Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
aa r X i v : . [ m a t h - ph ] M a y DESY 11–033DO–TH 11–12SFB/CPP-11-24LPN 11/24May 2011
Harmonic Sums and PolylogarithmsGenerated by Cyclotomic Polynomials
Jakob Ablinger a , Johannes Bl¨umlein b , and Carsten Schneider a a Research Institute for Symbolic Computation (RISC),Johannes Kepler University, Altenbergerstraße 69, A–4040, Linz, Austria b Deutsches Elektronen–Synchrotron, DESY,Platanenallee 6, D-15738 Zeuthen, Germany
Abstract
The computation of Feynman integrals in massive higher order perturbative calculationsin renormalizable Quantum Field Theories requires extensions of multiply nested harmonicsums, which can be generated as real representations by Mellin transforms of Poincar´e–iterated integrals including denominators of higher cyclotomic polynomials. We derive thecyclotomic harmonic polylogarithms and harmonic sums and study their algebraic andstructural relations. The analytic continuation of cyclotomic harmonic sums to complexvalues of N is performed using analytic representations. We also consider special valuesof the cyclotomic harmonic polylogarithms at argument x = 1, resp., for the cyclotomicharmonic sums at N → ∞ , which are related to colored multiple zeta values, derivingvarious of their relations, based on the stuffle and shuffle algebras and three multiple ar-gument relations. We also consider infinite generalized nested harmonic sums at roots ofunity which are related to the infinite cyclotomic harmonic sums. Basis representations arederived for weight w = 1,2 sums up to cyclotomy l = 20 . Dedicated to Martinus Veltman on the occasion of his 80th birthday.
Introduction
The analytic calculation of Feynman integrals requires the complete understanding of the asso-ciated mathematical structures at a given loop level. By the pioneering work in Refs. [1, 2] thishas been thoroughly achieved for one loop integrals occurring in renormalizable Quantum FieldTheories. The corresponding complete framework in case of higher order calculations is, how-ever, not yet available. Which mathematical functions are of relevance there is revealed stepwisein specific higher order calculations. Here higher transcendental functions, like the generalizedhypergeometric functions and their generalizations [3], play a central role. Their series expan-sion in the dimensional parameter [1, 4] leads to nested infinite sums over products of digammafunctions [5], cf. [6]. The nested harmonic sums [7, 8] are defined by S b,~a ( N ) = N X k =1 ( sign ( b )) k k | b | S ~a ( k ) , S ∅ ( N ) = 1 , b, a i ∈ Z \{ } . (1.1)They form a quasi-shuffle algebra [9]. Their values for N → ∞ are the multiple zeta values ζ ~a ,resp. Euler-Zagier values [10] defined by ζ b,~a = lim N →∞ S b,~a ( N ) , b = 1 , see [11]. In case of massless problems to 3–loop order the results for single scale quantities inMellin space can be written by polynomial expressions in terms of S ~a ( N ) and ζ ~a with coefficientsbeing from the rational function field Q ( N ), cf. e.g. [12]. In this context we consider the Mellintransform M [ f ( x )]( N ) = Z dx x N f ( x ) . (1.2)In most of the applications below we assume N ∈ N + = N \{ } .In computations at even higher orders in the coupling constant and through finite mass effects,however, generalizations of the nested harmonic sums contribute, at least in intermediary results.One extension concerns the so-called generalized harmonic sums [13, 14] given by S b,~a ( ζ , ~ξ ; N ) = N X k =1 ζ k k b S ~ξ ( ~r ; k ) , (1.3)with b, a i ∈ N + ; ζ , ξ i ∈ R ∗ = R \{ } . Known examples are related to the second index set ξ i ∈ { , − , / , − / , , − } , cf. [12, 15, 16].In case of single scale problems with two massive lines and m = m at 3-loop order summa-tion is also required over terms ( ± k (2 k + 1) n , (1.4)which is a special case of ( ± k ( l · k + m ) n , (1.5)with l, m, n ∈ N + . Note that we have deliberately chosen a real representation here, which is ofpractical importance in case of fast polynomial operations needed for solving nested summation2roblems [17] . Sums containing fractional terms m/l were considered in the context of coloredharmonic sums in [20–22]. It is expected that sums of this kind and their iterations will occur in awide class of massive calculations in higher order loop calculations in Quantum Electrodynamics,Quantum Chromodynamics, and other renormalizable Quantum Field Theories, in particularstudying single distributions, but also for more variable differential distributions. Usually objectsof this kind emerge first at intermediary steps and, at even higher orders, they occur in the finalresults. Therefore, these quantities have to be understood and methods have to be provided toperform these sums. This does not apply to integer values of the Mellin variable N only, but alsoto the analytic continuation of these quantities to N ∈ C , since the experimental applicationsrequire the Mellin inversion into momentum-fraction space.We show that the single sums (1.5) and their nested iterations can be obtained from linearcombinations of Mellin transforms of harmonic polylogarithms over an alphabet of letters con-taining x l / Φ k ( x ) , < l < deg(Φ k ( x )), where Φ k ( x ) denotes the k th cyclotomic polynomial [23].One may form words by Poincar´e-iterated integrals [25] over this alphabet, which leads to the cyclotomic harmonic polylogarithms H , forming a shuffle algebra. This class extends the harmonicpolylogarithms [26]. The Mellin transform of elements of H has support x ∈ [0 , N ∈ N , N → k · N , (1.6)where k denotes the index of Φ k ( x ). This assumption allows to associate nested harmonic sumsof the cyclotomic type by this Mellin transform. Special values are obtained by either thecyclotomic harmonic polylogarithms H ~a ( x ) at x = 1 or the associated nested harmonic sums for N → ∞ . They extend the multiple zeta values and Euler-Zagier values, cf. Section 5. In thepresent paper we investigate relations and representations of these three classes of quantities.Special emphasis has been put on the class of nested sums (1.4) where all derived algorithmshave been incorporated within the computer algebra package HarmonicSums [27].The paper is organized as follows. In Section 2 we establish the connection between the cyclo-tomic harmonic sums and the cyclotomic harmonic polylogarithms through the Mellin transform,at modified argument kN . The basic properties of the cyclotomic harmonic polylogarithms areinvestigated in Section 3. The cyclotomic harmonic polylogarithms obey a shuffle algebra. Thenested sums at finite values of N are studied in Section 4, including their algebraic and structuralrelations, generalizing [28–31]. Here are also three multiple argument relations of interest. Theanalytic continuation of the new harmonic sums to complex values of N is presented based onrecursion and asymptotic representations, similar to the case of the nested harmonic sum [29–32].The representation of the cyclotomic harmonic sums requires to know their values at N → ∞ ,which are equivalently given by the cyclotomic harmonic polylogarithms at x = 1. The setof special numbers spanning the Euler-Zagier and multiple zeta values [11] are extended. Westudy the cases of weight w = 1,2 up to cyclotomy l = 20 and derive the corresponding relationsbased on the stuffle and shuffle algebra and three multiple argument relations. Furthermore,we investigate the relations of the cyclotomic harmonic sums for N → ∞ extending for wordsresulting from the alphabet ( ± k /k, ( ± k / (2 k + 1), cf. Section 5. For the cyclotomic harmonicpolylogarithms, harmonic sums and their values at N → ∞ basis representations are derived.In Section 6 we study the relations of the infinite nested harmonic sums with numerators at l throot of unity, l ≤ for weight w = 1,2 . Here, also the distribution relation, cf. [19], is consideredbeyond the relations mentioned before. We also consider a relation valid for finite generalized Complex representations are related to the so-called colored harmonic sums S b,~a ( p, ~r ; N ) = P Nk =1 p k k b S ~a ( ~r ; k )with b, a i ∈ N + , p, r i ∈ ∪ Ml =2 { exp[2 πi ( n/l )] , n ∈ { , ..., l − }} , cf. [18, 19]. We consider cyclotomic harmonic sums defined by S { a ,b ,c } ,..., { a l ,b l ,c l } ( s , ..., s l ; N ) = N X k =1 s k ( a k + b ) c S { a ,b ,c } ; ... ; { a l ,b l ,c l } ( s , ..., s l ; k ) , S ∅ = 1 , (2.1)where a i , c i ∈ N + , b i ∈ N , s i = ± , a i > b i ; the weight of this sum is defined by c + · · · + c l and { a i , b i , c i } denote lists, not sets. One may generalize this case further allowing s i ∈ R ∗ , [14]. Ofspecial interest will be the infinite cyclotomic harmonic polylogarithms defined by σ { a ,b ,c } ,..., { a l ,b l ,c l } ( s , ..., s l ) = lim N →∞ S { a ,b ,c } ,..., { a l ,b l ,c l } ( s , ..., s l ; N )which diverge if c = 1 , s = 1. Sometimes we will use the notation S { a ,b ,c } ,..., { a l ,b l ,c l } ( s , ..., s l ; N ) = S { a ,b ,s c } ,..., { a l ,b l ,s l c l } ( N ) (2.2) σ { a ,b ,c } ,..., { a l ,b l ,c l } ( s , ..., s l ) = σ { a ,b ,s c } ,..., { a l ,b l ,s l c l } (2.3)as shortcut below.For further considerations, we rely on the following procedure which transforms the sums tointegrals. The denominators have the following integral representation( ± k ak + b = Z dx x ak + b − ( ± k (2.4)( ± k ( ak + b ) c = Z dx x Z x dx x ... Z x c − dx c − x c − Z x c − dx c x ak + b − c ( ± k , (2.5)and the sum over k yields l X k =1 ( ± k x ak + b − = x a + b − ( ± x a ) l +1 − ± x a ) − . (2.6)This representation is applied to the innermost sum ( a = a l , b = b l , c = c l ). One now mayperform the next sum in the same way, provided a l − | a l . If this is not the case, one transformsthe integration variables in (2.4, 2.5) such that the next denominator can be generated, etc. Inthis way the sums (2.1) can be represented in terms of linear combinations of Poincar´e-iteratedintegrals. Evidently, the representation of the cyclotomic harmonic sum (2.1) in terms of a(properly regularized) Mellin transform will be related to the Mellin variable kN , with k theleast common multiple of a , ..., a l .Let us illustrate the principle steps in case of the following example : S { , , } , { , , } (1 , − N ) = N X k =1 k + 2) k X l =1 ( − l (2 l + 1) . (2.7)4he first sum yields S { , , } , { , , } (1 , − N ) = N X k =1 Z dx x x + 1 ( − x ) k + 1(3 k + 2) . (2.8)Setting x = y one obtains S { , , } , { , , } (1 , − N ) = 12 Z dy y y + 1 N X k =1 ( − y ) k − k + 4) = 12 Z dy y y + 1 (Z y dzz Z z dt t ( − t ) N − t + 1 − y Z dzz Z z dt t t N − t − ) (2.9)= 12 Z dy y y + 1 Z y dzz Z z dt t ( − t ) N − t + 1 − (4 − π ) Z dzz Z z dt t t N − t − ) . (2.10)In general, the polynomials x a − a = 1 for which theexpression is Φ ( x ), see Section 3. Moreover, the polynomials x a + 1 = x a − x a − a = 2 n , n ∈ N or decompose into products of cyclotomic polynomialsin other cases, see Appendix A. All factors divide ( x a ) l −
1, resp. ( − x a ) l −
