On a Reciprocity Law for Finite Multiple Zeta Values

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.

1. Introduction

Let 𝐻_𝑛=∑^𝑛_𝑘=11/𝑘 denote the 𝑛th harmonic number and 𝐻_𝑛^(𝑠)=∑^𝑛_𝑘=11/𝑘^𝑠 the 𝑛th harmonic number of order 𝑠, with 𝑛,𝑠∈ℕ and 𝐻_𝑛=𝐻_𝑛⁽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:

^𝑗_𝑘=1𝐻^(𝑎)_𝑁−𝑘𝑘^𝑏+^{𝑁+1−𝑗}_𝑘=1𝐻^(𝑏)_𝑁−𝑘𝑘^𝑎1=−𝑗^𝑏(𝑁+1−𝑗)^𝑎+𝐻_𝑗^(𝑏)𝐻^(𝑎)_{𝑁+1−𝑗}+𝑅_𝑁^(𝑎,𝑏), (1.1) where 𝑅_𝑁^(𝑎,𝑏)=∑^𝑁_𝑘=1𝐻^(𝑎)_𝑁−𝑘/𝑘^𝑏, which can be evaluated into a finite analogue of the so-called Euler identity for 𝜁(𝑎)𝜁(𝑏) stated below,

𝑅_𝑁^(𝑎,𝑏)=^𝑎_𝑖=1𝜁𝑖+𝑏−2𝑏−1^𝑁(𝑖+𝑏−1,𝑎+1−𝑖)+^𝑏_𝑖=1𝜁𝑖+𝑎−2𝑎−1^𝑁(𝑖+𝑎−1,𝑏+1−𝑖), (1.2) where the multiple zeta values [4–9], and its finite counterpart are defined as follows:

𝜁(𝐚)=𝜁(𝑎₁,…,𝑎_𝑟)∶=_{𝑛1>𝑛2>⋯>𝑛𝑟≥1}1𝑛^𝑎1₁𝑛^𝑎2₂⋯𝑛^𝑎𝑟_𝑟,𝜁_𝑁(𝐚)=𝜁_𝑁(𝑎₁,…,𝑎_𝑟)∶=_{𝑁≥𝑛1>𝑛2>⋯>𝑛𝑟≥1}1𝑛^𝑎1₁𝑛^𝑎2₂⋯𝑛^𝑎𝑟_𝑟. (1.3) Note that 𝜁_𝑁(𝑎)=𝐻_𝑁^(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=∑^𝑟_𝑖=1𝑎_𝑖 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

^𝑗_𝑘=1𝜁_𝑘−1(𝑏₂,…,𝑏_𝑠)𝜁_𝑁−𝑘(𝑎₁,…,𝑎_𝑟)𝑘^𝑏1+^{𝑁+1−𝑗}_𝑘=1𝜁_𝑘−1(𝑎₂,…,𝑎_𝑟)𝜁_𝑁−𝑘(𝑏₁,…,𝑏_𝑠)𝑘^𝑎1, (1.4) instead of the previously considered sums ∑^𝑗_𝑘=1𝐻^(𝑎)_𝑁−𝑘/𝑘^𝑏 and ∑^{𝑁+1−𝑗}_𝑘=1𝐻^(𝑏)_𝑁−𝑘/𝑘^𝑎. 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 notations 𝐚=(𝑎₁,…,𝑎_𝑟), 𝐚₂=(𝑎₂,…,𝑎_𝑟), and 𝐛=(𝑏₁,…,𝑏_𝑠), 𝐛₂=(𝑏₂,…,𝑏_𝑠), respectively, with 𝑟,𝑠∈ℕ and 𝑎_𝑖,𝑏_𝑘∈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 values 𝜁_𝑁(𝐚)=𝜁_𝑁(𝑎₁,…,𝑎_𝑟), 𝜁_𝑁(𝐛)=𝜁_𝑁(𝑏₁,…,𝑏_𝑠) satisfy the following reciprocity relation. ^𝑗_𝑘=1𝜁_𝑘−1(𝑏₂,…,𝑏_𝑠)𝜁_𝑁−𝑘(𝑎₁,…,𝑎_𝑟)𝑘^𝑏1+^{𝑁+1−𝑗}_𝑘=1𝜁_𝑘−1(𝑎₂,…,𝑎_𝑟)𝜁_𝑁−𝑘(𝑏₁,…,𝑏_𝑠)𝑘^𝑎1=𝜁_{𝑁+1−𝑗}(𝐚)𝜁_𝑗𝜁(𝐛)−^𝑗−1(𝐛₂)𝜁_𝑁−𝑗(𝐚₂)𝑗^𝑏1(𝑁+1−𝑗)^𝑎1+𝑅_𝑁(𝐚;𝐛). (2.1) The quantity 𝑅_𝑁(𝐚;𝐛)=∑^𝑁_𝑘=1𝜁_𝑁−𝑘(𝐛)𝜁_𝑘−1(𝑎₂,…,𝑎_𝑟)/𝑘^𝑎1=𝑅_𝑁(𝐛;𝐚) can be written as a sum of finite multiple zeta values, all of them having weight 𝑤=∑^𝑟_𝑖=1𝑎_𝑟+∑^𝑠_𝑖=1𝑏_𝑖 and depth 𝑑=𝑟+𝑠.

