IJCTInternational Journal of Combinatorics1687-91711687-9163Hindawi Publishing Corporation15362110.1155/2010/153621153621Research ArticleOn a Reciprocity Law for Finite Multiple Zeta ValuesKubaMarkus1ProdingerHelmut2PanholzerAlois1Institut für Diskrete Mathematik und GeometrieTechnische Universität WienWiedner Hauptstr8-10/1041040 WienAustriatuwien.ac.at2Department of MathematicsUniversity of Stellenbosch7602 StellenboschSouth Africasun.ac.za201011022010201011102009140120102010Copyright © 2010This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 Hn=k=1n1/k denote the nth harmonic number and Hn(s)=k=1n1/ks the nth harmonic number of order s, with n,s and Hn=Hn(1). Kirschenhofer and Prodinger  analyzed the variance of the number of comparisons of the famous QUICKSELECT algorithm, also known as FIND  and derived a reciprocity relation for (first-order) harmonic numbers. Subsequently, the reciprocity relation of  was generalized , where the following identity was derived:

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,

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 , and its finite counterpart are defined as follows:

ζ(a)=ζ(a1,,ar):=n1>n2>>nr11n1a1n2a2nrar,ζN(a)=ζN(a1,,ar):=Nn1>n2>>nr11n1a1n2a2nrar. 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 , 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

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  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,,ar), a2=(a2,,ar), and b=(b1,,bs), b2=(b2,,bs), respectively, with r,s and ai,bk for 1ir and 1ks.

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 ζN(a)=ζN(a1,,ar), ζN(b)=ζN(b1,,bs) satisfy the following reciprocity relation. 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 quantity RN(a;b) satisfies a shuffle identity resembling the ordinary shuffle identity for multiple zeta values ζ(a)ζ(b)=ζ(ab); see Sections 3.1, 3.2 and Proposition 3.4 for details.

Corollary 2.3.

We obtain the complementary identity 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.

For N=2n+1, j=n+1, with a1,b1{1} and n, we obtain the following result: 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 Theorem 2.1, we proceed as follows (using the beforehand introduced shorthand notations).

k=1jζk-1(b2)ζN-k(a)kb1=k=1jζk-1(b2)kb1(ζN-j(a)+=N+1-jN-kζ-1(a2)a1)=ζN-j(a)ζj(b)+k=1jζk-1(b2)kb1=N+1-jN-kζ-1(a2)a1. After changing summations, we obtain

k=1jζk-1(b2)ζN-k(a)kb1=ζN-j(a)ζj(b)+=N+1-jN-1ζ-1(a2)a1k=1N-ζk-1(b2)kb1=ζN-j(a)ζj(b)+=N+1-jN-1ζ-1(a2)ζN-(b)a1. Using

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

k=1jζk-1(b2)ζN-k(a)kb1=ζN+1-j(a)ζj(b)-ζN-j(a2)ζj-1(b2)(N+1-j)a1jb1+=N+2-jNζ-1(a2)ζN-(b)a1, 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

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

RN(a;b)=k=1N(Rk(a;b)-Rk-1(a;b)). Since ζ0(b)=0, we have

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 .), which appears already in ,

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

k=1N-1ζk-1(a2,,ar)ζN-1-k(b2,,bs)(N-k)b1ka1=i=1a1k=1N-1(i+b1-2b1-1)ζk-1(a2,,ar)ζN-1-k(b2,,bs)Ni+b1-1ka1+1-i+i=1b1k=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):

RN(a;b)=i=1a1n1=1N(i+b1-2b1-1)n1i+b1-1Rn1-1(a1+1-i,a2,,ar;b2,,bs)+i=1b1n1=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 . 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

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 is a rational linear combination of words on 𝒜. The set of all such polynomials is denoted by 𝒜. Let the shuffle product of two words w,v𝒜*, with w=x1xn, v=xn+1xn+m, xi𝒜 for 1in+m, be defined as follows:

wv:=xσ(1)xσ(2)xσ(n+m), where the sum runs over all (n+mn) permutations σ𝔖n+m which satisfy σ-1(j)<σ-1(k) for all 1j<kn and n+1j<kn+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,v𝒜* can be defined in a recursive way:

w𝒜*,ɛw=wɛ=w,x,y𝒜,w,v𝒜*,xwyv=x(wyv)+y(xwv). The shuffle product extends to 𝒜 by linearity. Note that the set 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 .