1. We remark thatEq. (2.10) is not yet written in terms of a Mellin transform (1.2). To achieve this in an automaticfashion, the cyclotomic harmonic polylogarithms are introduced in Section 3. Furthermore, theirspecial values at x = 1 contribute to which we turn in Section 5. To account for the newly emerging sums (2.1) in perturbative calculations in Quantum FieldTheory we introduce Poincar´e-iterated integrals over the alphabet AA = (cid:26) x (cid:27) ∪ (cid:26) x l Φ k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) k ∈ N + , ≤ l < ϕ ( k ) (cid:27) , (3.1)where Φ k ( x ) denotes the k th cyclotomic polynomial [23], and ϕ ( k ) denotes Euler’s totient func-tion [24]. Φ n ( x ) = x n − Y d | n,d 1) which differs in sign from the corresponding letterin [26]. Numerical implementations were given in [33,34]. A few extensions of iterated integrals introduced in [26]based on linear denominator functions of different kind, which are used in quantum-field theoretic calculations,were made in [34, 35]. 6e form the Poincar´e iterated integrals C l ,...,l m k ,...,k m ( z ) = 1 m ! ln( x ) m if ( l , . . . , l m ) = (0 , . . . , , ( k , . . . , k m ) = (0 , . . . , ,C l m k m ( z ) = Z z dx f l m k m ( x ) if k m = 0 ,C l ,...,l m k ,...,k m ( z ) = Z z dx f l k ( x ) C l ,...,l m k ,...,k m ( x ) if ( k , . . . , k m ) = (0 , . . . , , and C ~l~a ( z ) denotes cyclotomic harmonic polylogarithms . They form a shuffle algebra [9, 41] bymultiplication C ~a ~a ( z ) · C ~b ~b ( z ) = C ~a ~a ( z ) ⊔⊔ C ~b ~b ( z ) = X h ~c ~c i ∈ h ~a ~a i ⊔⊔ (cid:20) ~b ~b (cid:21) C ~c ~c ( z ) (3.21)of M w elements at weight w , where M denotes the number of chosen letters from A . The shufflesymbol ⊔⊔ implies all combinations of indices a ij ∈ ~a i and b ij ∈ ~b j leaving the order in both setsunchanged and the brackets [ ] pair the the upper and lower indices forming a unity, cf. (3.17).The number of basis elements spanning the shuffle algebra are given by N basic ( w ) = 1 w X d | w µ (cid:16) w d (cid:17) M d , w ≥ µ denotes the M¨obius function [43]. The numberof basic cyclotomic harmonic polylogarithms in dependence of w and M is given in Table 1. lettersweight Number of basic cyclotomic harmonic polylogarithms in dependence of the number of lettersand weight. Now the cyclotomic harmonic sums (2.1) can be represented as Mellin transforms of cyclo-tomic harmonic polylogarithms. In the example (2.10) the iterated integrals have to be rewritten.We express respective denominators in terms of products of cyclotomic polynomials, (A.2-A.16),and perform partial fractioning in the respective integration variable, (A.27-A.38). In our con-crete example, the integrand of the y -integral in (2.9) has the representation y y + 1 = 13 (cid:2) f ( y ) − f ( y ) + 2 f ( y ) (cid:3) . (3.23)7ith integration by parts one obtains the following Mellin transforms of argument 6 N of cyclo-tomic harmonic polylogarithms C l ,...,l m k ,...,k m ( x ) weighted by the letters f kl ( x ) of the alphabet A : S { , , } , { , , } (1 , − N ) = 16 (4 − π ) Z dxx ( x N − (cid:2) f ( x ) − f ( x ) − f ( x ) − f ( x ) − f ( x ) + f ( x ) (cid:3) C ( x ) − Z dxx (cid:2) ( − N x N − (cid:3) (cid:2) − f ( x ) − f ( x ) + 2 f ( x ) (cid:3) C ( x ) − (cid:2) C , , (1) − C , , (1) + 2 C , , (1) (cid:3) Z dxx (cid:2) ( − N x N − (cid:3) × (cid:2) − f ( x ) − f ( x ) + 2 f ( x ) (cid:3) + 43 Z dxx (cid:2) ( − N x N − (cid:3) (cid:2) C , , ( x ) − C , , ( x ) + 2 C , , ( x ) (cid:3) × (cid:2) − f ( x ) − f ( x ) + 2 f ( x ) (cid:3) . (3.24)The constants C l ,...,l m k ,...,k m (1) are discussed in Section 5. In particular one obtains C , , = − C , (3.25)with C the Catalan number [44], and C , , and C , , are linear combinations of ψ ′ (1 / 12) and ψ ′ (5 / f ( z ) nor C ,l ...,l m ,k ,...,k m ( z ) are present, the z -independent terms in [( ± z k ) N − 1] can be integrated,since the other cyclotomic letters f kl ( z ) and C l,l ...,l m k,k ,...,k m ( z ) for k > z = 1.In general, linear combinations of the Mellin transforms M h f dc ( x ) · C ~b~a ( x ) i ( l N ) = Z dx x lN ( f dc ( x )) u · C ~b~a ( x ) , u ∈ { , } . (3.26)with l being the least common multiple of a , ..., a k allow to represent all cyclotomic harmonicsums. Summarizing, we can write S { a ,b ,c } ,..., { a k ,b k ,c k } ( N ) = s X n =1 e n Z dx x lN ( f β n α n ( x )) u n C ~δ n ~γ n ( x ) , (3.27)with e n ∈ R and u n ∈ { , } ; here e n is determined by polynomial expressions in terms ofcyclotomic harmonic polylogarithms evaluated at 1 with rational coefficients. Within this trans-formation and in the following the letter 1 /x plays a special role concerning the cyclotomicharmonic sums. We can exclude the case that the first letter in (3.26) is 1 /x since it would justshift the Mellin index by one unit. In case of Φ ( x ) and related functions the +-regularization M (cid:20)(cid:18) f ( x ) x − (cid:19) + (cid:21) ( N ) = Z dx x N − x − f ( x ) (3.28)is applied to (3.26). Only in case that f ( x ) = 1 / ( x − 1) and C ~b~a ( x ) do not vanish in the limit x → 1a +-function must occur. Iterations of cyclotomic letters f lk ( x ) , k ≥ , ≤ l < k for k ≥ x ∈ [0 , A ′ := (cid:8) f , f , f , f , f (cid:9) = (cid:26) x , ( x ) , ( x ) , ( x ) , x Φ ( x ) (cid:27) ⊆ A , (3.29)which allows one to express the cyclotomic harmonic sums S { a ,b ,c } ,..., { a l ,b l ,c l } ( s , ..., s l ; N ) with a i ∈ { , } , b i ∈ { , } and c i ∈ N + in terms of Mellin transforms of cyclotomic harmonicpolylogarithms in both directions; the implementation is available within the HarmonicSums package [27]. This transformation will be used, e.g., in Section 4.1.The Mellin transforms (3.26, 3.28) obey difference equations of order l in N , which can beused to define these functions specifying respective initial values for l moments. In the followingwe illustrate this for the words x l / Φ k ( x ). One obtains N k X n =0 c n,k φ k ( N + n − l ) = 1 N + 1 , (3.30)where Φ k ( x ) is given by Φ k ( x ) = N k X n =0 c n,k x n . (3.31)Here we define φ (0 , N ) = Z dx x N − x − φ k ( l, N ) = Z dx x N f lk ( x ) , φ k ( N ) = φ k (0 , N ) , k ≥ φ k ( l, N ) + = Z dx [ x N − f lk ( x ) , φ k ( N ) + = φ k (0 , N ) + . (3.34)For k < 105 for all coefficients c n,k ∈ {− , , } holds [45]. One derives the following firstorder difference equations (3.30) for Mellin transforms associated to the lowest order cyclotomicpolynomials : φ ( l, N + 1) − φ ( l, N ) = 1 N + l + 1 (3.35) φ ( l, N + 1) + φ ( l, N ) = 1 N + l + 1 (3.36) φ ( l, N + 3) − φ ( l, N ) = − N + l + 1)( N + l + 2) (3.37) φ ( l, N + 2) + φ ( l, N ) = 1 N + l + 1 (3.38) φ ( l, N + 5) − φ ( l, N ) = − N + l + 1)( N + l + 2) (3.39) φ ( l, N + 3) + φ ( l, N ) = 2( N + l ) + 3( N + l + 1)( N + l + 2) (3.40)9 ( l, N + 7) − φ ( l, N ) = − N + l + 1)( N + l + 2) (3.41) φ ( l, N + 4) − φ ( l, N ) = 1 N + l + 1 (3.42) φ ( l, N + 9) − φ ( l, N ) = − N + l + 1)( N + l + 4) (3.43) φ ( l, N + 5) + φ ( l, N ) = 3( N + l ) + 2( N + l + 1)( N + l + 2) (3.44) φ ( l, N + 11) − φ ( l, N ) = 1( N + l + 1)( N + l + 2) (3.45) φ ( l, N + 6) + φ ( l, N ) = 3( N + l ) + 2( N + l + 1)( N + l + 2) , etc . (3.46)Together with the corresponding initial values, these recurrence relations enable one to computeefficiently the values for N . In particular, due the special form of the recurrences, we get explicitrepresentations in terms of finite sums. E.g., for φ ( l, N ), we obtain φ ( l, N ) = ( − N N X i =1 ( − i (2(3 i + l ) − i + l ) − i + l ) − 1) + φ ( l, ! , (3.47)and φ ( l, N + 1) = φ ( l + 1 , N ) (3.48) φ ( l, N + 2 = φ ( l + 2 , N ) . (3.49)Looking atlim N →∞ φ ( l, N ) = lim N →∞ Z x N + l x − x − dx = lim N →∞ Z x N + l (1 + x − x − x + . . . ) dx = 0shows that φ ( l, 0) = − ∞ X i =1 ( − i [2(3 i + l ) − i + l − i + l − . Completely analogously, all the other functions (3.35–3.46) can be written in such a sum repre-sentation where the constants are the infinite versions of it multiplied with a minus sign. Notethat these constants can be written as a linear combination of the infinite sums σ { a,b,s } = ∞ X k =1 s k a k + b with s ∈ {− , } and a, b ∈ N with a = 0. The relevant values for this article will be worked outexplicitly in the Section 5.The φ k ( l, N ) + functions are preferred if the recurrences (3.35–3.46) form telescoping equa-tions. In this case, in particular if k is odd, the φ k ( l, r N ) for properly chosen r can be relatedto sums without any extra constant, e.g., φ ( l, N ) + = − N X i =1 i + l − i + l − . (3.50)10 and φ can be related to the single cyclotomic harmonic sums at weight w = 1 as follows φ ( l, N ) + = S ( N + l ) , (3.51) φ ( l, N ) = ( − N h S − ( N + l ) − ln(2) + S − ( l ) i . (3.52)For later considerations we use S ( N ) and S − ( N ) instead of φ ( N ) and φ ( N ). In the followingSection the single cyclotomic harmonic sums of weight w = 1 are expressed in terms of the Mellintransforms φ k ( l, r N ) for properly chosen r . In addition, the cyclotomic harmonic sums of higherweight and depth will be discussed. We consider the extension of the finite nested harmonic sums [7, 8] to those generated by thecyclotomic harmonic polylogarithms discussed in Section 3. First the single cyclotomic harmonicsums are considered and explicit representations are given. We derive their analytic continuationto complex values of N . Next the algebraic, differential and three multiple argument relations ofthe cyclotomic harmonic sums are discussed. These relations are used to represent these sumsover suitable bases. Finally we consider the nested sums over the alphabet { ( ± k /k, ( ± k / (2 k +1) } to higher weight deriving explicit relations up to w = 5 . The single cyclotomic harmonic sums are given by N X k =0 ( ± k ( l · k + m ) n . (4.1)Here N is either an even or an odd integer. In case one needs representations only for N ∈ N the following representations hold : N X k =0 ( − k l · k + m = " N X k =0 l ) · k + m − l ) · k + m + l , (4.2) N +1 X k =0 ( − k l · k + m = " N X k =0 l ) · k + m − l ) · k + m + l − l ) N + l + m , (4.3)with N = 2 N . However, one is interested in relations for general values of N , since for nestedsums more and more cases have to be distinguished. The single sums can be expressed in termsof the Mellin transforms φ k ( l, kN ), (3.32–3.33). Up to w = 6 one obtains : N X k =1 11 + 2 k = − N N + 1 − S ( N )2 + S (2 N ) , (4.4) N X k =1 ( − k k =( − N (cid:20) N + 1 − φ (2 N ) (cid:21) + σ { , , − } , (4.5) N X k =1 11 + 3 k = − N N + 1 − S ( N )6 + 12 S (3 N ) − φ (3 N ) + , (4.6)11 X k =1 ( − k k = 16 S − ( N ) − S − (3 N ) + ( − N (cid:20) N + 1 − φ (3 N ) (cid:21) + 13 σ { , , − } + σ { , , − } , (4.7) N X k =1 12 + 3 k = − N N + 2) − S ( N )6 + 12 S (3 N ) + 12 φ (3 N ) + , (4.8) N X k =1 ( − k k = − S − ( N ) + 12 S − (3 N ) + ( − N (cid:20) N + 2 − φ (3 N ) (cid:21) + 13 σ { , , − } + σ { , , − } + 12 , (4.9) N X k =1 11 + 4 k = − N N + 1 − S (2 N ) + 12 S (4 N ) − φ (4 N ) + 12 σ { , , − } + 12(4 N + 1) , (4.10) N X k =1 ( − k k =( − N (cid:20) N + 1 − φ (4 N ) (cid:21) + σ { , , − } , (4.11) N X k =1 13 + 4 k = − N N + 3) − S (2 N ) + 12 S (4 N ) + 12 φ (4 N ) − σ { , , − } − N + 3) , (4.12) N X k =1 ( − k k =( − N (cid:20) N + 3 − φ (2 , N ) (cid:21) + σ { , , − } , (4.13) N X k =1 11 + 