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Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
Jakob Ablinger, Johannes Blümlein, Carsten Schneider
TL;DR
Generalized harmonic sums arise in higher-order QCD and related calculations, but their broader weights and analytic properties require systematic treatment. The paper develops Mellin-based connections to generalized polylogarithms, continuation and expansion algorithms, and algebraic relations, with implementations in HarmonicSums. It also establishes that Mellin-transform representations expressed with S-sums at integer values extend to the complex plane.
Problem
Generalized harmonic sums with weights beyond ±1 arise in higher-order calculations, while their analytic and algorithmic properties need to be developed to a level comparable with harmonic sums.
Method
The paper develops Mellin and inverse Mellin transforms, generalized polylogarithm representations, asymptotic and series-expansion procedures, differentiation and argument relations, and implementations in HarmonicSums.
Results
Mellin-transform representations in terms of S-sums at integer values extend to the complex plane, while the paper provides algorithms for continuation, expansions, transformations, and relations.
Takeaways & Limitations
S-sums and generalized harmonic polylogarithms form mutually transformable representations supporting analytic continuation, compactification, and calculations involving infinite S-sums.
Takeaways & Limitations
Analytic continuation is not available for sums containing factors such as (−1)^n, and the treatment is limited to convergent combinations in the cited QCD application.
Abstract
from arXiv · showhide
In recent three--loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the so-called generalized harmonic sums (in short $S$-sums) arise. They are characterized by rational (or real) numerator weights also different from $\pm 1$. In this article we explore the algorithmic and analytic properties of these sums systematically. We work out the Mellin and inverse Mellin transform which connects the sums under consideration with the associated Poincaré iterated integrals, also called generalized harmonic polylogarithms. In this regard, we obtain explicit analytic continuations by means of asymptotic expansions of the $S$-sums which started to occur frequently in current QCD calculations. In addition, we derive algebraic and structural relations, like differentiation w.r.t. the external summation index and different multi-argument relations, for the compactification of $S$-sum expressions. Finally, we calculate algebraic relations for infinite $S$-sums, or equivalently for generalized harmonic polylogarithms evaluated at special values. The corresponding algorithms and relations are encoded in the computer algebra package {\tt HarmonicSums}.
1 Introduction
The paper addresses generalized harmonic sums arising in higher-order QCD and related calculations by extending harmonic-sum technologies to broader weights and analytic structures. It develops transformations, continuation methods, relations, and implementations connecting S-sums with generalized polylogarithms.
- Generalized harmonic sums arise in higher-order calculations, including QCD, combinatorics, statistics, and number theory.
- S-sums extend harmonic sums through real or rational numerator weights beyond ±1, while encompassing harmonic sums, multiple zeta values, and multiple polylogarithms as special cases.
- The paper connects S-sums and generalized harmonic polylogarithms through Mellin and inverse Mellin transforms, enabling analytic continuation and asymptotic expansions.
- It derives differentiation, algebraic, structural, duplication, shuffle, and duality relations to compactify finite and infinite S-sum expressions.
- The algorithms are implemented in the Mathematica package HarmonicSums.
2 Basic properties of S-Sums and generalized polylogarithms
This section defines S-sums and generalized polylogarithms, establishes their convergence and integral representations, and develops their quasi-shuffle and shuffle algebraic properties. These representations provide the basic framework for transformations between nested sums and iterated integrals.
- S-sums are nested sums with positive integer indices and nonzero field-valued weights, with the weight equal to the sum of their indices.
- S-sums converge absolutely at infinity exactly under conditions on the first weight and successive products of numerator weights.
- Products of S-sums with a common upper limit obey a quasi-shuffle relation that reduces products to linear combinations of lower-depth S-sums.
- Each S-sum admits an indefinite nested-integral representation, and sending the summation limit to infinity recovers multiple polylogarithms.
- Generalized polylogarithms are iterated integrals with real indices, whose length defines their weight and whose products satisfy shuffle relations.
- For arguments beyond the basic analytic range, trailing-zero extraction can fail because the resulting polylogarithms may be undefined or divergent.