In the following, we will restrict ourselves to the non-commutative alphabet 𝒜={ω0,ω1} and the arising shuffle algebra (𝒜,). Hoang and Petitot  derived a shuffle identity for words A=ω0a-1ω1, B=ω0b-1ω1, which is stated below.

Lemma 3.1.

For a,b, let A=ω0a-1ω1 and B=ω0b-1ω1 be words on the non-commutative alphabet 𝒜={ω0,ω1}. AB=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.

For r,s1 and ai,bj, 1ir, 1js, let A:=ω0a1-1ω1ω0ar-1ω1 and B:=ω0b1-1ω1ω0bs-1ω1 be words on the non-commutative alphabet 𝒜={ω0,ω1}. AB=i=1a1(i+b1-2b1-1)ω0i+b1-2ω1(AiB2)+i=1b1(i+a1-2a1-1)ω0i+a1-2ω1(A2Bi), with Ai:=ω0a1-iω1ω0a2-1ω1ω0ar-1ω1, Bi:=ω0b1-iω1ω0b2-1ω1ω0bs-1ω1 and further A2:=ω0a2ω1ω0ar-1ω1, B2:=ω0b2-1ω1ω0bs-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 .

3.2. The Shuffle Algebra and Finite Multiple Zeta Values

Let a denote an arbitrary r-tuple of positive integers a=(a1,,ar) with ai for 1ir and r1. 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:=ω0a1-1ω1ω0a2-1ω1ω0ar-1ω1. Let 𝒜* denote the set of all words over the alphabet 𝒜. Let (ZN)N1 denote a family of linear maps from the algebra 𝒜 to the rational numbers, ZN:𝒜, mapping words over the non-commutative alphabet 𝒜={ω0,ω1} to finite multiple zeta values in the following way. For words A:=ω0a1-1ω1ω0a2-1ω1ω0ar-1ω1𝒜*, with r,N1, we define

ZN(A)=ZN(ω0a1-1ω1ω0a2-1ω1ω0ar-1ω1)=ζN(a1,,ar)=ζN(a). Moreover, we additionally define Z0(A)=ζ0(a)=0 for all A𝒜*, and ZN(ɛ)=1 for all N1. The family of maps (ZN)N1 linearly extend to 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:=ω0a1-1ω1ω0a2-1ω1ω0ar-1ω1𝒜*, with r1 and a1,,ar1.

ZN(A)=ζN(a)=n1=1N1n1a1ζn1-1(a2,,ar)=n1=1N1n1a1Zn1-1(ω0a2-1ω1ω0ar-1ω1). We need the following result.

Lemma 3.3.