5 k = − N N + 1 − S ( N )20 + 14 S (5 N ) − φ (5 N ) + − φ (1 , N ) + − φ (2 , N ) + , (4.14) N X k =1 ( − k k =( − N (cid:20) − φ (5 N ) + 12 φ (1 , N ) − φ (2 , N ) + 15 N + 1 (cid:21) + 120 S − ( N ) − S − (5 N ) + 15 σ { , , − } + σ { , , − } , (4.15) N X k =1 12 + 5 k = − N N + 2) − S ( N )20 + 14 S (5 N ) + 14 φ (5 N ) + − φ (1 , N ) + − φ (2 , N ) + , (4.16) N X k =1 ( − k k =( − N (cid:20) − φ (5 N ) − φ (1 , N ) + 14 φ (2 , N ) + 15 N + 2 (cid:21) (4.17) − S − ( N ) + 14 S − (5 N ) − σ { , , − } + σ { , , − } , N X k =1 13 + 5 k = − N N + 3) − S ( N )20 + 14 S (5 N ) + 14 φ (5 N ) + + 12 φ (1 , N ) + − φ (2 , N ) + , (4.18) N X k =1 ( − k k =( − N (cid:20) φ (5 N ) − φ (1 , N ) − φ (2 , N ) + 15 N + 3 (cid:21) + 120 S − ( N ) − S − (5 N ) + 15 σ { , , − } + σ { , , − } , (4.19)12 X k =1 14 + 5 k = − N N + 4) − S ( N )20 + 14 S (5 N ) + 14 φ (5 N ) + + 12 φ (1 , N ) + + 34 φ (2 , N ) + , (4.20) N X k =1 ( − k k =( − N (cid:20) − φ (5 N ) + 12 φ (1 , N ) − φ (2 , N ) + 15 N + 4 (cid:21) − S − ( N ) + 14 S − (5 N ) + 35 σ { , , − } + σ { , , − } − σ { , , − } + σ { , , − } + 712 , (4.21) N X k =1 11 + 6 k = − N N + 1 + S ( N )12 − S (2 N ) − S (3 N ) + 12 S (6 N ) − φ (3 N ) + − φ (6 N ) + , (4.22) N X k =1 ( − k k =( − N (cid:20) − φ (6 N ) + 13 φ (2 N ) + 16 N + 1 (cid:21) + σ { , , − } , (4.23) N X k =1 15 + 6 k = − N N + 5) + S ( N )12 − S (2 N ) − S (3 N ) + 12 S (6 N )+ 14 φ (3 N ) + + 12 φ (6 N ) + , (4.24) N X k =1 ( − k k =( − N (cid:20) φ (6 N ) − φ (2 N ) − φ (6 N ) + 16 N + 5 (cid:21) + 43 σ { , , − } − σ { , , − } + 215 . (4.25)Taking the sum representations such as (3.50) and (3.47) given by the recurrence relations (3.35–3.46) and the corresponding initial values φ k ( l, 0) (which are expressible by the infinite sums σ { a,b, ± } , cf. Section 5), we used the summation package Sigma [17] to perform this transforma-tion. As a consequence, the single cyclotomic harmonic sums can be expressed in terms of thesums S − ( N ) , S − (3 N ) , S − (5 N ) , S ( N ) , S (2 N ) , S (3 N ) , S (4 N ) , S (5 N ) , S (6 N ) ,φ (3 N ) + , φ (6 N ) + , φ (2 N ) , φ (4 N ) , φ (6 N ) , φ (5 N ) + , φ (1 , N ) + ,φ (2 , N ) + , φ (3 N ) , φ (4 N ) , φ (2 , N ) , φ (5 N ) , φ (1 , N ) , φ (2 , N ) , φ (6 N ) . In particular, in the way how this construction is carried out it follows by the summation theoryof [17] that the sequences produced by these sums form an algebraic independent basis overthe ring of sequences generated by the elements from R ( N )[( − N ]. Note that we exploited thealgebraic relations (5.118–5.129) which implies that only the following constants σ { , , − } , σ { , , − } , σ { , , − } , σ { , , − } , σ { , , − } , σ { , , − } , σ { , , − } , σ { , , − } σ { , , − } from R appear in the representation found for the single cyclotomic harmonic sums with weight w = 1 .Due to (3.33, 3.34) the functions Φ k ( l, N ) , k ≥ N with poles at − n , n ∈ N . This also applies to φ k (0 , N ), (3.32). The latter function grows ∝ ln( N ) for N → ∞ , | arg( N ) | < π . The recursion13elations (3.30) allow one to shift φ k ( l, N ) in N → N + 1. To represent a function φ k ( l, N ) for N ∈ C one needs to know its asymptotic representation in addition.It is given in analytic form in terms of series involving the Stirling numbers of the 2ndkind [36, 37]. The corresponding representations read : φ (0 , N ) ∼ γ + ln( N ) + 12 N − N + 1120 N − N + 1240 N − N + 69132760 N + O (cid:18) N (cid:19) (4.26) φ (0 , N ) ∼ N − N + 18 N − N + 1716 N − N + 6918 N + O (cid:18) N (cid:19) (4.27) φ (0 , N ) ∼ N − N + 23 N − N + 161827 N − N + O (cid:18) N (cid:19) (4.28) φ (0 , N ) ∼ N − N + 52 N − N + 13852 N − N + O (cid:18) N (cid:19) (4.29) φ (0 , N ) ∼ N + 15 N − N − N + 3125 N + 675 N − N − N + 37795 N + 41275125 N − N − N + O (cid:18) N (cid:19) (4.30) φ (0 , N ) ∼ N − N + 22 N − N + 30742 N − N + O (cid:18) N (cid:19) (4.31) φ (0 , N ) ∼ N + 27 N − N − N + 56 N + 390049 N − N − N + 18099927 N + 45825007 N − N + O (cid:18) N (cid:19) (4.32) φ (0 , N ) ∼ N + 12 N − N − N + 572 N + 3612 N − N − N + 2507372 N + 28730412 N − N − N + O (cid:18) N (cid:19) (4.33) φ (0 , N ) ∼ N + 23 N − N − N + 343 N + 11723 N − N − N + 2073943 N + 149990123 N − N − N + O (cid:18) N (cid:19) (4.34) φ (0 , N ) ∼ N + 1 N − N − N + 151 N + 871 N − N − N + 1626151 N + 16922791 N − N − N + O (cid:18) N (cid:19) (4.35) φ (0 , N ) ∼ N + 411 N + 611 N − N − N + 306411 N + 2664611 N − N − N + 7618184 N + 13945859346121 N − N + O (cid:18) N (cid:19) (4.36) φ (0 , N ) ∼ N + 1 N − N − N + 305 N + 1681 N − N − N + 6815585 N + 67637281 N − N − N + O (cid:18) N (cid:19) (4.37)14 (2 , N ) ∼ N − N + 8625 N − N + 120945 N − N + O (cid:18) N (cid:19) (4.38) φ (2 , N ) ∼ N − N + 40 N − N + 177280 N − N + O (cid:18) N (cid:19) (4.39) φ (1 , N ) ∼ N + 1 N − N − N + 151 N + 871 N − N − N + 1626151 N + 16922791 N − N − N + O (cid:18) N (cid:19) (4.40) φ (2 , N ) ∼ N − N + 186 N − N + 2009946 N − N + O (cid:18) N (cid:19) . (4.41)The numerical accuracy of the asymptotic representations at a given suitably large value of N lowers with growing k , i.e., one has to choose larger values of N correspondingly to apply theasymptotic formulae. The above expansions can easily be extended to higher inverse powers of N . The recursion for φ k ( l, kN ) is given in (3.30), resp. (3.35–3.46) in a more compact form.Due to these for any N ∈ C at which φ k ( l, N ) is analytic one may map φ k ( l, N ) to values | N | ≫ , arg( N ) < π and use the asymptotic representations.The single cyclotomic harmonic sums of higher weight obey the representations N − X k =0 lk + m ) n = ( − n − ( n − Z dx ln n − ( x ) x m − x lN − x l − , m < l (4.42) N − X k =0 ( − k ( lk + m ) n = ( − n ( n − Z dx ln n − ( x ) x m − ( − x l ) N − x l + 1 , m < l, (4.43)for l, m, n ∈ N + , which may be expressed in terms of cyclotomic letters again. We note that ∂ n ∂m n Z dx x m − f ( x ) = Z dx x m − ln n ( x ) f ( x ) . (4.44)Therefore, (4.42, 4.43) can be expressed by the corresponding derivatives of φ k ( n, lN ) for N andcorresponding constants. x = 1 We consider the cyclotomic harmonic polylogarithm C c,~da,~b ( x ). Its value at x = 1 is given by C c,~da,~b (1) = Z dxf ca ( x ) C ~d~b ( x ) . (4.45)Let f ca ( x ) = x l / Φ k ( x ), with l < deg(Φ k ( x )) and n be the smallest integer such that Φ k ( x ) | ( x n − k ( x ) = Q j Φ j ( x ) x n − . (4.46)Since x ∈ [0 , x l Φ k ( x ) = − x l Y k Φ k ( x ) ∞ X j =0 x jn = w X m =0 a m ( l ) x m ∞ X j =0 x jn , a m ( l ) ∈ Z (4.47)15olds. Thus we get Z dx x l Φ k ( x ) = w X m =0 a m ( l ) ∞ X j =0 jn + m + 1 , (4.48)representing the value of the depth d=1 cyclotomic harmonic polylogarithms at x = 1 as a linearcombination of the corresponding infinite cyclotomic sums, see Section 5.We return now to Eq. (4.45). Integration by parts yields C c,~da,~b (1) = C ca (1) C ~d~b (1) − ∞ X j =0 w X m =0 a m Z dx x jn + m +1 jn + m + 1 C ~d~b ( x ) . (4.49)Since the Mellin transform of a cyclotomic harmonic polylogarithm can be represented by alinear combination of finite cyclotomic harmonic sums, the weighted infinite sum of the latterones in (4.49) is thus given as a polynomial of infinite cyclotomic harmonic sums, Section 5. The cyclotomic harmonic sums obey differentiation relations, cf. [7, 28], stuffle relations due totheir quasi–shuffle algebra, cf. [9], and multiple argument relations. As has been illustrated in Section 3, cyclotomic harmonic sums can be represented as linearcombinations of Mellin transforms (3.27) of cyclotomic harmonic polylogarithms. Based on thisrepresentation, the differentiation of these sums is defined by ∂ m ∂N m S { a ,b ,c } ,..., { a k ,b k ,c k } ( N ) = s X n =1 e n Z dx x lN l m ln m ( x )( f β n α n ( x )) u n C ~δ n ~γ n ( x ) . (4.50)The product ln m ( x ) C ~β n ~α n ( x ) = m ! C ,..., ,..., C ~β n ~α n ( x ) with m consecutive zeros may be transformed intoa linear combination of cyclotomic harmonic polylogarithms using the shuffle relation (3.21).Finally, using the inverse Mellin transform, the derivative (4.50) of a cyclotomic harmonic sumw.r.t. N is given as a polynomial expression in terms of cyclotomic harmonic sums and cyclotomicharmonic polylogarithms at x = 1. Together with the previous section, the derivative (4.50) canbe expressed as a polynomial expression with rational coefficients in terms of cyclotomic harmonicsums and their values at N → ∞ . The corresponding relations are denoted by ( D ). Furtherdetails on cyclotomic harmonic sums at infinity are given in Section 5.A given finite cyclotomic harmonic sum is determined for N ∈ C by its asymptotic represen-tation and the corresponding recursion from N → ( N − N is closely related to the original sum. For this reason one maycollect these derivatives in classes S ( D ) { a ,b ,c } ,..., { a k ,b k ,c k } ( N ) = (cid:26) ∂ n ∂N n S { a ,b ,c } ,..., { a k ,b k ,c k } ( N ); n ∈ N (cid:27) . (4.51) For the relations given in this Section we mostly present the results, giving for a few cases the proofs inAppendix B. The other proofs proceed in a similar manner. .3.2 Stuffle Algebra To derive the stuffle relations, cf. [11, 28], we consider the product of two denominator terms.For a , a , b , b , c , c , i ∈ N they are given by1( a i + b ) c ( a i + b ) c = ( − c c X j =1 ( − j (cid:18) c + c − j − c − (cid:19) a c a c − j ( a b − a b ) c + c − j a i + b ) j + ( − c c X j =1 ( − j (cid:18) c + c − j − c − (cid:19) a c − j a c ( a b − a b ) c + c − j a i + b ) j , (4.52)and 1( a i + b ) c ( a i + b ) c = (cid:18) a a (cid:19) c a i + b ) c + c , (4.53)if a b = a b .The product of two cyclotomic harmonic sums has the following representation. Let a i , b i , d i , e i , k, l, n ∈ N + and c i , f i ∈ Z \ { } . If a e = d b one has S { a ,b ,c } ,..., { a k ,b k ,c k } ( n ) S { d ,e ,f } ,..., { d l ,e l ,f l } ( n ) = n X i =1 sign ( c ) i ( a i + b ) | c | S { a ,b ,c } ,..., { a k ,b k ,c k } ( i ) S { d ,e ,f } ,..., { d l ,e l ,f l } ( i )+ n X i =1 sign ( f ) i ( d i + e ) | f | S { a ,b ,c } ,..., { a k ,b k ,c k } ( i ) S { d ,e ,f } ,..., { d l ,e l ,f l } ( i ) − n X i =1 ( − | c | | c | X j =1 ( − j (cid:18) | c | + | f | − j − | f | − (cid:19) a | f | d | c |− j a e − d b a i + b ) j +( − | f | | f | X j =1 ( − j (cid:18) | c | + | f | − j − | f | − (cid:19) a | f |− j d | c | d b − a e d i + e ) j × S { a ,b ,c } ,..., { a k ,b k ,c k } ( i ) S { d ,e ,f } ,..., { d l ,e l ,f l } ( i ) , (4.54)resp. for a e = d b one has S { a ,b ,c } ,..., { a k ,b k ,c k } ( n ) S { d ,e ,f } ,..., { d l ,e l ,f l } ( n ) = n X i =1 sign ( c ) i ( a i + b ) | c | S { a ,b ,c } ,..., { a k ,b k ,c k } ( i ) S { d ,e ,f } ,..., { d l ,e l ,f l } ( i )+ n X i =1 sign ( f ) i ( d i + e ) | f | S { a ,b ,c } ,..., { a k ,b k ,c k } ( i ) S { d ,e ,f } ,..., { d l ,e l ,f l } ( i ) − (cid:18) a d (cid:19) | f | n X i =1 (sign ( c ) sign ( f )) i ( a i + b ) | c | + | f | × S { a ,b ,c } ,..., { a k ,b k ,c k } ( i ) S { d ,e ,f } ,..., { d l ,e l ,f l } ( i ) . (4.55)Subsequently, the relations given by (4.54, 4.55) are denoted by ( A ).17 .3.3 Synchronization A first multiple argument relation is implied as follows. Let