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Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials

Jakob Ablinger, Johannes Blümlein, Carsten Schneider

arXiv:1105.6063v1math-phcs.SChep-phhep-thmath.AG

TL;DR

Higher-order massive Feynman-integral calculations require generalized nested harmonic sums, including analytic continuation beyond integer indices. The paper develops cyclotomic harmonic polylogarithms and sums, derives their algebraic relations and basis representations, and obtains explicit representations through weight 6 and cyclotomy 20 for selected cases.

  • Problem

    Higher-order massive calculations require generalized nested harmonic sums and methods for evaluating them beyond integer Mellin indices.

  • Method

    The paper constructs cyclotomic harmonic polylogarithms and sums, relates sums to Mellin transforms, and uses analytic continuation plus shuffle, quasi-shuffle, and multiple-argument relations.

  • Results

    Explicit basis representations were derived for infinite cyclotomic harmonic sums through weight 6 and for weight 1,2 sums with cyclotomy l ≤ 20.

  • Takeaways & Limitations

    The resulting algebraic and analytic framework supports reduction and continuation of cyclotomic harmonic sums arising in higher-order massive calculations.

  • Takeaways & Limitations

    Extending the methods to higher weight and cyclotomy requires correspondingly greater computational time and storage resources.

Abstract

from arXiv · show

The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré--iterated integrals including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of $N$ is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument $x=1$, resp., for the cyclotomic harmonic sums at $N \rightarrow \infty$, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle algebras and three multiple argument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight {\sf w = 1,2} sums up to cyclotomy {\sf l = 20}.

1 Introduction

Higher-order and massive Feynman-integral calculations require extensions of nested harmonic sums and their associated mathematical framework. The paper introduces cyclotomic harmonic polylogarithms and sums, establishes their Mellin-transform connection, and studies their algebraic, structural, and special-value relations.

  • Motivation: Higher-order calculations with finite-mass effects require generalizations of nested harmonic sums, including generalized harmonic sums and related fractional-term sums.These objects are expected in higher-order calculations across renormalizable Quantum Field Theories.
  • Representation: A real representation is chosen because it is practically important for fast polynomial operations in nested-summation problems.The representation is relevant to computations involving the displayed cyclotomic-type sums.
  • Analytic connections: The Mellin transform connects cyclotomic harmonic sums with cyclotomic harmonic polylogarithms, while special values at x = 1 and N →∞ extend multiple zeta and Euler-Zagier values.The paper investigates relations and representations among these three classes of quantities.
  • Paper organization: The paper studies shuffle, algebraic, and structural relations for cyclotomic harmonic polylogarithms and finite-N nested sums.The organization assigns these topics to Sections 3 and 4 after establishing the Mellin-transform connection in Section 2.

2 Basic Formalism

This section defines cyclotomic harmonic sums and their weights, then transforms them into linear combinations of Poincaré-iterated integrals. It also identifies the cyclotomic-polynomial structure underlying the denominators and notes a divergence condition for infinite sums.

  • Definitions: Cyclotomic harmonic sums use positive integer a_i and c_i, nonnegative integer b_i, signs s_i = ±1, and require a_i > b_i.Their weight is c_1 + ··· + c_l, and the parameters denote lists rather than sets; s_i may be generalized to nonzero real values.
  • Definitions: The infinite cyclotomic harmonic polylogarithms diverge when c_1 = 1 and s_1 = 1.
  • Integral representation: Integral representations of denominators and summation over k convert cyclotomic harmonic sums into linear combinations of Poincaré-iterated integrals.The procedure is applied from the innermost sum outward; subsequent summation proceeds directly when a_{l−1} divides a_l, otherwise integration variables are transformed.
  • Cyclotomic structure: The resulting polynomials decompose into cyclotomic-polynomial factors, with all factors dividing (x^a)^l − 1 or (−x^a)^l − 1.The case a = 1 yields Φ_1(x), while the other polynomial family is cyclotomic for a = 2^n and decomposes into cyclotomic products otherwise.

3 Cyclotomic Harmonic Polylogarithms

This section introduces cyclotomic harmonic polylogarithms as Poincaré iterated integrals over an alphabet built from cyclotomic polynomials and connects them to cyclotomic harmonic sums through Mellin transforms. It also develops their shuffle-algebra structure and recurrence-based evaluation, including reversible transformations implemented in HarmonicSums.