Remark 2.2. The quantity 𝑅_𝑁(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 ^𝑗−1_𝑘=1𝜁_𝑘(𝐛)𝜁_{𝑁−𝑘−1}(𝐚₂)(𝑁−𝑘)^𝑎1+^𝑁−𝑗_𝑘=1𝜁_𝑘(𝐚)𝜁_{𝑁−𝑘−1}(𝐛₂)(𝑁−𝑘)^𝑏1=𝜁_𝑗−1(𝐛)𝜁_𝑁−𝑗(𝐚₂)(𝑁+1−𝑗)^𝑎1+𝜁_𝑁−𝑗(𝐚)𝜁_𝑗−1(𝐛₂)𝑗^𝑏1−𝜁_{𝑁+1−𝑗}(𝐚)𝜁_𝑗𝜁(𝐛)+^𝑗−1(𝐛₂)𝜁_𝑛−𝑗(𝐚₂)𝑗^𝑏1(𝑁+1−𝑗)^𝑎1+𝑅_𝑁(𝐚;𝐛). (2.2)

Next we state an immediate asymptotic implication of the previous result.

Corollary 2.4. For 𝑁=2𝑛+1, 𝑗=𝑛+1, with 𝑎₁,𝑏₁∈ℕ⧵{1} and 𝑛→∞, we obtain the following result: lim_𝑛→∞(^𝑗_𝑘=1𝜁_𝑘−1(𝐛₂)𝜁_𝑁−𝑘(𝐚)𝑘^𝑏1+^{𝑁+1−𝑗}_𝑘=1𝜁_𝑘−1(𝐚₂)𝜁_𝑁−𝑘(𝐛)𝑘^𝑎1)=2𝜁(𝐚)𝜁(𝐛). (2.3)

3. The Proof of the Reciprocity Relation

In order to prove Theorem 2.1, we proceed as follows (using the beforehand introduced shorthand notations).

^𝑗_𝑘=1𝜁_𝑘−1(𝐛₂)𝜁_𝑁−𝑘(𝐚)𝑘^𝑏1=^𝑗_𝑘=1𝜁_𝑘−1(𝐛₂)𝑘^𝑏1(𝜁_𝑁−𝑗(𝐚)+^𝑁−𝑘_{ℓ=𝑁+1−𝑗}𝜁_ℓ−1(𝐚₂)ℓ^𝑎1)=𝜁_𝑁−𝑗(𝐚)𝜁_𝑗(𝐛)+^𝑗_𝑘=1𝜁_𝑘−1(𝐛₂)𝑘^{𝑏1𝑁−𝑘}_{ℓ=𝑁+1−𝑗}𝜁_ℓ−1(𝐚₂)ℓ^𝑎1. (3.1) After changing summations, we obtain

^𝑗_𝑘=1𝜁_𝑘−1(𝐛₂)𝜁_𝑁−𝑘(𝐚)𝑘^𝑏1=𝜁_𝑁−𝑗(𝐚)𝜁_𝑗(𝐛)+^𝑁−1_{ℓ=𝑁+1−𝑗}𝜁_ℓ−1(𝐚₂)ℓ^{𝑎1𝑁−ℓ}_𝑘=1𝜁_𝑘−1(𝐛₂)𝑘^𝑏1=𝜁_𝑁−𝑗(𝐚)𝜁_𝑗(𝐛)+^𝑁−1_{ℓ=𝑁+1−𝑗}𝜁_ℓ−1(𝐚₂)𝜁_𝑁−ℓ(𝐛)ℓ^𝑎1. (3.2) Using

𝜁_𝑁−𝑗(𝐚)𝜁_𝑗𝜁(𝐛)+^𝑁−𝑗(𝐚₂)𝜁_𝑗−1(𝐛)(𝑁+1−𝑗)^𝑎1=𝜁_𝑁−𝑗(𝐚)𝜁_𝑗𝜁(𝐛)+^𝑁−𝑗(𝐚₂)(𝑁+1−𝑗)^𝑎1(𝜁_𝑗𝜁(𝐛)−^𝑗−1(𝐛₂)𝑗^𝑏1)=𝜁_{𝑁+1−𝑗}(𝐚)𝜁_𝑗𝜁(𝐛)−^𝑁−𝑗(𝐚₂)𝜁_𝑗−1(𝐛₂)(𝑁+1−𝑗)^𝑎1𝑗^𝑏1, (3.3) and the fact that 𝜁₀(𝐛)=0 gives the intermediate result