a, b, k ∈ N , c ∈ Z \ { } , k ≥ 2. Then S { a,b,c } ( k · N ) = k − X i =0 sign ( c ) i S { k · a,b − a · i, sign( c ) k | c |} ( N ) . (4.56)For a i , b i , m, k ∈ N , c i ∈ Z \ { } , k ≥ 2, the general cyclotomic harmonic sums obey S { a m ,b m ,c m } , { a m − ,b m − ,c m − } ,..., { a ,b ,c } ( k · N ) = m − X i =0 N X j =1 S { a m − ,b m − ,c m − } ,..., { a ,b ,c } ( k · j − i ) sign ( c m ) k · j − i ( a m ( k · j − i ) + b ) | c m | . (4.57)Repeated application of (4.56, 4.57) allows to represent cyclotomic harmonic sums of argument kn by those of argument n . Subsequently, the resulting relations are denoted by ( M ). The usual duplication relation [11] holds also for cyclotomic harmonic sums : X ± S { a m ,b m , ± c m } , { a m − ,b m − , ± c m − } ,..., { a ,b , ± c } (2 N ) = 2 m S { a m ,b m ,c m } ,..., { a ,b ,c } ( N ) . (4.58)The proof of (4.58) is given in Appendix B.Similar to (4.58) one obtains a second duplication relation : X d i ∈{− , } d m d m − · · · d S { a m ,b m ,d m c m } , { a m − ,b m − ,d m − c m − } ,..., { a ,b ,d c } (2 N )= 2 m S { a m ,b m − a m ,c m } ,..., { a ,b − a ,c } ( N ) ;(4.59)its proof is given in Appendix B. The resulting algebraic relations of (4.58) and (4.59) are denotedby ( H ) and ( H ), respectively. We remark that more general relations arise for generalizedcyclotomic harmonic sums presented in Section 6, (6.12, 6.13). As an example we consider the class of cyclotomic sums, which occur in the physical applicationmentioned in Section 1. They are given by iteration of the summands1 k l , ( − k k l , k + 1) l , ( − k (2 k + 1) l (4.60)with N ≥ k ≥ , l i , k ∈ N + . As pointed out earlier, this class of cyclotomic harmonic sums canbe expressed by Mellin transforms in terms of cyclotomic harmonic polylogarithms generated bythe alphabet (3.29).We apply the relations in Section 4.2 to derive the corresponding bases for given weight w .The cyclotomic harmonic sums can be represented over the corresponding bases. Applying the18elations using computer algebra we find the pattern for the number of basis elements up to w= 5 given in Table 2. w N S H H , H H , M H , H , M D H , H , M, D A H , H , M, A A, D all Table 2: Reduction of the number of cyclotomic harmonic sums N S over the elements (4.60) at given weightw by applying the three multiple argument relations ( H , H , M ) , differentiation w.r.t. to the external sumindex N , ( D ) , and the algebraic relations ( A ) . A sequence of symbols corresponds to the combination ofthese relations. Due to the arguments given in Section 4.3.1 above, for any occurrence of a differential operator( ∂ m /∂N m ) S a ,b ,c , ..., a k , b k , c k ( N ) only one representative is counted.The total number of sums of weight w for the alphabet (4.60) is N S ( w ) = 4 · w − . (4.61)We mention that the single application of any of the multiple argument relations ( M, H , H )leads to the same number of basis sums N H = N H = N M . (4.62)This also applies to the combinations H M and H MN H ,M = N H ,M . (4.63)Explicit counting relations for the number of basis elements given Table 2 can be derived : N A ( w ) = 1 w X d | w µ (cid:16) wd (cid:17) d (4.64) N D ( w ) = N S ( w ) − N S ( w − 1) = 16 · w − (4.65) N H ( w ) = N S ( w ) − w − = 4 · w − − w − (4.66) N H H ( w ) = N S ( w ) − (2 · w − − 1) = 4 · w − − (2 · w − − 1) (4.67) N H M ( w ) = N S ( w ) − · w − = 4 · w − − · w − (4.68) N H H M ( w ) = N S ( w ) − (3 · w − − 1) = 4 · w − − (3 · w − − 1) (4.69) N AD ( w ) = N A ( w ) − N A ( w − w X d | w µ (cid:16) wd (cid:17) d − w − X d | w − µ (cid:18) w − d (cid:19) d (4.70) N AH H M ( w ) = 1 w X d | w µ (cid:16) wd (cid:17) d − · w X d | w µ (cid:16) wd (cid:17) d − (4.71) N DH H M ( w ) = N H H M ( w ) − N H H M ( w − 1) = 16 · w − − · w − (4.72) N ADH H M ( w ) = N AH H M ( w ) − N AH H M ( w − 1) (4.73)= 1 w X d | w µ (cid:16) wd (cid:17) (5 d − · d ) − w − X d | w − µ (cid:18) w − d (cid:19) (5 d − · d ) . (4.74)19ere µ denotes the M¨obius function [43].The analytic continuation of the cyclotomic harmonic sums can be performed as outlinedfor the case of the single sums in Section 4.1. Their representation as a Mellin transform ofthe cyclotomic harmonic polylogarithms (3.26) relates them to factorial series, except the case c = 1 , s = 1, cf. (2.1). If a sequential set of first indices c i = 1 , s i = 1 occurs, one may reducethe corresponding sum algebraically to convergent sums, separating factors, cf. [6]. I.e., alsoin the general case the poles of the cyclotomic harmonic sums are located at − k, k ∈ N . Therecursion relations of the cyclotomic harmonic sums (2.1) imply the shift relations N → N + 1in a hierarchic manner, referring to the sums of lower depth.To accomplish the analytic continuation in N , the asymptotic representations of the cyclo-tomic harmonic sums have to be computed. Since the cyclotomic harmonic sums are representedover respective bases, only the asymptotic representations for the basis elements have to be de-rived. One way consists in using iterated integration by parts Z dx x N f ( x ) = f (1) N + 1 − N + 1 Z dx x N [ xf ′ ( x )] , (4.75)with f ( x ) a linear combination of cyclotomic polylogarithms. Here f ( x ) is conveniently expressedin a power series to which we turn now.We illustrate the principle steps considering the alphabet (3.29) related to (4.60), cf. (3.16,3.17). In general the cyclotomic harmonic polylogarithms C ~b~a ( x ) over the alphabet (3.29) do nothave a regular Taylor series expansion, cf. [26]. This is due to the effect that trailing zeroes inthe index set may cause powers of ln( x ). Hence the proper expansion is one in terms of both x and ln( x ). For depth one and 0 < x < C ( x ) = ln( x ) (4.76) C ( x ) = − ∞ X i =1 ( − x ) i i = − ∞ X i =1 ( − x ) i i + ∞ X i =1 ( − x ) i +1 i + 1 (4.77) C ( x ) = − ∞ X i =1 x i i = ∞ X i =1 ( − x ) i i + ∞ X i =1 ( − x ) i +1 i + 1 (4.78) C ( x ) = − ∞ X i =1 ( − i x i − i − C ( x ) = ∞ X i =1 ( − i x i i . (4.80)Let C ~b~a ( x ) be a cyclotomic harmonic polylogarithm with depth d. Assume that its power seriesexpansion is of the form C ~b~a ( x ) = w X j =1 ∞ X i =1 σ i x i + c j (2 i + c j ) a S ~n j ( i ) (4.81)with the index sets ~a and ~b according to the iteration of the letters (4.76–4.80) and ~n j a corre-sponding index set of the cyclotomic harmonic sum, x ∈ (0 , w ∈ N and c j ∈ Z .Then the expansion of the cyclotomic harmonic polylogarithms of depth d + 1 is obtained byusing C ,~b ,~a ( x ) = w X j =1 ∞ X i =1 σ i x i + c j (2 i + c j ) a +1 S ~n j ( i ) (4.82)20 ,~b ,~a ( x ) = w X j =1 ∞ X i =1 x i + c j +1 (2 i + c j + 1) S { ,c j ,σa } ,~n j ( i ) + w X j =1 ∞ X i =1 x i + c j +2 (2 i + c j + 2) S { ,c j ,σa } ,~n j ( i )(4.83) C ,~b ,~a ( x ) = w X j =1 ∞ X i =1 x i + c j +1 (2 i + c j + 1) S { ,c j ,σa } ,~n j ( i ) − w X j =1 ∞ X i =1 x i + c j +2 (2 i + c j + 2) S { ,c j ,σa } ,~n j ( i )(4.84) C ,~b ,~a ( x ) = w X j =1 ∞ X i =1 ( − i x i + c j +1 (2 i + c j + 1) S { ,c j , − σa } ,~n j ( i ) (4.85) C , ~m , ~m ( x ) = w X j =1 ∞ X i =1 ( − i x i + c j +2 (2 i + c j + 2) S { ,c j , − σa } ,~n j ( i ) . (4.86)Sample proofs of these relations are given in Appendix B.The analytic continuation of the cyclotomic harmonic sums at larger depths to N ∈ C isperformed analogously to the case discussed in Section 4.1. Shifts parallel to the real axis areperformed with the recurrence relation induced by (2.1). The asymptotic relations for N →∞ , | arg( N ) | < π can be derived analytically to arbitrary precision. Examples are : S { , , } , { , , − } ( N ) ∼ C f ,f (1) + 4 C f (1) C f ,f ,f (1) − C f ,f ,f ,f (1) + (cid:18) π − N + 1 N − N + 34 N − N + 516 N − N + 764 N − N + 9256 N (cid:19) C f ,f (1) + ( − N (cid:18) N − N + 2764 N − N − N − N + 86991024 N (cid:19) + O (cid:18) N (cid:19) (4.87) S { , , } , { , , − } ( N ) ∼ − π C f (1) C f ,f ,f (1) − C f ,f ,f ,f (1) + 14 N − N + 1148 N − N + 127960 N − N + 2215376 N − N + 36715360 N − N + ( − N (cid:18) N − N + 25128 N − N − N + 221512 N + 15451024 N (cid:19) + C f ,f (1) (cid:18) C ,f (1) − π 8+ 14 N − N + 1148 N − N + 127960 N − N + 2215376 N − N + 36715360 N − N (cid:19) + O (cid:18) N (cid:19) . (4.88)Given a cyclotomic harmonic sum with the iterative denominators (4.60) and given the numberof desired terms, the corresponding expansion can be computed on demand by the HarmonicSums package. 21 Special Values The values of the cyclotomic harmonic polylogarithms at argument x = 1 and, related to it,the associated cyclotomic harmonic sums at N → ∞ occur in various relations of the finitecyclotomic harmonic sums and the Mellin transforms of cyclotomic harmonic polylogarithms.In this Section we investigate their relations and basis representations. The infinite cyclotomicharmonic sums extend the Euler-Zagier and multiple zeta values [11] and are related at lowerweight and depth to other known special numbers. We first consider the single non-alternatingand alternating sums up to cyclotomy l = 6 at general weight w . Next the relations of the infinitecyclotomic harmonic sums associated to the summands (4.60) up to weight w = 6 are workedout. Finally we investigate the sums of weight w = 1 and up to cyclotomy l = 20 . We consider the single sums of the type ∞ X k =0 ( ± k ( lk + m ) n , (5.1)with l, m, n ∈ N + , l > m, n ≥ 1. These sums are linearly related to the colored harmonic sumsby ∞ X k =1 e kl k n = l X m =1 e ml ∞ X k =0 lk + m ) n , e l = exp (cid:18) πil (cid:19) , (5.2)and similar linear relations for the nested sums. We will address the latter sums in Section 6. w = 1 For the non-alternating sums one obtains ∞ X k =0 lk + m = 1 l h σ − γ E − ψ (cid:16) ml (cid:17)i , (5.3)with σ = ∞ X k =1 k (5.4)denoting the divergence in (5.3). The corresponding sums, except σ can always be regularized.The reflection symmetry of the ψ -function interchanging the arguments x and (1 − x ), [36],implies ∞ X k =0 (cid:20) nk + l − nk + ( n − l ) (cid:21) = πn cot (cid:18) ln π (cid:19) . (5.5)The digamma-function at positive rational arguments obeys [36, 47] ψ (cid:18) pq (cid:19) = − γ E − ln(2 q ) − π (cid:18) pπq (cid:19) + 2 [( q − / X k =1 cos (cid:18) πkpq (cid:19) ln (cid:20) sin (cid:18) πkq (cid:19)(cid:21) (5.6) ψ (cid:18) n (cid:19) = − n ( γ E + ln( n )) − n X k =2 ψ (cid:18) kn (cid:19) . (5.7)22q. (5.6) is used to remove dependencies in (5.3, 5.5). If the regular q –polygon is constructible,the trigonometric functions in (5.6) are algebraic numbers. This is the case for q ∈ { , , , , , , , , , , , , , , , , , , , , , ... } , (5.8)[48] . Due to (5.6) the constants ln( d i ) with d i = 1, the divisors of q , and logarithms of furtheralgebraic numbers will occur. Theses are all transcendental numbers [50].Let us list the first few of these relations, see also [51] : − (cid:20) γ E + ψ (cid:18) (cid:19)(cid:21) = ln(2) (5.9) − (cid:20) γ E + ψ (cid:18) (cid:19)(cid:21) = 12 ln(3) + π √ − (cid:20) γ E + ψ (cid:18) (cid:19)(cid:21) = 12 ln(3) − π √ − (cid:20) γ E + ψ (cid:18) (cid:19)(cid:21) = 34 ln(2) + π − (cid:20) γ E + ψ (cid:18) (cid:19)(cid:21) = 34 ln(2) − π − (cid:20) γ E + ψ (cid:18) (cid:19)(cid:21) = √ h