  • Cyclotomic alphabet: Poincaré iterated integrals over an alphabet involving cyclotomic-polynomial denominators generate the newly emerging cyclotomic harmonic sums.The alphabet extends that used for ordinary harmonic polylogarithms.
  • Algebraic structure: Cyclotomic harmonic polylogarithms form a shuffle algebra, with basis-element counts determined by the first Witt formula and tabulated by weight and alphabet size.The number of basic elements depends on the weight w and the number M of chosen letters.
  • Mellin representation: Cyclotomic harmonic sums are represented as Mellin transforms of cyclotomic harmonic polylogarithms after rewriting denominators as products of cyclotomic polynomials and applying partial fractioning.The construction also provides a general representation using a least common multiple of the relevant cyclotomic indices.
  • Mellin representation: The transformation between cyclotomic harmonic sums and Mellin transforms of cyclotomic harmonic polylogarithms is reversible for a specified alphabet and is implemented in the HarmonicSums package.The stated implementation covers sums with ai ∈ {1, 2}, bi ∈ {0, 1}, and ci ∈ N+.
  • Recurrence relations: Mellin transforms satisfy difference equations of order l in N, and corresponding initial values enable efficient computation and explicit finite-sum representations.For φ6(l, N), the special recurrence form yields such a finite-sum representation.

4 Cyclotomic Harmonic Sums · 4.1 The Single Sums

The paper extends finite nested harmonic sums to cyclotomic harmonic sums, deriving explicit representations, analytic continuation in N, and structural relations. These relations support basis representations, while higher-weight sums admit further cyclotomic-polylogarithmic forms.

  • 4 Cyclotomic Harmonic Sums: Cyclotomic harmonic sums extend finite nested harmonic sums generated by cyclotomic harmonic polylogarithms and are studied through algebraic, differential, and multiple-argument relations.The relations are used to represent the sums over suitable bases.
  • 4.1 The Single Sums: The single sums are explicitly represented for even or odd integer N, with specialized forms available when N belongs to the natural numbers.General N is needed because nested sums require increasingly many cases.
  • 4.1 The Single Sums: For weights up to w = 6, single sums can be expressed through Mellin transforms φk(l, kN) and associated recurrence-based representations.The transformation uses sum representations and initial values derived from recurrence relations.
  • 4.1 The Single Sums: The resulting representation uses a finite collection of harmonic sums and Mellin-transform quantities, including S1(jN), S−1(jN), and cyclotomic functions through φ12(6N).The listed basis elements include S−1(N), S−1(3N), S1(N) through S1(6N), and φ-functions at cyclotomic arguments.
  • 4.1 The Single Sums: The sequences generated by these sums form an algebraically independent basis over the ring generated by R(N)[(−1)N].Algebraic relations among constants were exploited in constructing the representation.
  • 4.1 The Single Sums: At weight w = 1, only the specified constants from R appear in the representations of the single cyclotomic harmonic sums.These constants are reduced using algebraic relations (5.118–5.129).
  • 4.1 The Single Sums: The functions Φk(l, N) and φk(0, N) are meromorphic in N with poles at −n, while φk(0, N) grows proportionally to ln(N) as N approaches infinity.The meromorphic representations follow from factorial-series expansions.
  • 4.1 The Single Sums: Asymptotic expansions and recursions enable analytic continuation by shifting or mapping φk(l, N) to sufficiently large complex N, with larger starting N required as k grows.The asymptotic formulas can be extended to higher inverse powers of N.

4.2 Cyclotomic harmonic polylogarithms at x = 1

This section develops representations of cyclotomic harmonic polylogarithms at x = 1, linking depth-one values and Mellin transforms to infinite cyclotomic sums. Integration by parts further expresses weighted infinite sums as polynomials in infinite cyclotomic harmonic sums.

  • The construction assumes a(x) = x^l/Φ_k(x), with l < deg(Φ_k(x)), and chooses the smallest n satisfying Φ_k(x)|(x^n−1).
  • Depth-one cyclotomic harmonic polylogarithms at x = 1 are represented as linear combinations of corresponding infinite cyclotomic sums.
  • Integration by parts is used to transform the relevant expression before deriving the x = 1 representations.
  • Mellin transforms of cyclotomic harmonic polylogarithms reduce to linear combinations of finite cyclotomic harmonic sums, making the associated weighted infinite sum a polynomial in infinite cyclotomic harmonic sums.

4.3 Relations of Cyclotomic Harmonic Sums

Cyclotomic harmonic sums satisfy differentiation, stuffle, multiple-argument, and duplication relations. Their differentiation follows from Mellin-transform representations and can be expressed polynomially using sums and cyclotomic harmonic polylogarithms at x = 1.