^𝑗_𝑘=1𝜁_𝑘−1(𝐛₂)𝜁_𝑁−𝑘(𝐚)𝑘^𝑏1=𝜁_{𝑁+1−𝑗}(𝐚)𝜁_𝑗𝜁(𝐛)−^𝑁−𝑗(𝐚₂)𝜁_𝑗−1(𝐛₂)(𝑁+1−𝑗)^𝑎1𝑗^𝑏1+^𝑁_{ℓ=𝑁+2−𝑗}𝜁_ℓ−1(𝐚₂)𝜁_𝑁−ℓ(𝐛)ℓ^𝑎1, (3.4) Add the sum ∑^{𝑁+1−𝑗}_𝑘=1𝜁_𝑘−1(𝐚₂)𝜁_𝑁−𝑘(𝐛)/𝑘^𝑎1 to both sides of the equation above. This proves the first part of Theorem 2.1 and

𝑅_𝑁(𝐚;𝐛)=^𝑁_𝑘=1𝜁_𝑁−𝑘(𝐛)𝜁_𝑘−1(𝑎₂,…,𝑎_𝑟)𝑘^𝑎1. (3.5) For the evaluation of 𝑅_𝑁(𝐚;𝐛), we note that 𝑅₀(𝐚;𝐛)=0, and further

𝑅_𝑁(𝐚;𝐛)=^𝑁_𝑘=1(𝑅_𝑘(𝐚;𝐛)−𝑅_𝑘−1(𝐚;𝐛)). (3.6) Since 𝜁₀(𝐛)=0, we have

𝑅_𝑁(𝐚;𝐛)−𝑅_𝑁−1(𝐚;𝐛)=^𝑁_𝑘=1𝜁_𝑁−𝑘(𝐛)𝜁_𝑘−1(𝑎₂,…,𝑎_𝑟)𝑘^𝑎1−^𝑁−1_𝑘=1𝜁_{𝑁−1−𝑘}(𝐛)𝜁_𝑘−1(𝑎₂,…,𝑎_𝑟)𝑘^𝑎1=^𝑁−1_𝑘=1(𝜁_𝑁−𝑘(𝐛)−𝜁_{𝑁−1−𝑘}(𝐛)𝜁_𝑘−1(𝑎₂,…,𝑎_𝑟)𝑘^𝑎1=^𝑁−1_𝑘=1𝜁_{𝑁−1−𝑘}(𝑏₂,…,𝑏_𝑠)𝜁_𝑘−1(𝑎₂,…,𝑎_𝑟)(𝑁−𝑘)^𝑏1𝑘^𝑎1. (3.7) 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],

1𝑘^𝑎(𝑁−𝑘)^𝑏=^𝑎_𝑖=1(^{𝑖+𝑏−2}_𝑏−1)𝑁^{𝑖+𝑏−1}𝑘^{𝑎+1−𝑖}+^𝑏_𝑖=1(^{𝑖+𝑎−2}_𝑎−1)𝑁^{𝑖+𝑎−1}(𝑁−𝑘)^{𝑏+1−𝑖}, (3.8) and obtain

^𝑁−1_𝑘=1𝜁_𝑘−1(𝑎₂,…,𝑎_𝑟)𝜁_{𝑁−1−𝑘}(𝑏₂,…,𝑏_𝑠)(𝑁−𝑘)^𝑏1𝑘^𝑎1=^𝑎1_𝑖=1^𝑁−1_𝑘=1(^𝑖+𝑏1_𝑏⁻²₁₋₁)𝜁_𝑘−1(𝑎₂,…,𝑎_𝑟)𝜁_{𝑁−1−𝑘}(𝑏₂,…,𝑏_𝑠)𝑁^{𝑖+𝑏1−1}𝑘^{𝑎1+1−𝑖}+^𝑏1_𝑖=1^𝑁−1_𝑘=1(^𝑖+𝑎1_𝑎⁻²₁₋₁)𝜁_𝑘−1(𝑎₂,…,𝑎_𝑟)𝜁_{𝑁−1−𝑘}(𝑏₂,…,𝑏_𝑠)𝑁^{𝑖+𝑎1−1}(𝑁−𝑘)^{𝑏1+1−𝑖}. (3.9) Consequently, by summing up according to (3.6), we get the following recurrence relation for 𝑅_𝑁(𝐚;𝐛):

𝑅_𝑁(𝐚;𝐛)=^𝑎1𝑁_{𝑖=1𝑛1=1}(^𝑖+𝑏1_𝑏⁻²₁₋₁)𝑛^𝑖+𝑏1₁⁻¹𝑅_𝑛1−1(𝑎₁+1−𝑖,𝑎₂,…,𝑎_𝑟;𝑏₂,…,𝑏_𝑠)+^𝑏1𝑁_{𝑖=1𝑛1=1}(^𝑖+𝑎1_𝑎⁻²₁₋₁)𝑛^𝑖+𝑎1₁⁻¹𝑅_𝑛1−1(𝑎₂,…,𝑎_𝑟;𝑏₁+1−𝑖,𝑏₂,…,𝑏_𝑠). (3.10) This recurrence relation suggests that there exists an evaluation of 𝑅_𝑁(𝐚;𝐛) into sums of finite multiple zeta values, all of them having weight 𝑤=∑^𝑟_𝑖=1𝑎_𝑟+∑^𝑠_𝑖=1𝑏_𝑖 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 𝑅_𝑁(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 𝑅_𝑁(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