ln(2) − ln( √ − i + 14 ln(5) + p 25 + 10 √ π (5.14) − (cid:20) γ E + ψ (cid:18) (cid:19)(cid:21) = − √ h ln(2) − ln( √ − i + 14 ln(5) + p − √ " − √ π (5.15) − (cid:20) γ E + ψ (cid:18) (cid:19)(cid:21) = − √ h ln(2) − ln( √ − i + 14 ln(5) − p − √ " − √ π (5.16) − (cid:20) γ E + ψ (cid:18) (cid:19)(cid:21) = √ h ln(2) − ln( √ − i + 14 ln(5) − p 25 + 10 √ π (5.17) − (cid:20) γ E + ψ (cid:18) (cid:19)(cid:21) = 13 ln(2) + 14 ln(3) + π √ − (cid:20) γ E + ψ (cid:18) (cid:19)(cid:21) = 13 ln(2) + 14 ln(3) − π √ , etc . (5.19) w = 1 The alternating sums at w = 1 have the representation : ∞ X k =0 ( − k lk + m = 12 l (cid:20) ψ (cid:18) m + l l (cid:19) − ψ (cid:16) m l (cid:17)(cid:21) , (5.20)with the reflection relation ∞ X k =0 (cid:20) ( − k nk + l − ( − k nk + ( n − l ) (cid:21) = 12 n " ψ (cid:18) − l n (cid:19) + ψ (cid:18) 12 + l n (cid:19) − ψ (cid:18) − l n (cid:19) − ψ (cid:18) l n (cid:19) . (5.21) See [49] for special values of the trigonometric functions occurring in (5.6). w = 1 one obtains : ∞ X k =0 ( − k k + 1 = π ∞ X k =0 ( − k k + 1 = 13 (cid:20) π √ (cid:21) (5.23) ∞ X k =0 ( − k k + 2 = 13 (cid:20) π √ − ln(2) (cid:21) (5.24) ∞ X k =0 ( − k k + 1 = 12 √ h π − ln( √ − i (5.25) ∞ X k =0 ( − k k + 3 = 12 √ h π √ − i (5.26) ∞ X k =0 ( − k k + 1 = 15 h √ i ln(2) − √ √ − 1) + 1 + √ p 10 + 2 √ π (5.27) ∞ X k =0 ( − k k + 2 = − h − √ i ln(2) − √ √ − 1) + √ − p − √ π (5.28) ∞ X k =0 ( − k k + 3 = 15 h − √ i ln(2) + 1 √ √ − 1) + √ − p − √ π (5.29) ∞ X k =0 ( − k k + 4 = − h √ i ln(2) + 1 √ √ − 1) + 1 + √ p 10 + 2 √ π (5.30) ∞ X k =0 ( − k k + 1 = π √ (cid:20) 12 ln(2) − ln( √ − (cid:21) (5.31) ∞ X k =0 ( − k k + 5 = π − √ (cid:20) 12 ln(2) − ln( √ − (cid:21) , etc. (5.32) These sums obey representations which are obtained by repeated differentiation of (5.3) and(5.20) for m : ∞ X k =0 lk + m ) n = 1 l n ζ H (cid:16) n, ml (cid:17) = 1Γ( n ) (cid:18) − l (cid:19) n ψ ( n − (cid:16) ml (cid:17) (5.33) ∞ X k =0 ( − k ( lk + m ) n = ( − n − Γ( n ) 1(2 l ) n (cid:20) ψ ( n − (cid:18) m + l l (cid:19) − ψ ( n − (cid:16) m l (cid:17)(cid:21) . (5.34)Here ζ H is the Hurwitz ζ –function [52] with the serial representation ζ H ( s, a ) = ∞ X n =0 n + a ) s . (5.35)24e consider the representations of the polygamma functions at rational arguments for p/q , p ∤ q, q ∈ N + , q ≤ One obtains : ψ ( l ) (cid:18) (cid:19) = ( − l +1 l ! (cid:0) l +1 − (cid:1) ζ l +1 , (5.36)with ζ n = ( − n − (2 π ) n n )! B n , (5.37)where B n denote the Bernoulli numbers [54, 55]. They are generated by x exp( x ) − ∞ X n =0 B n n ! x n , B n +1 = 0 for n > . (5.38)For odd values of l no new basis elements occur due to (5.36).The reflection formula [36] ψ ( n ) (1 − z ) = ( − n (cid:20) ψ n ( z ) + π d n dz n cot( πz ) (cid:21) (5.39)implies the relations for the argument p/q and ( q − p ) /q . Likewise, the multiplication formulafor the Hurwitz zeta function [52] holds, ζ H ( s, kz ) = 1 k s k − X n =0 ζ H (cid:16) s, z + nk (cid:17) , k ∈ N + , (5.40) m l +1 ψ ( l ) ( mz ) = m − X k =0 ψ ( l ) (cid:18) z + km (cid:19) , (5.41)with the special relation n !( − n +1 ζ n +1 (cid:2) m n +1 − (cid:3) = m − X k =1 ψ ( n ) (cid:18) km (cid:19) . (5.42)One obtains, cf. also [56], ψ ( l ) (cid:18) (cid:19) = ( − l +1 l ! (cid:0) l +1 − (cid:1) ζ l +1 − ψ ( l ) (cid:18) (cid:19) . (5.43)For even values of l , ψ ( l ) (1 / 3) and ψ ( l ) (2 / 3) are linear in π l +1 √ ζ l +1 , with ψ (2) (cid:18) (cid:19) = − √ π − ζ (5.44) ψ (2) (cid:18) (cid:19) = 49 √ π − ζ (5.45) ψ (4) (cid:18) (cid:19) = − √ π − ζ (5.46) ψ (4) (cid:18) (cid:19) = 163 √ π − ζ , etc. (5.47) Special examples were also considered in [53]. l one new basis element due to ψ (2 l +1) (1 / 3) contributes.The two values at argument 1 / / ψ (2 l − (cid:18) (cid:19) = 4 l − l (cid:2) π l (2 l − | B l | + 2(2 l )! β D (2 l ) (cid:3) (5.48) ψ (2 l − (cid:18) (cid:19) = 4 l − l (cid:2) π l (2 l − | B l | − l )! β D (2 l ) (cid:3) (5.49) ψ (2 l ) (cid:18) (cid:19) = − l − (cid:2) π l +1 | E l | + 2(2 l )!(2 l +1 − ζ l +1 (cid:3) (5.50) ψ (2 l ) (cid:18) (cid:19) = +2 l − (cid:2) π l +1 | E l | − l )!(2 l +1 − ζ l +1 (cid:3) . (5.51)Here, E l denote the Euler numbers [58–61]. They are generated by2exp( x ) + exp( − x ) = ∞ X n =0 E n n ! x n , E n +1 = 0 for n > . (5.52) β D is the Dirichlet β –function [52, 62] β D ( l ) = ∞ X k =0 ( − k (2 k + 1) l = Ti l (1) , l ∈ N + , (5.53)which is also given by the inverse tangent integral Ti l ( x ) [63, 64], being related to Clausenintegrals [65]. The special value for l = 2 yields Catalan’s constant [44] ∞ X k =0 ( − k (2 k + 1) = C . (5.54)At even values of l no new constants appear. Odd values contribute with Ti(2 l ).For z = 1 / , / , / , / ψ ( l ) (cid:18) (cid:19) = ( − l (cid:20) ψ ( l ) (cid:18) (cid:19) + π d l dz l cot( πz ) | z =2 / (cid:21) (5.55) ψ ( l ) (cid:18) (cid:19) = ( − l (cid:20) ψ ( l ) (cid:18) (cid:19) + π d l dz l cot( πz ) | z =1 / (cid:21) (5.56) l !( − l +1 ζ l +1 (cid:2) l +1 − (cid:3) = ψ ( l ) (cid:18) (cid:19) + ψ ( l ) (cid:18) (cid:19) + ψ ( l ) (cid:18) (cid:19) + ψ ( l ) (cid:18) (cid:19) (5.57)hold. For even values of l , ψ ( l ) (2 / 5) is thus dependent on ψ ( l ) (1 / z = 1 / , / ψ ( l ) (cid:18) (cid:19) = l !( − l +1 ζ l +1 (cid:2) l +1 − l +1 − l +1 + 1 (cid:3) − ψ ( l ) (cid:18) (cid:19) (5.58)holds. The reflection formula (5.39) relates ψ ( l ) (1 / 6) and ψ ( l ) (5 / 6) for even values of l to linearcombinations of π l +1 √ ζ l +1 . For odd l the ψ -values can be expressed by ψ ( l ) (1 / / l +1 ψ ( l ) (2 z ) = ψ ( l ) ( z ) + ψ ( l ) (cid:18) z + 12 (cid:19) (5.59)26mplies ψ ( l ) (cid:18) (cid:19) = (cid:0) l +1 + 1 (cid:1) ψ ( l ) (cid:18) (cid:19) + ( − l l ! (cid:0) l +1 − (cid:1) ζ l +1 , (5.60)and relates all values to ψ ( l ) (1 / z = 1 / , / , / , / ψ ( l ) (cid:18) (cid:19) + ψ ( l ) (cid:18) (cid:19) + ψ ( l ) (cid:18) (cid:19) + ψ ( l ) (cid:18) (cid:19) = l !( − l +1 ζ l +1 l +1 (cid:0) l +1 − (cid:1) . (5.61)Due to (5.41) and (5.39) the corresponding ψ -values can be expressed by ψ ( l ) (1 / ψ ( l ) (cid:18) (cid:19) = 2 l +1 ψ ( l ) (cid:18) (cid:19) − ( − l (cid:20) ψ ( l ) (cid:18) (cid:19) + π d l dz l cot( πz ) | z =1 / (cid:21) (5.62) ψ ( l ) (cid:18) (cid:19) = 2 l +1 ψ ( l ) (cid:18) (cid:19) − ψ ( l ) (cid:18) (cid:19) (5.63) ψ ( l ) (cid:18) (cid:19) = ( − l (cid:20) ψ ( l ) (cid:18) (cid:19) + π d l dz l cot( πz ) | z =1 / (cid:21) . (5.64)For z = 1 / , / , / , / 10 two reflection relations and ψ ( l ) (cid:18) (cid:19) + ψ ( l ) (cid:18) (cid:19) + ψ ( l ) (cid:18) (cid:19) + ψ ( l ) (cid:18) (cid:19) = l !( − l +1 ζ l +1 (cid:2) l +1 − l +1 − l +1 + 1 (cid:3) (5.65)hold. By the shift relation (5.59) one obtains ψ ( l ) (cid:18) (cid:19) = 2 l +1 ψ ( l ) (cid:18) (cid:19) − ψ ( l ) (cid:18) (cid:19) (5.66) ψ ( l ) (cid:18) (cid:19) = 2 l +1 ψ ( l ) (cid:18) (cid:19) − ψ ( l ) (cid:18) (cid:19) (5.67) ψ ( l ) (cid:18) (cid:19) = 2 l +1 ψ ( l ) (cid:18) (cid:19) − ψ ( l ) (cid:18) (cid:19) (5.68) ψ ( l ) (cid:18) (cid:19) = 2 l +1 ψ ( l ) (cid:18) (cid:19) − ψ ( l ) (cid:18) (cid:19) . (5.69)No new constants contribute.For z = 1 / , / , / , / 12 two reflection relations and ψ ( l ) (cid:18) (cid:19) + ψ ( l ) (cid:18) (cid:19) + ψ ( l ) (cid:18) (cid:19) + ψ ( l ) (cid:18) (cid:19) = l !( − l +1 ζ l +1 l +1 × (cid:0) l +1 − l +1 − l +1 + 1 (cid:1) (5.70)hold. All values can be expressed by ψ ( l ) (1 / ψ ( l ) (cid:18) (cid:19) = ( − l (cid:20) l +1 ψ ( l ) (cid:18) (cid:19) − ψ ( l ) (cid:18) (cid:19) + π d l dz l cot( πz ) | z =7 / (cid:21) (5.71) ψ ( l ) (cid:18) (cid:19) = 2 l +1 ψ ( l ) (cid:18) (cid:19) − ψ ( l ) (cid:18) (cid:19) (5.72) ψ ( l ) (cid:18) (cid:19) = ( − l (cid:20) ψ ( l ) (cid:18) (cid:19) + π d l dz l cot( πz ) | z =1 / (cid:21) , etc. (5.73)27q. (5.41) yields ψ ( l ) (cid:18) (cid:19) = 3 l +1 ψ ( l ) (cid:18) (cid:19) − ψ ( l ) (cid:18) (cid:19) − ψ ( l ) (cid:18) (cid:19) . (5.74)For odd values of l one obtains ψ ( l ) (cid:18) (cid:19) = 2 l ψ ( l ) (cid:18) (cid:19) + 12 (cid:20) l +1 ψ ( l ) (cid:18) (cid:19) − ψ ( l ) (cid:18) (cid:19) + π d l dz l cot( πz ) | z =7 / (cid:21) . (5.75)Eq. (5.70) does not imply a further relation. ψ ( l ) (1 / 12) contributes as new constant for evenvalues of l .For even values of l = 2 q the Hurwitz ζ -function (5.5) obeys the representation [66] ∞ X k =0 qk + 2 p − n = 1(2 q ) n ζ H (cid:18) n, p − q (cid:19) = 1 q q X k =1 (cid:20) C n (cid:18) kq (cid:19) cos (cid:18) (2 p − kπq (cid:19) + S n (cid:18) kq (cid:19) sin (cid:18) (2 p − kπq (cid:19)(cid:21) , (5.76)where C ν and S ν are represented by the Legendre χ -function [67] (5.77), χ ν ( z ) = 12 [Li ν ( z ) − Li ν ( − z )] (5.77)with C ν ( x ) = Re χ ν (exp( iπx )) S ν ( x ) = Im χ ν (exp( iπx )) . (5.78)For ν ∈ N , (5.78) can be represented by Euler polynomials [68] and powers of π , C n (cid:18) pq (cid:19) = ( − n n − π n E n − (cid:18) pq (cid:19) (5.79) S n +1 (cid:18) pq (cid:19) = ( − n n )! π n +1 E n (cid:18) pq (cid:19) , p, q ∈ N + , p ≤ q, (5.80)with 2 exp( xt )exp( t ) + 1 = ∞ X n =0 E n ( x ) t n n ! . (5.81)Since ψ ( m − (cid:18) l (cid:19) = ( − m l m ( m − ∞ X k =0 lk + 1) m , (5.82)in particular ψ ( l ) (1 / 8) and ψ ( l ) (1 / , l ∈ N + , l ≥ π and a second term ∝ Re ( Im )( χ l +1 ( r )) , r ∈ Q according to (5.76, 5.78). We corrected typos in Eq. (10) of Ref. [66]. 28n conclusion, the representation of the single sums of weight w ≥ and l ≤ ζ k +1 , ψ (2 k +1) (cid:18) (cid:19) , Ti k (1) , ψ ( k ) (cid:18) (cid:19) , ψ (2 k +1) (cid:18) (cid:19) , ψ ( k ) (cid:18) (cid:19) , ψ (2 k ) (cid:18) (cid:19) , k ∈ N + . (5.83)Finally we would like to comment on a relation in Ramanujan’s notebooks [69] Chapter 9,(11.3), which was claimed to involve a cyclotomic harmonic sum, G (1) = 18 ∞ X r =1 r r X s =1 l − H (1) = π ∞ X r =0 ( − r (4 r + 1) − π √ ∞ X r =0 r + 1) , (5.85)with G (1) = H (1) . (5.86)This relation is unfortunately incorrect, cf. [69], p. 257. G (1) may be given by the followingrepresentation in multiple zeta values introducing nested harmonic sums [70] G (1) = 716 σ , + 12 σ − , = − ζ − ζ ln (2) + 78 ζ ln(2) + 124 ln (2) + Li (cid:18) (cid:19) ≃ . H (1) is given by H (1) = − π (cid:26) (cid:20) π + 28 (cid:18) √ (cid:19) ζ (cid:21) + ψ (2) (cid:18) (cid:19)(cid:27) ≃ . . (5.88) As a further extension of the infinite cyclotomic harmonic sums [11] we consider the iteratedsummation of the terms (4.60). The corresponding sums diverge if the first indices have thepattern c i = 1 , s i = 1, (2.1). However, these divergences can be regulated by polynomials in σ and cyclotomic harmonic sums, which are convergent for N → ∞ , very similar to the caseof the usual harmonic sums [7, 8]. We study the relations given in Section 4.3 supplementedby those of the shuffle algebra (SH) of the cyclotomic harmonic polylogarithms (at argument x = 1), Eq. (3.21). In Table 3 we present the number of basis elements obtained applying therespective relations up to weight w = 6 . The representation of all sums were computed by meansof computer algebra in explicit form. We derive the cumulative basis, quoting only the newelements in the next weight. Up to w = 4 we derive also suitable integral