  • Stuffle relations: Products of cyclotomic harmonic sums satisfy stuffle relations induced by the quasi-shuffle algebra, with separate forms under specified denominator conditions.The relations are obtained by analyzing products of denominator terms and are collectively denoted (A).
  • Differentiation relations: Cyclotomic harmonic sums obey differentiation relations derived from their Mellin-transform representations as cyclotomic harmonic polylogarithms.Differentiation with respect to N is closely related to the original sum through differentiated asymptotic representations and recursions.
  • Differentiation relations: The derivative with respect to N becomes a polynomial expression in cyclotomic harmonic sums and cyclotomic harmonic polylogarithms evaluated at x = 1.Shuffle relations transform polylogarithmic expressions before applying the inverse Mellin transform.
  • Multiple argument relations: Repeated application of multiple-argument relations represents cyclotomic harmonic sums with argument kn in terms of sums with argument n.The resulting relations are denoted (M).
  • Duplication relations: The usual duplication relation extends to cyclotomic harmonic sums, and a second duplication relation yields algebraic relation classes (H1) and (H2).More general analogous relations arise for generalized cyclotomic harmonic sums.

4.4 Sums of Higher Depth and Weight

This section develops higher-depth cyclotomic harmonic sums, representing them through cyclotomic harmonic polylogarithms and reducing them to weight-dependent bases. It also gives counting relations for these bases and describes analytic continuation to complex N using Mellin-transform and asymptotic methods.

  • Construction: Cyclotomic harmonic sums arise by iterating summands and can be expressed through Mellin transforms of cyclotomic harmonic polylogarithms over alphabet (3.29).The sums are considered for N ≥ k ≥ 1 with positive indices.
  • Basis reduction: Relations from Section 4.2 represent cyclotomic harmonic sums over corresponding bases, with computer algebra revealing the basis-count pattern through weight w = 5.Table 2 records reductions using multiple argument relations, differentiation with respect to N, and algebraic relations.
  • Counting relations: The multiple argument relations M, H1, and H2 each yield the same number of basis sums when applied singly, as do the combinations H1M and H2M.Explicit formulas include ND(w) = 16 · 5^(w−2), NH1(w) = 4 · 5^(w−1) − 2^(w−1), and NDH1H2M(w) = 16 · 5^(w−2) − 3 · 2^(w−2).
  • Analytic continuation: Analytic continuation to complex N uses Mellin-transform representations related to factorial series, with algebraic reductions for sequential occurrences of c_i = 1 and s_i = 1.The exceptional case is c1 = 1, s1 = 1; divergent structures can be separated into convergent sums and factors.
  • Analytic continuation: For larger depths, continuation shifts parallel to the real axis use recurrence relations, while asymptotic expansions for N → ∞ with |arg(N)| < π can be derived to arbitrary precision.The required asymptotic representations are obtained from basis elements, with expansions involving powers of x and ln(x) when trailing zeroes occur.

5 Special Values

This section studies special values of cyclotomic harmonic polylogarithms at x = 1 and associated infinite cyclotomic harmonic sums, deriving their relations, regularizations, and basis representations. It also analyzes generalized sums and establishes basis-count consistency through weight 2 and cyclotomy 20.

  • Special values: Special values at x = 1 and N →∞ are investigated through relations and basis representations extending Euler-Zagier and multiple zeta values.These values enter relations among finite cyclotomic harmonic sums and Mellin transforms of cyclotomic harmonic polylogarithms.
  • Relations and regularization: Single cyclotomic sums are linearly related to colored harmonic sums, while analogous relations for nested sums are deferred to Section 6.The section also treats divergences and regularization for selected infinite sums.
  • Rational special values: For rational arguments, reflection, shift, and multiplication relations determine dependencies among polygamma and Hurwitz-zeta values, with new constants appearing only in specified parity and denominator cases.Examples include no new basis elements for odd cyclotomy l in one relation, while ψ^(2l+1)(1/3), odd Ti(2l), and ψ^(l)(1/12) for even l contribute new elements.
  • Infinite sums: Infinite cyclotomic sums with divergent index patterns can be regulated by polynomials in σ0 and convergent cyclotomic harmonic sums, using stuffle, shuffle, and multiple-argument relations.The resulting basis representations include higher-weight integral forms and explicit generalized nested-sum identities.
  • Basis representations: At most l + 1 basis elements are required, and analytic representations combined with shuffle, stuffle, and multiple-argument relations yield the same count for l ≤6 in both non-alternating and alternating cases.The study additionally covers weights w = 2 and cyclotomies l = 20.

6 Generalized Harmonic Sums at Roots of Unity

The section relates infinite generalized harmonic sums at roots of unity to infinite cyclotomic harmonic sums and develops their polylogarithmic representations. It derives relations and basis elements for weights 1 and 2 through cyclotomy 20.