Let 𝒜 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⋯𝑥_𝑛, 𝐯=𝑥_𝑛+1⋯𝑥_𝑛+𝑚, 𝑥_𝑖∈𝒜 for 1≤𝑖≤𝑛+𝑚, be defined as follows:

𝐰⊔⊔𝐯∶=𝑥_𝜎(1)𝑥_𝜎(2)⋯𝑥_{𝜎(𝑛+𝑚)}, (3.11) where the sum runs over all (_𝑛^𝑛+𝑚) permutations 𝜎∈𝔖_𝑛+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,…,𝑥_𝑛 and 𝑥_𝑛+1,…,𝑥_𝑛+𝑚 are preserved. Equivalently, the shuffle product of two words 𝐰,𝐯∈𝒜^∗ can be defined in a recursive way:

∀𝐰∈𝒜^∗,𝜀⊔⊔𝐰=𝐰⊔⊔𝜀=𝐰,∀𝑥,𝑦∈𝒜,𝐰,𝐯∈𝒜^∗,𝑥𝐰⊔⊔𝑦𝐯=𝑥(𝐰⊔⊔𝑦𝐯)+𝑦(𝑥𝐰⊔⊔𝐯). (3.12) 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 alphabet 𝒜={𝜔₀,𝜔₁} and the arising shuffle algebra (ℚ⟨𝒜⟩,⊔⊔). Hoang and Petitot [16] derived a shuffle identity for words 𝐴=𝜔₀^𝑎−1𝜔₁, 𝐵=𝜔₀^𝑏−1𝜔₁, which is stated below.

Lemma 3.1. For 𝑎,𝑏∈ℕ, let 𝐴=𝜔₀^𝑎−1𝜔₁ and 𝐵=𝜔₀^𝑏−1𝜔₁ be words on the non-commutative alphabet 𝒜={𝜔₀,𝜔₁}. 𝐴⊔⊔𝐵=^𝑎−1_𝑖=0𝜔𝑏−1+𝑖𝑏−1^{0𝑏−1+𝑖}𝜔₁𝜔₀^{𝑎−1−𝑖}𝜔₁+^𝑏1−1_𝑖=0𝜔𝑎−1+𝑖𝑎−1^{0𝑎−1+𝑖}𝜔₁𝜔₀^{𝑏−1−𝑖}𝜔₁. (3.13)

We will use a slight extension of this identity, which easily follows from the recursive definition of the shuffle product.

Lemma 3.2. For 𝑟,𝑠≥1 and 𝑎_𝑖,𝑏_𝑗∈ℕ, 1≤𝑖≤𝑟, 1≤𝑗≤𝑠, let 𝐴∶=𝜔^𝑎1₀⁻¹𝜔₁⋯𝜔^𝑎𝑟₀⁻¹𝜔₁ and 𝐵∶=𝜔^𝑏1₀⁻¹𝜔₁⋯𝜔^𝑏𝑠₀⁻¹𝜔₁ be words on the non-commutative alphabet 𝒜={𝜔₀,𝜔₁}. 𝐴⊔⊔𝐵=^𝑎1_𝑖=1𝑖+𝑏₁𝑏−2₁𝜔−1^{𝑖+𝑏10−2}𝜔₁(𝐴^′_𝑖⊔⊔𝐵₂)+^𝑏1_𝑖=1𝑖+𝑎₁𝑎−2₁𝜔−1^{𝑖+𝑎10−2}𝜔₁(𝐴₂⊔⊔𝐵^′_𝑖), (3.14) with 𝐴^′_𝑖∶=𝜔^𝑎1₀^−𝑖𝜔₁𝜔^𝑎2₀⁻¹𝜔₁⋯𝜔^𝑎𝑟₀⁻¹𝜔₁, 𝐵^′_𝑖∶=𝜔^𝑏1₀^−𝑖𝜔₁𝜔^𝑏2₀⁻¹𝜔₁⋯𝜔^𝑏𝑠₀⁻¹𝜔₁ and further 𝐴₂∶=𝜔^𝑎2₀𝜔₁⋯𝜔^𝑎𝑟₀⁻¹𝜔₁, 𝐵₂∶=𝜔^𝑏2₀⁻¹𝜔₁⋯𝜔^𝑏𝑠₀⁻¹𝜔₁.

Note that the partial fraction decomposition (3.8) of 1/𝑘^𝑎(𝑁−𝑘)^𝑏 somewhat mimics the shuffle identity for words 𝐴=𝜔₀^𝑎−1𝜔₁, 𝐵=𝜔₀^𝑏−1𝜔₁, derived by Hoang and Petitot [16].