representations overknown functions. One possible choice of basis elements is : w = 1: σ { , , } = σ (5.89) σ { , , − } = − ln(2) (5.90) σ { , , − } = − π eight N S A SH A + SH A + sh + H A + SH + H + H A + SH + H + H + M Table 3: Basis representations of the infinite cyclotomic harmonic sums over the alphabet { ( ± k /k, ( ± k / (2 k + 1) } after applying the stuffle (A), shuffle (SH) relations, their combination,and their application together with the three multiple argument relations (H , H , M), as far theselead not to new quantities. In the latter case we quote the cumulative number of basis elementsappearing at the new weight.w = 2: σ { , , − } = − C (5.92) w = 3: σ { , , } = ζ (5.93) σ { , , − } , { , , − } = π − π 48 + 12 Z dx √ xx + 1 Li ( x ) (5.94) σ { , , − } , { , , − } = − C ln(2) + Z dx 11 + x χ ( √ x ) √ x (5.95) w = 4: σ { , , − } , { , , } , { , , } , { , , } = − Li (cid:18) (cid:19) (5.96) σ { , , − } = − − iχ ( i ) (5.97) σ { , , − } , { , , − } = i (cid:16) − π (cid:17) χ ( i ) + 12 Z dx x + 1 χ ( √ x ) (5.98) σ { , , − } , { , , − } , { , , − } = − (cid:18) π ln(2) − ζ (cid:19) (cid:16) − π (cid:17) + 12 Z √ x x "(cid:18) Li ( − x ) + π (cid:19) ln(1 − x )+ 12 S , ( x ) − S , ( x ) − S , ( − x ) (5.99) σ { , , − } , { , , − } , { , , } = − Z dx √ x x (cid:2) ln (cid:0) − √ x (cid:1) − ln (cid:0) √ x (cid:1)(cid:3) Li ( x ) − ln(2) Z dx √ x x Li ( x ) + π 24 [ln(2) π − C ] (5.100) σ { , , − } , { , , − } , { , , } = − Z dx √ x x ln(1 − x )Li ( x ) − 12 [1 − ln(2)] Z dx √ x x Li ( x )30 π (cid:20) π (cid:20) − 12 ln(2) (cid:21) − C (cid:21) . (5.101)Here S , ( x ) denotes a Nielsen integral [71], S , ( x ) = 12 Z x dzz ln (1 − z ) . (5.102)At weight w = 5,6 we give only a few integral representations. They can in general beobtained form the Mellin transforms setting the kernel x N → 1. The following basis elementsare obtained : w = 5: σ { , , } = ζ , (5.103) σ { , , − } , { , , } , { , , } , { , , } , { , , } = − Li (cid:18) (cid:19) , (5.104) σ { , , − } , { , , − } = 7720 π − π + 12 Z dx √ x x Li ( x ) , (5.105) σ { , , } , { , , − } = − π 90 + π 360 + 12 Z dx √ x x Li ( − x ) , (5.106) σ { , , − } , { , , − } , σ { , , − } , { , , − } , { , , } , σ { , , − } , { , , − } , { , , − } , σ { , , } , { , , − } , { , , − } ,σ { , , − } , { , , − } , { , , } , σ { , , − } , { , , − } , { , , − } , { , , − } , σ { , , − } , { , , − } , { , , − } , { , , − } ,σ { , , − } , { , , − } , { , , } , { , , } , σ { , , − } , { , , − } , { , , } , { , , } . (5.107) w = 6: σ { , , − } , { , , − } = σ − , − , (5.108) σ { , , − } , { , , } , { , , } , { , , } , { , , } , { , , } = − Li (cid:18) (cid:19) , (5.109) σ { , , − } = − − iχ ( i ) , (5.110) σ { , , − } , { , , − } , { , , } , { , , } , { , , } , σ { , , − } , { , , − } , { , , } , { , , } , { , , } ,σ { , , − } , { , , − } , { , , − } , { , , − } , { , , } , σ { , , − } , { , , − } , { , , − } , { , , − } , { , , } ,σ { , , − } , { , , − } , { , , − } , { , , − } , { , , − } , σ { , , } , { , , − } , { , , − } , { , , } ,σ { , , − } , { , , − } , { , , } , { , , } , σ { , , − } , { , , − } , { , , } , { , , } , σ { , , − } , { , , − } , { , , − } , { , , − } ,σ { , , − } , { , , − } , { , , − } , { , , − } , σ { , , − } , { , , − } , { , , − } , { , , − } , σ { , , − } , { , , − } , { , , − } , { , , − } ,σ { , , } , { , , − } , { , , − } , σ { , , } , { , , − } , { , , − } , σ { , , } , { , , − } , { , , − } ,σ { , , − } , { , , − } , { , , } , σ { , , − } , { , , − } , { , , } , σ { , , − } , { , , − } , { , , } ,σ { , , − } , { , , − } , { , , } , σ { , , − } , { , , − } , { , , − } , σ { , , − } , { , , − } ,σ { , , − } , { , , − } . (5.111)Recently, infinite sums of a type proposed in [72, 73] were studied in [75], Eqs. (11a-c).They can be expressed in terms of sums studied in this Section : J ( r ) = 12 ∞ X k =0 [ ψ ( k + 1 / − ψ (1 / k + 1) r = σ { , ,r } − σ { , ,r +1 } + σ { , ,r +1 } , { , , } (5.112) For similar sums see [74]. ( r ) = 12 ∞ X k =0 [ ψ ( k + 1 / − ψ (1 / k ) r = 12 r σ { , ,r } , { , − , } (5.113) M ( r ) = 12 ∞ X k =0 [ ψ ( k + 1) + γ ](2 k + 1) r = 12 σ { , ,r } , { , , } . (5.114)Moreover, values of usual harmonic polylogarithms [26] at x = 1, R dt [Li p ( ± t ) − Li p ( ± / (1 ± t ),are discussed. In part these integrals refer to all three letters of the corresponding alphabet. Thecorresponding representations involve infinite harmonic sums of depth d > naturally, as e.g.for Z dx Li ( x )1 + x = H − , , , , (1) = − ζ ln(2) + σ − , − (5.115)and in similar cases of higher weight, see Ref. [8, 11, 26]. Let us now consider more cyclotomic letters. We study the sums up to weight w = 2 andcyclotomy l = 20 , based on the sets of the non-alternating and alternating sums using theletters ( ± k ( lk + m ) n , ≤ n ≤ , ≤ l ≤ , m < l . (5.116)We use the stuffle (quasi-shuffle) relations for the sums, the shuffle relations on the side of theassociated cyclotomic harmonic polylogarithms, and the multiple argument relations for thesesums, cf. Section 4.3. In some cases the latter request to include sums which are outside theabove pattern. In this case the corresponding relations are not accounted for. At w = 1 therespective numbers of basis elements is summarized in Table 4. l sums basis new basis sums Table 4: The number of the w = 1 cyclotomic harmonic sums (5.116) up to l = 20, the basiselements at fixed value of l, and the new basis elements in ascending sequence. The reflection relation (5.39) for the ψ -functions for x ↔ (1 − x ) implies that there are at most l + 1 basis elements. We showed that the use of the above analytic representations and theshuffle, stuffle, and multiple argument relations lead to the same number of basis elements for l ≤ w = 1 up to l = 6 are : σ { , , } , σ { , , − } , σ { , , − } , σ { , , } , σ { , , − } , σ { , , − } , σ { , , − } , σ { , , } , σ { , , − } ,σ { , , − } , σ { , , − } , σ { , , − } , (5.117) Relations between colored nested infinite harmonic sums have been investigated also in Refs. [76,77] recently. l = 6 are σ { , , } = − − σ { , , − } + 12 σ { , , } (5.118) σ { , , } = − − σ { , , − } − σ { , , − } + σ { , , } (5.119) σ { , , − } = 12 + 23 σ { , , − } + σ { , , − } (5.120) σ { , , } = − − σ { , , − } + 14 σ { , , } + 12 σ { , , − } (5.121) σ { , , } = − − σ { , , − } + 14 σ { , , } − σ { , , − } (5.122) σ { , , } = 15 σ { , , − } + σ { , , } − σ { , , − } (5.123) σ { , , } = − − σ { , , − } − σ { , , − } + σ { , , } (5.124) σ { , , } = − − σ { , , − } − σ { , , − } + σ { , , } − σ { , , − } (5.125) σ { , , − } = 712 + 45 σ { , , − } + σ { , , − } − σ { , , − } + σ { , , − } (5.126) σ { , , } = − σ { , , − } + 12 σ { , , − } + 12 σ { , , } (5.127) σ { , , } = − − σ { , , − } − σ { , , − } + 12 σ { , , } (5.128) σ { , , − } = 215 + 43 σ { , , − } − σ { , , − } , etc. (5.129)The remaining sums are related to those given in (5.117–5.129). The following counting relationsfor the basis elements were tested up to l = 700 using computer algebra methods. Let p, p i , q bepairwise distinct primes > , and let k, k i be positive integers. The number of basis elements at w = 1 and cyclotomy l are given by ϕ ( l ) = l + 1 , l = 1 or l = 2 k ( p − p k − + 2 , l = p k ϕ (cid:0) k − Q ni =1 p k i i (cid:1) − n − , l = 2 k Q ni =1 p k i i ( q − ϕ (cid:0)Q ni =1 p k i i (cid:1) − n ( q − − q + 3 , l = q Q ni =1 p k i i q ϕ (cid:0) q k − Q ni =1 p k i i (cid:1) − ( n + 2)( q − , l = q k Q ni =1 p k i i , k > . (5.130)Let us now consider the case w = 2 . Applying the relations given in Section 4.3 and theshuffle algebra of the cyclotomic harmonic polylogarithms at argument x = 1 the results givenin Table 5 are obtained for the number of basis elements. Again we solved the correspondinglinear systems using computer algebra methods and derived the representations for the dependentsums analytically.The number of the weight w = 2 infinite sums for cyclotomy l is N S = 2 l (2 l + 1) . (5.131)One may guess, based on the results for l ≤ , counting relations for the length of the baseslisted in Table 5. We found for all but the last column : N A ( l ) = l (2 l + 1) (5.132)33 N S SH A A + SH A + SH + H A + SH + H + H A + SH + H + H + M Number of basis elements of the w = 2 cyclotomic harmonic sums (5.116) up to cyclotomyl = 20 after applying the quasi-shuffle algebra of the sums (A), the shuffle algebra of the cyclotomicharmonic polylogarithms (SH), and the three multiple argument relations (H , H , M) of the sums.N SH ( l ) = (5 l + 3) l N A , SH ( l ) = 6 l + 1 − ( − l N A , SH , H ( l ) = 6 l − l + 7 − ( − l N A , SH , H , H ( l ) = 6 l − l + 3(1 − ( − l )8 . (5.136)The latter relation (5.136) has been derived prior to us by D. Broadhurst : N A , SH , H , H ( l ) = 34 l − l + if(modp(l,2) = 0 , , / 4) (5.137)and the corresponding generating function f ( x ) = (cid:20) x x (cid:21) − x ) = ∞ X l =0 a ( l ) x l . (5.138)We conjecture that in case of N A , SH , H , H ( l ) the ( M )-relations lead to a reduction of one in thebasis for l being a prime. Otherwise quite significant reductions are obtained for which we donot know an explicit counting relation. The corresponding sequence is also not recorded yet inthe data base [78]. We would like to thank D. Broadhurst for communicating this relation to us. Generalized Harmonic Sums at Roots of Unity In Section 5 we considered real representations for the infinite cyclotomic harmonic sums. Theseare related to the infinite generalized harmonic sums at the roots of unity. We definelim N →∞ S k ,...,k m ( x , ..., x m ; N ) = σ k ,...,k m ( x , ..., x m ) , (6.1)with S k ,...,k m ( x , ..., x m ; N ) a generalized harmonic sum (1.3), see also [13,14], and x j ∈ C n , n ≥ C n ∈ { e n | e nn = 1 , e n ∈ C } ; k = 1 for x = 1.We seek the relations between the sums of w = 1,2 . They can be expressed in terms ofpolylogarithms by : σ w ( x ) = Li w ( x ) , w ∈ N , w ≥ (6.2) σ ( x ) = Li ( x ) = − ln(1 − x ) (6.3) σ , ( x, y ) = Li ( x ) + 12 ln (1 − x ) + Li (cid:18) − x (1 − y )1 − x (cid:19) (6.4) σ , ( x, x ∗ ) = Li ( x ) + 12 ln (1 − x ) (6.5)Li w ( x ) = Li ∗ w ( x ∗ ) (6.6)and ∗ denotes complex conjugation. Furthermore, the symmetric combination σ , ( x, y ) + σ , ( y, x ) is given by [13, 14] σ , ( x, y ) + σ , ( y, x ) = ln(1 − x ) ln(1 − y ) + Li ( xy ) . (6.7)Knowing the representations at w = 1 and the dilogarithms at the corresponding roots of unityone term σ , ( y, x ) may be expressed by (6.7).In analyzing the functions σ k ,...,k m ( x , ..., x m ) with x lk = 1 one may use the real representa-tions of nested cyclotomic sums for N → ∞ , accounting for the respective sub-cycles at d, d | l and complex conjugation.An example is given by n e k (cid:12)(cid:12) k =1 o ≡ (cid:8) e , e , e , e , e , e , e ∗ , e ∗ , e ∗ , e ∗ , e ∗ , e (cid:9) . (6.8)Here we labeled the elements occurring in sub-cycles accordingly.The polylogarithms Li ( e kl ) and Li ( e kl ) obey : Im (cid:2) Li ( e kl ) (cid:3) = l − k l π (6.9) Re (cid:2) Li ( e kl ) (cid:3) = 6 k ( k − l ) + l l π . (6.10)More generally, PSLQ [79] tests let conjecture that Im (cid:2) Li n ( e kl ) (cid:3) = r n,l,k π n for n odd and Re (cid:2) Li n ( e kl ) (cid:3) = r n,l,k π n for n odd with r n,l,k ∈ Q .Now we extend Proposition 2.3 of Ref. [19], where we consider generalized harmonic sums S k ,...,k m ( x , ..., x m ; N ) with N ∈ N , k i ∈ N + , x i ∈ C , | x i | ≤ 1. Let l ∈ N + and y li = x i (6.11)then S k ,...,k m ( x , ..., x m ; N ) = m Y j =1 l k j − X y li = x i S k ,...,k m ( y , ..., y m ; lN ) . (6.12)35ere the sum is over the l th roots of x i for i ∈ [1 , m ]. Eq. (6.12) is called Distribution Relation .It follows from the Vieta’s theorem [80] for (6.11) and properties of symmetric polynomials [81].Eq. (6.12) contains the well–known duplication relation, cf. Eq. (2.15) [11].If also x = 1 for k = 1 the limit σ k ,...,k m ( x , ..., x m ) = lim N →∞ S k ,...,k m ( x , ..., x m ; N ) (6.13)exists. One may apply (6.12,6.13) to roots of unity x i and y j , i.e. x i = exp(2 πin i /m i ) and y jk = exp(2 πikn i / ( m i l )) , k = 1 ... ( l − , n i , m i ∈ N + . Let us now consider the cases w = 1, 2 inmore detail. w = 1 The first element is real and occurs at l = 2 Li ( e ) = − ln(2) , (6.14)representing the simplest alternating multiple zeta value, cf. e.g. [11]. At l = 3 we get thecomplex conjugate numbers Li ( e ) = − 12 ln(3) + πi ( e ) = − 12 ln(3) − πi . (6.16)Due to (6.9), ln(3) and iπ are considered as basis elements from this level on. For all highervalues of l one thus needs only to consider the real part of the w = 1 sums and one may workwith the real representations given in Section 5. Let us consider the exampleLi ( e ) = − ln(1 − e )= e ∞ X k =1 k − e ∞ X k =1 k − e ∗ ∞ X k =1 k − e ∞ X k =1 k = e ∞ X k =1 (cid:18) k − − k (cid:19) + e ∞ X k =1 (cid:18) k − − k (cid:19) + e ∗ ∞ X k =1 (cid:18) k − − k (cid:19) . (6.17)Eq. (6.17) follows from n − X k =1 e kn = 0 . (6.18)The type of sums occurring in (6.17) leads to digamma-functions and one may use their relationsgiven before to find the corresponding basis representations. The first terms are given by :Li ( e ) = − 12 ln(2) + πi ( e ) = Li ∗ ( e ) (6.20)Li ( e ) = 12 ln √ ! − 14 ln(5) + i π (6.21)36i ( e ) = 12 ln √ − ! − 14 ln(5) + i π (6.22) Re (Li ( e )) + Re (Li ( e )) = − 12 ln(5) (6.23)Li ( e ) = πi Re (Li ( e )) = − 14 ln(2) − 12 ln( √ − 1) (6.25) Re (Li ( e )) = 12 ln(2) − ln( √ − . (6.26)In Table 6 we summarize the number of basis elements. l basis Ref. [82] newelements .Table 6: The number of the basis elements spanning the w=1 cyclotomic harmonic polylogarithmsat l th roots of unity up to 20. At cyclotomy l = 9 we find one basis element less than reported in [82]. The new elementscontributing at the respective level of cyclotomy for l ≤ are : l = ln(2) (6.27) l = ln(3) , π (6.28) l = − (6.29) l = (Li ( e )) , Re (Li ( e )) (6.30) l = − (6.31) l = (Li ( e k )) (cid:12)(cid:12) k =1 (6.32) l = (Li ( e )) (6.33) l = (Li ( e )) , Re (Li ( e )) (6.34) l = − (6.35) l = 11 Re (Li ( e k )) (cid:12)(cid:12) k =1 (6.36) l = 12 Re (Li ( e )) (6.37) l = 13 Re (Li ( e )) | k =1 (6.38) l = − (6.39) l = 15 Re (Li ( e )) , Re (Li ( e )) (6.40) l = 16 Re (Li ( e )) , Re (Li ( e )) (6.41) l = 17 Re (Li ( e )) | k =1 (6.42) l = − (6.43) l = 19 Re (Li ( e )) | k =1 (6.44) l = 20 Re (Li ( e )) , Re (Li ( e )) . (6.45)37 .2 w = 2 We first consider the relations for Li ( e kn ). The following well–known representations for thefunction holds, cf. [83, 84] : Li (cid:0) e iθ (cid:1) = π ¯ B (cid:18) θ π (cid:19) + i Cl ( θ ) , (6.46)with Cl ( θ ) = ∞ X k =1 sin( kθ ) k (6.47)¯ B ( x ) = − π ∞ X k =1 cos(2 πkx ) k . (6.48)¯ B denotes the second modified Bernoulli polynomial. Due to (6.10) only the imaginary partshave to be considered. The number π occurs at w = 1, l = 3 only. The first terms are given by :Li ( e ) = π ( e ) = − π 12 (6.50) Im (Li ( e )) = √ (cid:20) ψ (1) (cid:18) (cid:19) − π (cid:21) (6.51) Im (Li ( e )) = C (6.52)Li ( e ) = Li ∗ ( e ) (6.53) Im (Li ( e )) = 5 √ (s √ (cid:20) ψ (1) (cid:18) (cid:19) − π (cid:18) √ (cid:19)(cid:21) + s − √ (cid:20) ψ (1) (cid:18) (cid:19) − π (cid:18) − √ (cid:19)(cid:21)) (6.54) Im (Li ( e )) = i √ (s − √ (cid:20) ψ (1) (cid:18) (cid:19) − π (cid:18) √ (cid:19)(cid:21) − s √ (cid:20) ψ (1) (cid:18) (cid:19) − π (cid:18) − √ (cid:19)(cid:21)) (6.55)Li ( e ) = Li ∗ ( e ) (6.56)Li ( e ) = Li ∗ ( e ) (6.57) Im (Li ( e )) = 32 Im (Li ( e )) (6.58)Li ( e ) = Li ∗ ( e ) (6.59) Im Li ( e ) = √ ψ ′ (cid:18) (cid:19) + 14 (cid:16) − √ (cid:17) C − (cid:16) √ (cid:17) π (6.60) Im Li ( e ) = √ ψ ′ (cid:18) (cid:19) + 23 C − √ π . (6.61)38he new basis elements spanning the dilogarithms of the l th roots of unity for l ≤ are : l = , − (6.62) l = (Li ( e )) (6.63) l = C (6.64) l = (Li ( e )) , Im (Li ( e )) (6.65) l = − (6.66) l = (Li ( e k )) (cid:12)(cid:12) k =1 (6.67) l = (Li ( e )) (6.68) l = (Li ( e )) , Im (Li ( e )) (6.69) l = − (6.70) l = 11 Im (Li ( e k )) (cid:12)(cid:12) k =1 (6.71) l = 12 Im (Li ( e )) (6.72) l = 13 Im (Li ( e )) | k =1 (6.73) l = − (6.74) l = 15 Im (Li ( e )) (6.75) l = 16 Im (Li ( e )) , Im (Li ( e )) (6.76) l = 17 Im (Li ( e )) | k =1 (6.77) l = − (6.78) l = 19 Im (Li ( e )) | k =1 (6.79) l = 20 Im (Li ( e )) . (6.80)Let us now turn to all convergent sums σ , ( x, y ), x, y ∈ C n . These sums belong to cyclotomy l if x = e k l , y = e k l and k , k ∈ N + , k < l, k < l .We consider first the case x = 1 , y = 1, σ , ( x, 1) = Li ( x ) + 12 Li ( x )= 12 Re (Li ( x )) + (cid:18) r − r (cid:19) π + i [ r π Re (Li ( x )) + Im (Li ( x ))] , (6.81)with r = Im ( − ln(1 − x )) (6.82) r = Re (Li ( x )) . (6.83)Including the basis elements of w = 1 up to l , no new basis element is obtained. Furthermore, σ , ( e , x ) = − π + 12 ln (2) + Li (cid:18) − x (cid:19) (6.84) σ , ( x, e ) = − Li ( x ) ln(2) + 12 Li ( x ) − Li ( x ) + 12 (cid:2) π + ln (2) (cid:3) − Li (cid:18) − x (cid:19) (6.85)hold. Li ((1 + x ) / 2) may yield a new basis element. In some cases besides x also − x is elementof the the cycle of the roots of unity which have to be considered. Here, however, Li ((1 + x ) / (cid:18) x (cid:19) = − Li (cid:18) − x (cid:19) + π − Li ( x )Li ( − x ) − ln (2) − ln(2) [Li ( x ) + Li ( − x )] . (6.86)Also the elements x = e kn and e − x ≡ − x occur in the cycles, for whichLi (1 − e kn ) = − Li ( e kn ) + 2 πi kn Li ( e kn ) + π l = 2 one obtains σ , ( e , 1) = − π + 12 ln (2) . (6.88)Because of (6.50) two basis elements contribute. If a corresponding special number has occuredalready at w = 1 at the same value of l , it is not counted as new. π occurs at w = 2, l = 1 ,unlike for w = 1 where it contributes first at l = 3 . No new basis element occurs at l=2 . For w= 3,4 the new basis elements occur for Li ( e kn ) only. We apply the relations, cf. e.g. [64],Li (cid:18) 11 + x (cid:19) = Li ( − x ) + ln( x ) ln(1 + x ) − 12 ln (1 + x ) + ζ (6.89)Li (cid:18) x x (cid:19) = − Li ( − x ) − 12 ln (1 + x ) . (6.90)At l = 5 the above relations lead to corresponding reductions and the two functionsLi (cid:18) − e (1 − e )1 − e (cid:19) , Li (cid:18) − e e (cid:19) (6.91)remain. For the first function the identityLi (cid:18) − e (1 − e )1 − e (cid:19) = Li (cid:18) e e (cid:19) = − Li ( − e ) − 12 ln (1 + e ) (6.92)holds, through which the corresponding sum σ , ( e , e ) can be expressed by σ , ( e , e ) = 12 Li ∗ ( e ) − Li ∗ ( e ) − (cid:0) Li ( e ) (cid:1) ∗ + Li ( e ) ∗ Li ( e ) ∗ . (6.93)The second dilogarithm can be transformed in the following way :Li (cid:18) − e e (cid:19) = − Li (cid:18) e + e e (cid:19) − ln (cid:18) − e e (cid:19) ln (cid:18) e + e e (cid:19) + ζ . (6.94)Furthermore, Li (cid:18) e + e e (cid:19) = Li ( − e ) (6.95)holds, through whichLi (cid:18) − e e (cid:19) = − 12 Li ( e ) + Li ∗ ( e ) + 19150 π − iπ (cid:2) Li ( e ) − Li ( e ) (cid:3) (6.96)40s obtained.The representations at w = 6 were given in [20], with Li (1 / 2) the new basis element. TheClausen function Cl ( π/ 3) used there is given byCl n ( x ) = (cid:26) i [Li n (exp( − ix )) − Li n (exp( ix ))] , n even [Li n (exp( − ix )) + Li n (exp( ix ))] , n odd (6.97)Cl (cid:16) π (cid:17) = Ls (0)2 (cid:16) π (cid:17) = 32 Im (Li ( e )) (6.98)with Ls ( k ) j ( θ ) = − Z θ dtt k ln j − k − (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (6.99)cf. [85]. In Table 7 we summarize the number of basis elements found at w = 2 using the relationsin Section 4.3, the distribution relations (6.12, 6.13), and the relations for dilogarithms givenabove. We also list the number of basis elements for the class of dilogarithms at roots of unity,and in both cases the number of new elements beyond those being obtained at w = 1 at thesame value of l . l basis new basis Ref. [82] newelements .Table 7: The number of the basis elements spanning the dilogarithms resp. w = 2 cyclotomicharmonic sums at l th roots of unity up to 20. At w = 2 the respective new basis elements are : l = π (6.100) l = − (6.101) l = (Li ( e )) (6.102) l = C (6.103) l = (Li ( e )) , Im (Li ( e )) (6.104) l = Li (cid:18) (cid:19) (6.105) l = (Li ( e k )) (cid:12)(cid:12) k =1 , σ , ( e , e ) (6.106) l = (Li ( e )) , σ , ( e , e ) , σ , ( e , e ) (6.107) l = (Li ( e )) , Im (Li ( e )) , σ , ( e , e ) , σ , ( e , e ) , σ , ( e , e ) (6.108) l = σ , ( e , e ) , σ , ( e , e ) , σ , ( e , e ) , σ , ( e , e ) (6.109) l = 11 Im (Li ( e k )) (cid:12)(cid:12) k =1 , σ , ( e , e k ) (cid:12)(cid:12) k =2 , σ , ( e , e k ) (cid:12)(cid:12) k =3 , etc. (6.110)We mention that counting relations for majorants of the motivic numbers, which are claimed tobe related to the bases of the sums σ k ,...,k m ( x , ..., x m ) , x j ∈ C n , were given in [77]. The dimension41f the respective Q -vector space is majorized by the expansion coefficients D n ( w ) of G w ( x ) = ∞ X n =0 D n ( w ) x n (6.111)with G ( x ) = 11 − x − x G ( x ) = 11 − x − x G k ( x ) = 11 − [ ϕ ( k ) + ν ] x − ( ν − x , k ≥ . (6.112)Here, ϕ ( k ) denotes Euler’s totient function [24] and ν is the number of prime factors of k . In evaluating massive 3-loop integrals with local operator insertions nested sums occur, contain-ing denominators which belong to the class of harmonic sums generated by cyclotomic polyno-mials. To simplify the computations, the relations between these quantities have to be knowand used in computer algebra codes such as Sigma [17] and HarmonicSums [27] to allow for anefficient reduction in the corresponding summation problem. In integrals of similar structure weexpect even more general terms (1.5) to occur.The usual harmonic sums [7, 8] and polylogarithms [26] were thus generalized to cover thenewly occurring structures. We started with the harmonic polylogarithms extending the usualalphabet of denominators ranging to Φ ( x ) to general cyclotomic polynomials Φ n ( x ) , n ≥ 3. Thecyclotomic harmonic polylogarithms form a shuffle algebra. They have support x ∈ [0 , 1] andone may define a Mellin transform, usually of argument kN , with k, N ∈ N + which span thefinite cyclotomic harmonic sums (2.1) together with special values as the cyclotomic harmonicsums in the limit N → ∞ .The cyclotomic harmonic sums are meromorphic functions with poles at the non-positiveintegers. They obey recurrence relations in terms of sums of lower depth and one may deriveanalytical asymptotic representations. In