∞ζj(a)ζN+1-j(b)=limn→∞ζn+1(a)ζn+1(b)=ζ(a)ζ(b),limn→∞ζn(b2,…,bs)ζn(a2,…,ar)(n+1)a1+b1=0,limn→∞RN(a;b)=limn→∞ζ2n+1(a⧢b)=ζ(a)ζ(b),
and the stated result follows.


## 5. The Reciprocity Relation for Weighted Multiple Zeta Values

Results similar to Theorem2.1 and Corollary 2.4 can be obtained for products of weighted finite multiple zeta values, ζN(a1,a2,…,ar;σ1,…,σr), σi∈ℝ∖{0} for 1≤i≤r, defined as follows:(5.1)ζN(a,σ)=ζN(a1,a2,…,ar;σ1,…,σr)=∑N≥n1>n2>⋯>nr≥11∏i=1rniaiσini.
Of particular interest are the cases σi∈{±1} corresponding to a mixture of alternating and nonalternating signs, which are of particular importance in particle physics. We only state the result generalizing Theorem 2.1, with respect to the notations a2=(a2,…,ar), σ2=(σ2,…,σr), and the corresponding notations for b2 and τ2, and leave the generalizations of Corollaries 2.3 and 2.4 to the reader.Theorem 5.1.
The multiple zeta valuesζN(a,σ) and ζN(b,τ) with weights σ and τ satisfy the following reciprocity relation:
(5.2)∑k=1jζk-1(b2,τ2)ζN-k(a,σ)kb1τ1k+∑k=1N+1-jζk-1(a2,σ2)ζN-k(b,τ)ka1σ1k=ζN+1-j(a,σ)ζj(b,τ)-ζj-1(b2,τ2)ζN-j(a2,σ2)τ1jjb1σ1N+1-j(N+1-j)a1+RN(a,σ;b,τ).
Here RN(a,σ;b,τ)=∑k=1NζN-k(b,τ)ζk-1(a2,σ2)/σ1kka1=RN(b,τ;a,σ) satisfies an analogue of the shuffle identity with respect to the weights σ and τ.The proof of Theorem2.1 can easily be adapted to the weighted case. Hence, we only elaborate on the main new difficulty, namely, the evaluation of the quantity(5.3)RN(a,σ;b,τ)=∑k=1NζN-k(b,τ)ζk-1(a2;σ2)σ1kka1.
Proceeding as before, that is, taking differences and using partial fraction decomposition, we obtain the recurrence relation(5.4)RN(a,σ;b,τ)=∑i=1a1∑n1=1N(i+b1-2b1-1)n1i+b1-1τ1n1Rn1-1(a1+1-i,a2,τ1σ1,σ2;b2,τ2)+∑i=1b1∑n1=1N(i+a1-2a1-1)n1i+a1-1σ1n1Rn1-1(a2,σ2;b1+1-i,b2,σ1τ1,τ2).
Consequently, the value RN(a,σ;b,τ) can be evaluated into sums of weighted finite multiple zeta values according to a shuffle identity with respect to the weights σ and τ. We omit the precise definition of this generalization and leave the details to the interested reader.


## 6. Conclusion

We presented a reciprocity relation for finite multiple zeta values, extending the previous results of [1, 3]. The reciprocity relation involves a shuffle product identity for (finite) multiple zeta values, for which we gave a proof using only partial fraction decomposition and the combinatorial properties of the shuffle product. Moreover, we also presented the reciprocity relation for weighted finite multiple zeta values.



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