3.2. The Shuffle Algebra and Finite Multiple Zeta Values

Let 𝐚 denote an arbitrary 𝑟-tuple of positive integers 𝐚=(𝑎₁,…,𝑎_𝑟) with 𝑎_𝑖∈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∶=𝜔^𝑎1₀⁻¹𝜔₁𝜔^𝑎2₀⁻¹𝜔₁⋯𝜔^𝑎𝑟₀⁻¹𝜔₁. Let 𝒜^∗ denote the set of all words over the alphabet 𝒜. Let (𝑍_𝑁)_𝑁≥1 denote a family of linear maps from the algebra ℚ⟨𝒜⟩ to the rational numbers, 𝑍_𝑁∶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∶=𝜔^𝑎1₀⁻¹𝜔₁𝜔^𝑎2₀⁻¹𝜔₁⋯𝜔^𝑎𝑟₀⁻¹𝜔₁∈𝒜^∗, with 𝑟,𝑁≥1, we define

𝑍_𝑁(𝐴)=𝑍_𝑁(𝜔^𝑎1₀⁻¹𝜔₁𝜔^𝑎2₀⁻¹𝜔₁⋯𝜔^𝑎𝑟₀⁻¹𝜔₁)=𝜁_𝑁(𝑎₁,…,𝑎_𝑟)=𝜁_𝑁(𝐚). (3.15) Moreover, we additionally define 𝑍₀(𝐴)=𝜁₀(𝐚)=0 for all 𝐴∈𝒜^∗, and 𝑍_𝑁(𝜀)=1 for all 𝑁≥1. The family of maps (𝑍_𝑁)_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 𝑍_𝑁 in a recursive way. Let 𝐴∶=𝜔^𝑎1₀⁻¹𝜔₁𝜔^𝑎2₀⁻¹𝜔₁⋯𝜔^𝑎𝑟₀⁻¹𝜔₁∈𝒜^∗, with 𝑟≥1 and 𝑎₁,…,𝑎_𝑟≥1.

𝑍_𝑁(𝐴)=𝜁_𝑁(𝐚)=^𝑁_𝑛1=11𝑛^𝑎1₁𝜁_𝑛1−1(𝑎₂,…,𝑎_𝑟)=^𝑁_𝑛1=11𝑛^𝑎1₁𝑍_𝑛1−1(𝜔^𝑎2₀⁻¹𝜔₁⋯𝜔^𝑎𝑟₀⁻¹𝜔₁). (3.16) We need the following result.

Lemma 3.3. For 𝑟,𝑠≥1 and 𝑎_𝑖,𝑏_𝑗∈ℕ, 1≤𝑖≤𝑟, 1≤𝑗≤𝑠, let 𝐴∶=𝜔^𝑎1₀⁻¹𝜔₁⋯𝜔^𝑎𝑟₀⁻¹𝜔₁ and 𝐵∶=𝜔^𝑏1₀⁻¹𝜔₁⋯𝜔^𝑏𝑠₀⁻¹𝜔₁ be words on the non-commutative alphabet 𝒜={𝜔₀,𝜔₁}. Then, 𝑍_𝑁(𝐴⊔⊔𝐵)=^𝑎1𝑁_{𝑖=1𝑛1=1}(^𝑖+𝑏1_𝑏⁻²₁₋₁)𝑛^𝑖+𝑏1₁⁻¹𝑍_𝑛1−1(𝐴^′_𝑖⊔⊔𝐵₂)+^𝑏1𝑁_{𝑖=1𝑛1=1}(^𝑖+𝑎1_𝑎⁻²₁₋₁)𝑛^𝑖+𝑎1₁⁻¹𝑍_𝑛1−1(𝐴₂⊔⊔𝐵^′_𝑖). (3.17) The depths 𝑑=𝑟+𝑠 and the weights 𝑤=∑^𝑟_𝑖=1𝑎_𝑖+∑^𝑠_𝑘=1𝑏_𝑘 of the arising finite multiple zeta values are all the same.