  • Definitions: Infinite generalized harmonic sums converge as N→∞ to σ_k1,...,km(x1,...,xm) for root-of-unity arguments, with k1 ≠ 1 when x1 = 1.The arguments satisfy xj ∈ C_n, where C_n consists of nth roots of unity.
  • Polylogarithmic representations: For weight w ≥ 1, σ_w(x) equals Li_w(x), including σ_1(x) = Li_1(x) = −ln(1 − x).These formulas provide the basic polylogarithmic representation for the infinite sums.
  • Weight-two relations: The symmetric weight-two combination satisfies σ_1,1(x,y) + σ_1,1(y,x) = ln(1 − x)ln(1 − y) + Li_2(xy).Thus, once weight-one representations and the relevant dilogarithms are known, one σ_1,1 term can be obtained from the other.
  • Distribution relations: Distribution relations extend the harmonic-sum framework to roots of unity and include the well-known duplication relation.They follow from Vieta’s theorem and properties of symmetric polynomials.
  • Weight-one bases: At cyclotomy l = 9, the weight-one basis contains one fewer element than previously reported, while new basis elements are listed for l ≤ 20.For higher l, only real parts of weight-one sums need to be considered, using real representations from Section 5.

7 Conclusions

The paper generalizes harmonic sums and polylogarithms to cyclotomic structures, establishing their algebraic relations and analytic continuation. It derives special-value and basis representations, including weight 1,2 cases through cyclotomy 20, and outlines extensions to higher weight and cyclotomy.

  • Cyclotomic generalization: Cyclotomic harmonic polylogarithms extend the usual denominator alphabet to general cyclotomic polynomials, form a shuffle algebra, and generate finite cyclotomic harmonic sums through Mellin transforms.The functions have support on x ∈ [0,1], with Mellin-transform arguments kN for k,N ∈ N+.
  • Analytic structure: Cyclotomic harmonic sums are meromorphic with poles at non-positive integers and admit recurrence, asymptotic, differentiation, and analytic-continuation representations forming a quasi-shuffle algebra.These methods continue the sums from integer N to complex N.
  • Special values: Special values at N →∞ and x = 1 are linearly related to infinite nested harmonic sums with roots of unity and introduce constants beyond ordinary multiple zeta values.The analysis uses shuffle, stuffle, and multiple-argument relations.
  • Basis representations: Weight w = 6 iterations and weight w = 1,2 sums for cyclotomy l ≤20 received explicit basis representations and counting relations computed with HarmonicSums.The explicit representations cover all infinite cyclotomic harmonic sums considered.
  • Extensions and limitations: The methods extend to higher weight and cyclotomy for finite N and N →∞, although computational time and storage requirements increase accordingly.This affects explicit representations of all sums over the corresponding bases.
  • Roots of unity: Generalized harmonic sums at lth roots of unity for 1 ≤l ≤20 have dilogarithmic representations and basis reductions using shuffle, stuffle, distribution, and dilogarithm relations.These sums possess more symmetries than infinite cyclotomic harmonic sums at fixed weight and cyclotomy.

Appendix

The appendix collects technical ingredients for representing expressions in the paper, focusing on cyclotomic-polynomial decompositions and the construction of cyclotomic harmonic polylogarithms. It establishes the structure of factors of the form x^a + 1 and gives partial-fraction-based expressions up to cyclotomy l = 6.

  • The appendix summarizes technical aspects needed to represent expressions used throughout the paper.
  • Cyclotomic polynomials: Cyclotomic-polynomial decompositions are developed for cyclotomy l ≤ 20, including cases involving odd indices, prime divisors, and power rescaling.
  • Cyclotomic polynomials: The only cyclotomic polynomials of the form x^a + 1 are those with a = 2^k, k ∈ N+.
  • Cyclotomic harmonic polylogarithms: Partial fractioning defines the cyclotomic harmonic polynomials through words f_l^k(x), with corresponding cyclotomic harmonic polylogarithm expressions provided up to l = 6.

B Appendix

The appendix proves Eqs. (4.58), (4.59), and (4.82–4.86) through induction, abbreviations, and integral manipulations. The derivations conclude by summing over j and treating analogous cases.

  • Proofs of Eqs. (4.58) and (4.59): Eqs. (4.58) and (4.59) are proved by induction on m, beginning with m = 1 and then assuming the result for m.The proofs introduce abbreviations for the nested sums before carrying out the induction step.
  • Proofs of Eqs. (4.82–4.86): The proof of Eqs. (4.82–4.86) considers integrals involving shifted factors such as (2i + cj)aSn (i) and (2i + cj + 2)aSn (i + 1)dy.Several equivalent integral forms are used for the relevant indexed sums.
  • Proofs of Eqs. (4.82–4.86): Summing over j yields the desired result for the considered cases, including the cases Cf1 and Cf0.The remaining case 4,⃗m(x) is stated to follow analogously.
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