this way the cyclotomic harmonic sums are analyticallycontinued from integer values of the sum index N to N ∈ C . The cyclotomic harmonic sums forma quasi-shuffle algebra, cf. [9]. After the analytic continuation they obey differentiation relations,accounting for their values at N → ∞ . Furthermore, three multiple argument relations apply.Using these relations one may reduce the number of cyclotomic harmonic sums vastly growingwith the weight to smaller bases. We study the case of the sums following from iteration of thedenominators (4.60) up to weight w = 5 . Corresponding counting relations for the number ofbasis elements are obtained.The values of the cyclotomic harmonic sums for N → ∞ , resp. the values of the cyclotomicharmonic polylogarithms at x = 1, are of special interest. They are linearly related to the infinitenested harmonic sums with roots of unity as numerator weight factors. Already in case of thesingle infinite cyclotomic harmonic sums a large number of new constants contribute beyondthose know in the case of multiple zeta values and Euler-Zagier values [11]. The quasi-shuffleand multiple argument relations of the cyclotomic harmonic sums and the shuffle relations ofthe cyclotomic harmonic polylogarithms allow to derive basis representations induced by theserelations. We studied in this respect the case of the infinite cyclotomic harmonic sums based42n the iteration of the summands (4.60) to weight w = 6 and the sums of weight w = 1,2 forcyclotomy l ≤ . Using computer algebra methods the explicit representation of all infinitecyclotomic harmonic sums were derived as well. For wide classes of relations explicit countingrelations of the basis elements were given. The corresponding representations were derived withthe Mathematica -based computer algebra system HarmonicSums [27].The present investigations can be readily extended to cyclotomic harmonic sums and poly-logarithms of higher weight and cyclotomy, both in the case of finite values of N and for N → ∞ ,using the present methods. The requested computational time and storage resources grow ac-cordingly. This applies in particular to the derivation of the explicit representations of all sumsover the corresponding bases.We finally considered also the generalized harmonic sums of weight w = 1,2 at l th roots ofunity for 1 ≤ l ≤ 20. They obey dilogarithmic reprentations. Besides the shuffle and stufflerelations, they obey distribution relations and the known relations for dilogarithms. We usedthese relations to derive corresponding basis representations. Compared to the case of theinfinite cyclotomic harmonic sums these sums obey more symmetries. Thus at a given weightand cyclotomy l they are represented by a lower number of basis elements. We compared toresults in literature. Acknowledgment. We thank D. Broadhurst for providing us the relations (5.137) and (5.138).For discussions we would like to thank D. Broadhurst, F. Brown and D. Kreimer, and N.J.A.Sloan and M. Kauers for remarks on special issues. We would like to thank H. Kawamura forRef. [54]. This work has been supported in part by DFG Sonderforschungsbereich Transregio9, Computergest¨utzte Theoretische Teilchenphysik, by the Austrian Science Fund (FWF) grantP20162-N18, and by the EU Network LHCPHENOnet PITN-GA-2010-264564.43 Appendix In the following we summarize some technical aspects needed to represent expressions used inthe present paper.First we summarize some aspects on cyclotomic polynomials, [23]. We give the decomposi-tions of the polynomials x l + 1 , l ∈ N \{ } (A.1)in terms of cyclotomic polynomials for l ≤ 20 : x + 1 = Φ ( x )Φ ( x ) (A.2) x + 1 = Φ ( x )Φ ( x ) (A.3) x + 1 = Φ ( x )Φ ( x ) (A.4) x + 1 = Φ ( x )Φ ( x ) (A.5) x + 1 = Φ ( x )Φ ( x )Φ ( x ) (A.6) x + 1 = Φ ( x )Φ ( x ) (A.7) x + 1 = Φ ( x )Φ ( x ) (A.8) x + 1 = Φ ( x )Φ ( x ) (A.9) x + 1 = Φ ( x )Φ ( x ) (A.10) x + 1 = Φ ( x )Φ ( x ) (A.11) x + 1 = Φ ( x )Φ ( x )Φ ( x )Φ ( x ) (A.12) x + 1 = Φ ( x )Φ ( x ) (A.13) x + 1 = Φ ( x )Φ ( x )Φ ( x ) (A.14) x + 1 = Φ ( x )Φ ( x ) (A.15) x + 1 = Φ ( x )Φ ( x ) . (A.16)For odd values of n , Φ n ( x ) = Φ n ( − x ) (A.17)holds. The decomposition x k +1 − x − Y i Φ i ( x ) (A.18)results thus into x k +1 + 1 = Φ ( x ) Y i Φ i ( x ) . (A.19)From Φ n ( x ) | ( x n − 1) and Φ n ( x ) ( x n − 1) (A.20)it follows Φ n ( x ) | ( x n + 1) . (A.21)44f p is a prime and p | n then [86] Φ pn ( x ) = Φ n ( x p ) . (A.22)For n = 2 k , k ∈ N + it follows that all Φ k ( x ) are cyclotomic polynomials. In (A.6,A.12,A.14)more factors than just Φ n ( x ) occur. They originate due to power-rescaling, i.e., x + 1 x + 1 = y + 1 y + 1 = Φ ( y ) (A.23) x + 1 x + 1 = y + 1 y + 1 = Φ ( y ) . (A.24)Therefore, all the factors of ( x + 1) and ( x + 1) have to emerge in the decomposition, andsimilarly for other N with more non-trivial divisors. For N = 2 k · n where the integer n > x k · n + 1 x k + 1 = y n + 1 y + 1 = Φ n ( y ) . (A.25)The argument remains valid if n is a product of odd primes. Therefore the only cyclotomicpolynomials of the structure x a + 1 are those with a = 2 k , k ∈ N + .For a proper definition of the cyclotomic harmonic polynomials which appear in sum repre-sentations like (2.6) 1 x l ± , l ∈ N + (A.26)we perform partial fractioning. In the following we provide the corresponding expressions interms of the words f lk ( x ) forming the cyclotomic harmonic polylogarithms up to l = 6 :( x − − = f ( x ) (A.27)( x + 1) − = f ( x ) (A.28)( x − − = 12 (cid:2) f ( x ) − f ( x ) (cid:3) (A.29)( x + 1) − = f ( x ) (A.30)( x − − = 13 (cid:2) f ( x ) − f ( x ) − f ( x ) (cid:3) (A.31)( x + 1) − = 13 (cid:2) f ( x ) − f ( x ) + 2 f ( x ) (cid:3) (A.32)( x − − = 14 (cid:2) f ( x ) − f ( x ) − f ( x ) (cid:3) (A.33)( x + 1) − = f ( x ) (A.34)( x − − = 15 (cid:20) f ( x ) − f ( x ) − f ( x ) − f ( x ) − f ( x ) (cid:21) (A.35)( x + 1) − = 15 (cid:20) f ( x ) + 45 f ( x ) − f ( x ) + 25 f ( x ) − f ( x ) (cid:21) (A.36)( x − − = 16 (cid:2) f ( x ) − f ( x ) − f ( x ) − f ( x ) − f ( x ) + f ( x ) (cid:3) (A.37)( x + 1) − = 13 (cid:2) f ( x ) + 2 f ( x ) − f ( x ) (cid:3) etc. (A.38)45 Appendix Proof of Eq. (4.58). We proceed by induction on m. Let m = 1 : S { a ,b ,c } (2 n ) + S { a ,b , − c } (2 n ) = n X i =1 a i + b ) c + n X i =1 ( − i ( a i + b ) c = n X i =1 (cid:18) a i + b ) c + 1( a (2 i − 1) + b ) c (cid:19) + n X i =1 (cid:18) a i + b ) c − a (2 i − 1) + b ) c (cid:19) = 2 n X i =1 a i + b ) c = 2 S { a ,b ,c } ( n ) . In the following we use the abbreviation: A ( n ) := X S { a m ,b m , ± c m } ,..., { a ,b , ± c } ( n ) . Now we assume that (4.58) holds for m : X S { a m +1 ,b m +1 , ± c m +1 } ,..., { a ,b , ± c } (2 n )= n X i =1 a m +1 i + b m +1 ) c m +1 X S { a m ,b m , ± c m } ,..., { a ,b , ± c } ( i )+ n X i =1 ( − i ( a m +1 i + b m +1 ) c m +1 X S { a m ,b m , ± c m } ,..., { a ,b , ± c } ( i )= n X i =1 A ( i )( a m +1 i + b m +1 ) c m +1 + n X i =1 ( − i A ( i )( a m +1 i + b m +1 ) c m +1 = n X i =1 (cid:18) A (2 i )(2 a m +1 i + b m +1 ) c m +1 + A (2 i − i − a m +1 + b m +1 ) c m +1 (cid:19) + n X i =1 (cid:18) ( − i A (2 i )(2 a m +1 i + b m +1 ) c m +1 + ( − i − A (2 i − i − a m +1 + b m +1 ) c m +1 (cid:19) = 2 n X i =1 A (2 i )(2 a m +1 i + b m +1 ) c m +1 = 2 n X i =1 a m +1 i + b m +1 ) c m +1 m S { a m ,b m ,c m } ,..., { a ,b ,c } ( i )= 2 m +1 S { a m +1 ,b m +1 ,c m +1 } ,..., { a ,b ,c } ( n ) (cid:3) roof of Eq. (4.59). We proceed by induction on m. Let m = 1 : S { a ,b ,c } (2 n ) − S { a ,b , − c } (2 n ) = n X i =1 a i + b ) c + n X i =1 ( − i ( a i + b ) c = n X i =1 (cid:18) a i + b ) c + 1( a (2 i − 1) + b ) c (cid:19) − n X i =1 (cid:18) a i + b ) c − a (2 i − 1) + b ) c (cid:19) = 2 n X i =1 i − a + b ) c = 2 S { a ,b − a ,c } ( n ) . In the following we use the abbreviation: A ( n ) := X d m · · · d S { a m ,b m ,d m c m } ,..., { a ,b ,d c } ( n ) . Note that A (2 n − 1) = A (2 n ) − X d m · · · d d nm S { a m − ,b m − ,d m − c m − } ,..., { a ,b ,d c } (2 n )( a m n + b m ) c m = A (2 n ) − X d m − · · · d S { a m − ,b m − ,d m − c m − } ,..., { a ,b ,d c } (2 n )( a m n + b m ) c m + X d m − · · · d S { a m − ,b m − ,d m − c m − } ,..., { a ,b ,d c } (2 n )( a m n + b m ) c m = A (2 n ) . Now we assume that (4.59) holds for m : X d m +1 · · · d S { a m +1 ,b m +1 ,d m +1 c m +1 } ,..., { a ,b ,d c } (2 n )= n X i =1 a m +1 i + b m +1 ) c m +1 X d m · · · d S { a m ,b m ,d m c m } ,..., { a ,b ,d c } ( i ) − n X i =1 ( − i ( a m +1 i + b m +1 ) c m +1 X d m · · · d S { a m ,b m ,d m c m } ,..., { a ,b ,d c } ( i )= n X i =1 A ( i )( a m +1 i + b m +1 ) c m +1 − n X i =1 ( − i A ( i )( a m +1 i + b m +1 ) c m +1 = n X i =1 (cid:18) A (2 i )(2 a m +1 i + b m +1 ) c m +1 + A (2 i − i − a m +1 + b m +1 ) c m +1 (cid:19) − n X i =1 (cid:18) ( − i A (2 i )(2 a m +1 i + b m +1 ) c m +1 + ( − i − A (2 i − i − a m +1 + b m +1 ) c m +1 (cid:19) = 2 n X i =1 A (2 i − i − a m +1 + b m +1 ) c m +1 = 2 n X i =1 A (2 i )((2 i − a m +1 + b m +1 ) c m +1 = 2 n X i =1 a m +1 i + b m +1 − a m +1 ) c m +1 m S { a m ,b m − a m ,c m } ,..., { a ,b − a ,c } ( n )47 2 m +1 S { a m +1 ,b m +1 − a m +1 ,c m +1 } ,..., { a ,b − a ,c } ( n ) (cid:3) Proof of Eq. (4.82–4.86). We start with C f , ~m ( x ). We consider integrals of the form Z x y ∞ X i =1 σ i y i + c j (2 i + c j ) a S n ( i ) dy = ∞ X i =1 σ i (2 i + c j ) a S n ( i ) Z x y i + c j − dy = ∞ X i =1 σ i (2 i + c j ) a S n ( i ) x i + c j i + c j dy = ∞ X i =1 σ i x i + c j (2 i + c j ) a +1 S n ( i ) . Summing over j yields the desired result.For C f , ~m ( x ) we consider the integrals Z x 11 + y ∞ X i =1 σ i y i + c j (2 i + c j ) a S n ( i ) dy = Z x ∞ X i =1 ( − i y i ∞ X i =0 σ i +1 y i + c j +2 (2 i + c j + 2) a S n ( i + 1) dy = Z x ∞ X i =1 i X k =0 ( − i − k y i − k σ k +1 y k + c j +2 (2 k + c j + 2) a S n ( k + 1) dy = Z x ∞ X i =1 ( − i +1 y i + c j +2 i X k =0 ( − σ ) k +1 (2 k + c j + 2) a S n ( k + 1) dy = ∞ X i =1 ( − i +1 y i + c j +3 i + c j + 3 i +1 X k =1 ( − σ ) k (2 k + c j ) a S n ( k )= ∞ X i =1 ( − i x i + c j +1 (2 i + c j + 1) S { ,c j , − σa } , n ( i ) . Summing over j yields the desired result. The case C f , ~m ( x ) follows analogously.For C f , ~m ( x ) we consider the integrals of the form Z x 11 + y ∞ X i =1 σ i y i + c j (2 i + c j ) a S n ( i ) dy = Z x − y − y ∞ X i =1 σ i y i + c j (2 i + c j ) a S n ( i ) dy = Z x − y ∞ X i =1 σ i y i + c j (2 i + c j ) a S n ( i ) dy − Z x y − y ∞ X i =1 σ i y i + c j (2 i + c j ) a S n ( i ) dy = Z x ∞ X i =0 y i ∞ X i =0 σ i +1 y i + c j +2 (2 i + c j + 2) a S n ( i + 1) dy Z x ∞ X i =0 y i +1 ∞ X i =0 σ i +1 y i + c j +2 (2 i + c j + 2) a S n ( i + 1) dy = Z x ∞ X i =0 i X k =0 y i − k σ k +1 y k + c j +2 (2 k + c j + 2) a S n ( k + 1) dy − Z x ∞ X i =0 i X k =0 y i − k +1 σ k +1 y k + c j +2 (2 k + c j + 2) a S n ( k + 1) dy = Z x ∞ X i =0 y i + c j +1 S { ,c j ,σa } , n ( i + 1) dy − Z x ∞ X i =0 y i + c j +3 S { ,c j ,σa } , n ( i + 1) dy = ∞ X i =1 x i + c j +1 (2 i + c j + 1) S { ,c j ,σa } , n ( i ) − ∞ X i =1 x i + c j +2 (2 i + c j + 2) S { ,c j ,σa } , n ( i ) . Summing over j yields the desired result. 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