Proof. By linearity of the maps 𝑍_𝑁 and Lemma 3.2, we get first 𝑍_𝑁(𝐴⊔⊔𝐵)=^𝑎1_𝑖=1𝑖+𝑏₁𝑏−2₁𝑍−1^𝑁(𝜔^𝑖+𝑏1₀⁻²𝜔₁(𝐴^′_𝑖⊔⊔𝐵₂+))_𝑏1𝑖=1𝑖+𝑎₁𝑎−2₁𝑍−1^𝑁(𝜔^𝑖+𝑎1₀⁻²𝜔₁(𝐴₂⊔⊔𝐵^′_𝑖)), (3.18) using the notations of Lemma 3.2 for 𝐴^′_𝑖,𝐵^′_𝑖,𝐴₂,𝐵₂. By definition of the shuffle product, 𝐴^′_𝑖⊔⊔𝐵₂∈ℚ⟨𝒜⟩ and 𝐴₂⊔⊔𝐵^′_𝑖∈ℚ⟨𝒜⟩ are rational linear combinations of words over 𝒜. Let {𝐴^′_𝑖⊔⊔𝐵₂} and {𝐴₂⊔⊔𝐵^′_𝑖} denote the sets of different words generated by the shuffles 𝐴^′_𝑖⊔⊔𝐵₂ and 𝐴₂⊔⊔𝐵^′_𝑖. Using the set notation, we write 𝐴^′_𝑖⊔⊔𝐵₂=_{𝐰∈{𝐴′𝑖⊔⊔𝐵2}}𝑞_𝐰𝐰,𝐴₂⊔⊔𝐵^′_𝑖=_{𝐰∈{𝐴2⊔⊔𝐵′𝑖}}𝑞_𝐰𝐰, (3.19) with 𝑞_𝐰∈ℚ and 𝐰∈𝒜^∗, which helps to obtain a simple presentation of the subsequent calculations. We have 𝑍_𝑁(𝐴⊔⊔𝐵)=^𝑎1_𝑖=1𝑖+𝑏₁𝑏−2₁𝑍−1^𝑁(𝜔^𝑖+𝑏1₀⁻²𝜔_{1𝐰∈{𝐴′𝑖⊔⊔𝐵2}}𝑞_𝐰+𝐰)_𝑏1𝑖=1𝑖+𝑎₁𝑎−2₁𝑍−1^𝑁(𝜔^𝑖+𝑎1₀⁻²𝜔_{1𝐰∈{𝐴2⊔⊔𝐵′𝑖}}𝑞_𝐰𝐰). (3.20) Using the linearity of the maps 𝑍_𝑁 and the fact that we can recursively describe their images, we get further 𝑍_𝑁(𝐴⊔⊔𝐵)=^𝑎1_𝑖=1𝑖+𝑏₁𝑏−2₁−1_𝐰∈{𝐴^′_{𝑖⊔⊔𝐵2}}𝑞_𝐰^𝑁_𝑛1=11𝑛^𝑖+𝑏1₁⁻¹𝑍_𝑛1−1+(𝐰)_𝑏1𝑖=1𝑖+𝑎₁𝑎−2₁−1_{𝐰∈{𝐴2⊔⊔𝐵}^′_𝑖}𝑞_𝐰^𝑁_𝑛1=11𝑛^𝑖+𝑎1₁⁻¹𝑍_𝑛1−1(𝐰). (3.21) Interchanging the latter summations gives the stated result. 𝑍_𝑁(𝐴⊔⊔𝐵)=^𝑎1_𝑖=1𝑖+𝑏₁𝑏−2₁−1^𝑁𝑛₁⁼¹1𝑛^𝑖+𝑏1₁⁻¹_{𝐰∈{𝐴′𝑖⊔⊔𝐵2}}𝑞_𝐰𝑍_𝑛1−1+(𝐰)_𝑏1𝑖=1𝑖+𝑎₁𝑎−2₁−1^𝑁𝑛₁⁼¹1𝑛^𝑖+𝑎1₁⁻¹_{𝐰∈{𝐴2⊔⊔𝐵′𝑖}}𝑞_𝐰𝑍_𝑛1−1=(𝐰)_{𝑎1𝑁𝑖=1𝑛1=1}(^𝑖+𝑏1_𝑏⁻²₁₋₁)𝑛^𝑖+𝑏1₁⁻¹𝑍_𝑛1−1(𝐴^′_𝑖⊔⊔𝐵₂)+^𝑏1𝑁_{𝑖=1𝑛1=1}(^𝑖+𝑎1_𝑎⁻²₁₋₁)𝑛^𝑖+𝑎1₁⁻¹𝑍_𝑛1−1(𝐴₂⊔⊔𝐵^′_𝑖). (3.22) 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 of 𝑅_𝑁(𝐚;𝐛).

Proposition 3.4. For arbitrary 𝑟,𝑠≥1, let 𝐚 and 𝐛 be given by 𝐚=(𝑎₁,…,𝑎_𝑟) and 𝐛=(𝑏₁,…,𝑏_𝑠), with 𝑎_𝑖,𝑏_𝑗∈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∶=𝜔^𝑎1₀⁻¹𝜔₁⋯𝜔^𝑎𝑟₀⁻¹𝜔₁ and 𝐵∶=𝜔^𝑏1₀⁻¹𝜔₁⋯𝜔^𝑏𝑠₀⁻¹𝜔₁. Then, for arbitrary 𝑁≥1, 𝑅_𝑁(𝐚;𝐛)=𝑍_𝑁(𝐴⊔⊔𝐵). (3.23)