For r,s1 and ai,bj, 1ir, 1js, let A:=ω0a1-1ω1ω0ar-1ω1 and B:=ω0b1-1ω1ω0bs-1ω1 be words on the non-commutative alphabet 𝒜={ω0,ω1}. Then, ZN(AB)=i=1a1n1=1N(i+b1-2b1-1)n1i+b1-1Zn1-1(AiB2)+i=1b1n1=1N(i+a1-2a1-1)n1i+a1-1Zn1-1(A2Bi). 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 maps ZN and Lemma 3.2, we get first ZN(AB)=i=1a1(i+b1-2b1-1)ZN(ω0i+b1-2ω1(AiB2))+i=1b1(i+a1-2a1-1)ZN(ω0i+a1-2ω1(A2Bi)), using the notations of Lemma 3.2 for Ai,Bi,A2,B2. By definition of the shuffle product, AiB2𝒜 and A2Bi𝒜 are rational linear combinations of words over 𝒜. Let {AiB2} and {A2Bi} denote the sets of different words generated by the shuffles AiB2 and A2Bi. Using the set notation, we write AiB2=w{AiB2}qww,A2Bi=w{A2Bi}qww, with qw and w𝒜*, which helps to obtain a simple presentation of the subsequent calculations. We have ZN(AB)=i=1a1(i+b1-2b1-1)ZN(ω0i+b1-2ω1w{AiB2}qww)+i=1b1(i+a1-2a1-1)ZN(ω0i+a1-2ω1w{A2Bi}qww). Using the linearity of the maps ZN and the fact that we can recursively describe their images, we get further ZN(AB)=i=1a1(i+b1-2b1-1)w{AiB2}qwn1=1N1n1i+b1-1Zn1-1(w)+i=1b1(i+a1-2a1-1)w{A2Bi}qwn1=1N1n1i+a1-1Zn1-1(w). Interchanging the latter summations gives the stated result. ZN(AB)=i=1a1(i+b1-2b1-1)n1=1N1n1i+b1-1w{AiB2}qwZn1-1(w)+i=1b1(i+a1-2a1-1)n1=1N1n1i+a1-1w{A2Bi}qwZn1-1(w)=i=1a1n1=1N(i+b1-2b1-1)n1i+b1-1Zn1-1(AiB2)+i=1b1n1=1N(i+a1-2a1-1)n1i+a1-1Zn1-1(A2Bi). 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 RN(a;b).

Proposition 3.4.

For arbitrary r,s1, let a and b be given by a=(a1,,ar) and b=(b1,,bs), with ai,bj for 1ir, 1js. Let A=A(a) and B=A(b) denote the words associated to a and b by A:=ω0a1-1ω1ω0ar-1ω1 and B:=ω0b1-1ω1ω0bs-1ω1. Then, for arbitrary N1, RN(a;b)=ZN(AB).

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 . Now assume that d3. Using the recurrence relation (3.10) for RN(a;b), we get RN(a;b)=i=1a1n1=1N(i+b1-2b1-1)n1i+b1-1Rn1-1(a1+1-i,a2,,ar;b2,,bs)+i=1b1n1=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(AB) for arbitrary r,s1 such that r+s<d and arbitrary N1. By the recurrence relation for RN(a;b), we can reduce RN(a;b) to values of the types Rn1-1(a1+1i,a2,,ar;b2,,bs) and Rn1-1(a2,,ar;b1+1i,b2,,bs), which are of depth smaller than d=r+s. Hence, we get by the induction hypothesis RN(a;b)=i=1a1n1=1N(i+b1-2b1-1)n1i+b1-1Zn1-1(AiB2)+i=1b1n1=1N(i+a1-2a1-1)n1i+a1-1Zn1-1(A2Bi). By Lemma 3.3, using the notations for Ai,Bi,A2,B2 of Lemma 3.2, we get i=1a1n1=1N(i+b1-2b1-1)n1i+b1-1Zn1-1(AiB2)+i=1b1n1=1N(i+a1-2a1-1)n1i+a1-1Zn1-1(A2Bi)=ZN(AB). Consequently, RN(a;b)=ZN(AB). This proves the stated result for RN(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 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 given a=(a1,,ar) and b=(b1,,bs), one may define the shuffle product ζN(ab) 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:=ω0a1-1ω1ω0ar-1ω1 and B:=ω0b1-1ω1ω0bs-1ω1,

ζN(ab):=ZN(AB). 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 Lia(z) denote the (multiple) polylogarithm function with parameters a1,,ar, defined by

Lia(z)=Lia1,,ar(z)=n1>n2>>nr1zn1n1a1n2a2nrar. The value RN(a;b) can be obtained by coefficient extraction in the following way:

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:

RN(a;b)=[zN]Liab(z)1-z. Here the shuffle product for polylogarithm functions Liab(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 , 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  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 . The shuffle identity for polylogarithm functions is due to the iterated Drinfeld integral representation of polylogarithm functions and multiple zeta values due to Kontsevich . As remarked in , 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

limnζj(a)ζN+1-j(b)=limnζn+1(a)ζn+1(b)=ζ(a)ζ(b),limnζn(b2,,bs)ζn(a2,,ar)(n+1)a1+b1=0,limnRN(a;b)=limnζ2n+1(ab)=ζ(a)ζ(b), 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, ζN(a1,a2,,ar;σ1,,σr), σi{0} for 1ir, defined as follows:

ζN(a,σ)=ζN(a1,a2,,ar;σ1,,σr)=Nn1>n2>>nr11i=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 values ζN(a,σ) and ζN(b,τ) with weights σ and τ satisfy the following reciprocity relation: 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 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

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

RN(a,σ;b,τ)=i=1a1n1=1N(i+b1-2b1-1)n1i+b1-1τ1n1Rn1-1(a1+1-i,a2,τ1σ1,σ2;b2,τ2)+i=1b1n1=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.

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.

KirschenhoferP.ProdingerH.Comparisons in Hoare's find algorithmCombinatorics, Probability and Computing19987111112010.1017/S0963548397003325MR1611049ZBL0892.68021HoareC. A. R.Algorithm 64: quicksortCommunications of the ACM19614732132210.1145/366622.366644KubaM.ProdingerH.SchneiderC.Generalized reciprocity laws for sums of harmonic numbersIntegers: The Electronic Journal of Combinatorial Number Theory20088A17120MR2425615BorweinJ. M.BradleyD. M.BroadhurstD. J.LisonekP.Combinatorial aspects of multiple zeta valuesElectronic Journal of Combinatorics19985R38MR1637378ZBL0904.05012BorweinJ. M.BradleyD. M.BroadhurstD. J.LisonekP.Special values of multiple polylogarithmsTransactions of the American Mathematical Society2001353390794110.1090/S0002-9947-00-02616-7MR1709772ZBL1002.11093BowmanD.BradleyD. M.Multiple polylogarithms: a brief surveyQ-Series with Applications to Combinatorics, Number Theory, and Physics20012917192Contemporary MathematicsMR1874522ZBL0998.33013BowmanD.BradleyD. M.The algebra and combinatorics of shuffles and multiple zeta valuesJournal of Combinatorial Theory A2002971436110.1006/jcta.2001.3194MR1879045ZBL1021.11026BowmanD.BradleyD. M.RyooJ. H.Some multi-set inclusions associated with shuffle convolutions and multiple zeta valuesEuropean Journal of Combinatorics200324112112710.1016/S0195-6698(02)00117-8MR1957970ZBL1016.11035ZagierD.Values of zeta functions and their applicationsFirst European Congress of Mathematics Vol. II1994Boston, Mass, USABirkhäuser497512BlümleinJ.KurthS.Harmonic sums and Mellin transforms up to two-loop orderPhysical Review D199960014018VermaserenJ. A. M.Harmonic sums, Mellin transforms and integralsInternational Journal of Modern Physics A199914132037207610.1142/S0217751X99001032MR1693541ZBL0939.65032MochS.UwerP.WeinzierlS.Nested sums, expansion of transcendental functions, and multiscale multiloop integralsJournal of Mathematical Physics20024363363338610.1063/1.1471366MR1902486ZBL1060.33007HoffmanM. E.Quasi-shuffle productsJournal of Algebraic Combinatorics2000111496810.1023/A:1008791603281MR1747062ZBL0959.16021KoornwinderT. H.SchlosserM.On an identity by Chaundy and BullardIndagationes Mathematicae. New Series200819223926110.1016/S0019-3577(08)80002-XMR2489329NielsenN.Handbuch der Theorie der Gamma Funktion1965New York, NY, USAChelseaMR0185152HoangN. M.PetitotM.Lyndon words, polylogarithms and the Riemann ζ functionDiscrete Mathematics20002171–327329210.1016/S0012-365X(99)00267-8MR1766271ZBL0959.68144HoangN. M.PetitotM.Van Der HoevenJ.Shuffle algebra and polylogarithmsProceedings of the 10th Conference on Formal Power Series and Algebraic Combinatorics (FPSAC '98)June 1998Toronto, Canada1210.1016/S0012-365X(99)00267-8ZBL0965.68129