Proof. We use induction with respect to 𝑑=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 𝑅_𝑁(𝐚;𝐛), we get 𝑅_𝑁(𝐚;𝐛)=^𝑎1𝑁_{𝑖=1𝑛1=1}(^𝑖+𝑏1_𝑏⁻²₁₋₁)𝑛^𝑖+𝑏1₁⁻¹𝑅_𝑛1−1(𝑎₁+1−𝑖,𝑎₂,…,𝑎_𝑟;𝑏₂,…,𝑏_𝑠)+^𝑏1𝑁_{𝑖=1𝑛1=1}(^𝑖+𝑎1_𝑎⁻²₁₋₁)𝑛^𝑖+𝑎1₁⁻¹𝑅_𝑛1−1(𝑎₂,…,𝑎_𝑟;𝑏₁+1−𝑖,𝑏₂,…,𝑏_𝑠). (3.24) The induction hypothesis states that 𝑅_𝑁(𝐚;𝐛)=𝑍_𝑁(𝐴⊔⊔𝐵) for arbitrary 𝑟,𝑠≥1 such that 𝑟+𝑠<𝑑 and arbitrary 𝑁≥1. By the recurrence relation for 𝑅_𝑁(𝐚;𝐛), we can reduce 𝑅_𝑁(𝐚;𝐛) to values of the types 𝑅_𝑛1−1(𝑎₁+1 – 𝑖,𝑎₂,…,𝑎_𝑟;𝑏₂,…,𝑏_𝑠) and 𝑅_𝑛1−1(𝑎₂,…,𝑎_𝑟;𝑏₁+1 – 𝑖,𝑏₂,…,𝑏_𝑠), which are of depth smaller than 𝑑=𝑟+𝑠. Hence, we get by the induction hypothesis 𝑅_𝑁(𝐚;𝐛)=^𝑎1𝑁_{𝑖=1𝑛1=1}(^𝑖+𝑏1_𝑏⁻²₁₋₁)𝑛^𝑖+𝑏1₁⁻¹𝑍_𝑛1−1(𝐴^′_𝑖⊔⊔𝐵₂)+^𝑏1𝑁_{𝑖=1𝑛1=1}(^𝑖+𝑎1_𝑎⁻²₁₋₁)𝑛^𝑖+𝑎1₁⁻¹𝑍_𝑛1−1(𝐴₂⊔⊔𝐵^′_𝑖). (3.25) By Lemma 3.3, using the notations for 𝐴^′_𝑖,𝐵^′_𝑖,𝐴₂,𝐵₂ of Lemma 3.2, we get ^𝑎1𝑁_{𝑖=1𝑛1=1}(^𝑖+𝑏1_𝑏⁻²₁₋₁)𝑛^𝑖+𝑏1₁⁻¹𝑍_𝑛1−1(𝐴^′_𝑖⊔⊔𝐵₂)+^𝑏1𝑁_{𝑖=1𝑛1=1}(^𝑖+𝑎1_𝑎⁻²₁₋₁)𝑛^𝑖+𝑎1₁⁻¹𝑍_𝑛1−1(𝐴₂⊔⊔𝐵^′_𝑖)=𝑍_𝑁(𝐴⊔⊔𝐵). (3.26) Consequently, 𝑅_𝑁(𝐚;𝐛)=𝑍_𝑁(𝐴⊔⊔𝐵). (3.27) This proves the stated result for 𝑅_𝑁(𝐚;𝐛) and the corresponding statement of Theorem 2.1.

Corollary 2.3 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 𝑅_𝑁(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 given 𝐚=(𝑎₁,…,𝑎_𝑟) and 𝐛=(𝑏₁,…,𝑏_𝑠), one may define the shuffle product 𝜁_𝑁(𝐚⊔⊔𝐛) in terms of the images of the maps 𝑍_𝑁 using the words 𝐴=𝐴(𝐚) and 𝐵=𝐴(𝐛) associated to 𝐚 and 𝐛 by 𝐴∶=𝜔^𝑎1₀⁻¹𝜔₁⋯𝜔^𝑎𝑟₀⁻¹𝜔₁ and 𝐵∶=𝜔^𝑏1₀⁻¹𝜔₁⋯𝜔^𝑏𝑠₀⁻¹𝜔₁,

𝜁_𝑁(𝐚⊔⊔𝐛)∶=𝑍_𝑁(𝐴⊔⊔𝐵). (4.1) 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].

Let Li_𝐚(𝑧) denote the (multiple) polylogarithm function with parameters 𝑎₁,…,𝑎_𝑟, defined by

Li_𝐚(𝑧)=Li_{𝑎1,…,𝑎𝑟}(𝑧)=_{𝑛1>𝑛2>⋯>𝑛𝑟≥1}𝑧^𝑛1𝑛^𝑎1₁𝑛^𝑎2₂⋯𝑛^𝑎𝑟_𝑟. (4.2) The value 𝑅_𝑁(𝐚;𝐛) can be obtained by coefficient extraction in the following way:

𝑅_𝑁(𝐚;𝐛)=^𝑁_𝑘=1𝜁_𝑘−1(𝑎₂,…,𝑎_𝑟)𝜁_𝑁−𝑘(𝑏₁,…,𝑏_𝑠)𝑘^𝑎1=[𝑧^𝑁]Li_𝐚(𝑧)Li_𝐛(𝑧)1−𝑧. (4.3) On the other hand, by the finite shuffle identity (3.23) for 𝑅_𝑁(𝐚;𝐛), one can show the following representation:

𝑅_𝑁(𝐚;𝐛)=[𝑧^𝑁]Li_{𝐚⊔⊔𝐛}(𝑧).1−𝑧 (4.4) Here the shuffle product for polylogarithm functions Li_{𝐚⊔⊔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 𝑅_𝑁(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 Corollary 2.4. For 𝑁=2𝑛+1 and 𝑗=𝑛+1 and 𝑛→∞, we have

lim_𝑛→∞𝜁_𝑗(𝐚)𝜁_{𝑁+1−𝑗}(𝐛)=lim_𝑛→∞𝜁_𝑛+1(𝐚)𝜁_𝑛+1(𝐛)=𝜁(𝐚)𝜁(𝐛),lim_𝑛→∞𝜁_𝑛(𝑏₂,…,𝑏_𝑠)𝜁_𝑛(𝑎₂,…,𝑎_𝑟)(𝑛+1)^𝑎1+𝑏1=0,lim_𝑛→∞𝑅_𝑁(𝐚;𝐛)=lim_𝑛→∞𝜁_2𝑛+1(𝐚⊔⊔𝐛)=𝜁(𝐚)𝜁(𝐛), (4.5) and the stated result follows.

5. The Reciprocity Relation for Weighted Multiple Zeta Values

Results similar to Theorem 2.1 and Corollary 2.4 can be obtained for products of weighted finite multiple zeta values, 𝜁_𝑁(𝑎₁,𝑎₂,…,𝑎_𝑟;𝜎₁,…,𝜎_𝑟), 𝜎_𝑖∈ℝ⧵{0} for 1≤𝑖≤𝑟, defined as follows:

𝜁_𝑁(𝐚,𝝈)=𝜁_𝑁(𝑎₁,𝑎₂,…,𝑎_𝑟;𝜎₁,…,𝜎_𝑟)=_{𝑁≥𝑛1>𝑛2>⋯>𝑛𝑟≥1}1∏^𝑟_𝑖=1𝑛^𝑎𝑖_𝑖𝜎^𝑛𝑖_𝑖. (5.1) Of particular interest are the cases 𝜎_𝑖∈{±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,…,𝑎_𝑟), 𝝈₂=(𝜎₂,…,𝜎_𝑟), 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 values 𝜁_𝑁(𝐚,𝝈) and 𝜁_𝑁(𝐛,𝝉) with weights 𝝈 and 𝝉 satisfy the following reciprocity relation: ^𝑗_𝑘=1𝜁_𝑘−1(𝐛₂,𝝉₂)𝜁_𝑁−𝑘(𝐚,𝝈)𝑘^𝑏1𝜏^𝑘₁+^{𝑁+1−𝑗}_𝑘=1𝜁_𝑘−1(𝐚₂,𝝈₂)𝜁_𝑁−𝑘(𝐛,𝝉)𝑘^𝑎1𝜎^𝑘₁=𝜁_{𝑁+1−𝑗}(𝐚,𝝈)𝜁_𝑗𝜁(𝐛,𝝉)−^𝑗−1(𝐛₂,𝝉₂)𝜁_𝑁−𝑗(𝐚₂,𝝈₂)𝜏^𝑗₁𝑗^𝑏1𝜎₁^{𝑁+1−𝑗}(𝑁+1−𝑗)^𝑎1+𝑅_𝑁(𝐚,𝝈;𝐛,𝝉). (5.2) Here 𝑅_𝑁(𝐚,𝝈;𝐛,𝝉)=∑^𝑁_𝑘=1𝜁_𝑁−𝑘(𝐛,𝝉)𝜁_𝑘−1(𝐚₂,𝝈₂)/𝜎^𝑘₁𝑘^𝑎1=𝑅_𝑁(𝐛,𝝉;𝐚,𝝈) satisfies an analogue of the shuffle identity with respect to the weights 𝝈 and 𝝉.

The proof of Theorem 2.1 can easily be adapted to the weighted case. Hence, we only elaborate on the main new difficulty, namely, the evaluation of the quantity

𝑅_𝑁(𝐚,𝝈;𝐛,𝝉)=^𝑁_𝑘=1𝜁_𝑁−𝑘(𝐛,𝝉)𝜁_𝑘−1(𝐚₂;𝝈₂)𝜎^𝑘₁𝑘^𝑎1. (5.3) Proceeding as before, that is, taking differences and using partial fraction decomposition, we obtain the recurrence relation

𝑅_𝑁(𝐚,𝝈;𝐛,𝝉)=^𝑎1𝑁_{𝑖=1𝑛1=1}(^𝑖+𝑏1_𝑏⁻²₁₋₁)𝑛^𝑖+𝑏1₁⁻¹𝜏^𝑛1₁𝑅_𝑛1−1(𝑎₁+1−𝑖,𝐚₂,𝜏₁𝜎₁,𝝈₂;𝐛₂,𝝉₂)+^𝑏1𝑁_{𝑖=1𝑛1=1}(^𝑖+𝑎1_𝑎⁻²₁₋₁)𝑛^𝑖+𝑎1₁⁻¹𝜎^𝑛1₁𝑅_𝑛1−1(𝐚₂,𝝈₂;𝑏₁+1−𝑖,𝐛₂,𝜎₁𝜏₁,𝝉₂). (5.4) Consequently, the value 𝑅_𝑁(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.

Acknowledgments

The authors thank the referees for very valuable comments improving the presentation of this work, clearifying several points, and for providing additional references. The first author was supported by the Austrian Science Foundation FWF, Grant S9608-N13. The second author was supported by the South African Science Foundation NRF, Grant 2053748.