3 Identities between Multiple Polylogarithms of Related Arguments
The section develops recursive identities that transform generalized polylogarithms among shifted, reflected, scaled, and sign-reversed arguments. Shuffle linearization and lower-weight recursion control the resulting expressions and their analytic domains.
- Argument transformations rewrite generalized polylogarithms at shifted or reflected arguments in terms of values at simpler arguments and evaluations at constants.
- The transformations are subject to domain conditions ensuring that the original and transformed generalized polylogarithms are defined.
- The recursive construction proceeds by weight, applying known lower-weight transformations and then linearizing products with the shuffle product.
- For transformations of Hm1,...,mw(b−x), only one resulting polylogarithm at argument x retains the full weight; the other x-dependent terms have lower weights.
- Scaling and sign-reversal lemmas extend the transformation framework to arguments k·x and −x under corresponding definition conditions.
4 Power Series Expansion of Multiple Polylogarithms
The section generalizes series-expansion algorithms from harmonic to generalized polylogarithms and provides inverse constructions between series coefficients and S-sums. It also derives asymptotic behavior by transforming large arguments to expansions near zero.
- Generalized polylogarithms with trailing zeroes require separating a log(x) part from a log-free part before series expansion.
- For Hm1,...,mw(x) with mw≠0, the Taylor expansion on [0,q) has coefficients given explicitly in terms of S-sums.
- The construction is reversible: suitable infinite series can be converted into linear combinations of weighted generalized polylogarithms.
- Asymptotic behavior at large x is obtained by transforming the argument to y=1/x, expanding around y=0, and translating back.
5 Values of Multiple Polylogarithms Expressed by S-Sums at Infinity
The section develops an iterative algorithm for rewriting finite generalized polylogarithms at constants as S-sums at infinity, with a reversible transformation for the converse direction.
- The method handles trailing zeroes by extracting them when possible and rewriting powers of H0(c) iteratively.
- For suitable indices, power-series expansions evaluated at c convert generalized polylogarithms into S-sums.
- The remaining unresolved cases include certain polylogarithms with trailing zeroes at c > 1 and constrained index ranges.
- When direct rewriting fails, argument transformations reduce the evaluation point and simplify the negative indices until conversion becomes possible.
- Finite generalized polylogarithms can be rewritten in terms of S-sums at infinity.
- The reverse process rewrites finite S-sums at infinity as generalized polylogarithms, with a complete algorithm referenced elsewhere.
6 Analytic Continuation of S-sums
The section establishes analytic continuation of S-sums through Carlson-type integral representations. It treats positive and general real weights, including separate continuations for even and odd subsequences.
- Carlson’s theorem makes a C-type analytic continuation unique when two such functions agree at every nonnegative integer.
- Integral representations of S-sums with positive weights define C-type functions and therefore uniquely determine their analytic continuations.
- The example S1,2,2 cannot be continued directly because an analytic continuation of (−1)^n does not exist within the required C-type class.
- For this example, separate even and odd subsequences can nevertheless be continued analytically.
- For arbitrary nonzero real weights, the transformed integrals f(2z) and f(2z + 1) are C-type.
- Consequently, even and odd S-sums have uniquely determined continuations through the corresponding integral representations at 2z and 2z + 1.
7 The Mellin Transform and Inverse Mellin Transform Between Multiple Polylogarithms and S-sums
The section develops Mellin and inverse Mellin transformations connecting generalized polylogarithms and S-sums, including analytic continuation and algorithmic conversion in both directions.
- Mellin-transform correspondence: Generalized polylogarithm Mellin transforms can be expressed using a subclass of S-sums and constants, while inverse Mellin transforms express S-sums through generalized polylogarithms.The correspondence extends the harmonic-sum construction to generalized indices and weighted polylogarithms.
- Mellin-transform correspondence: The modified Mellin transform handles kernels such as 1/(a−x), whose ordinary Mellin integrals may diverge when a lies in (0,1).The transform is defined for generalized polylogarithms with specified index and parameter ranges.
- Analytic continuation: Integer-point representations extend to the complex plane through unique even and odd analytic continuations of C-type, justified using Carlson’s theorem.The constructed S-sum representation agrees with the analytically continued Mellin transform.
- Inverse Mellin transform: For the relevant subclass, properly weighted generalized polylogarithms and S-sums are related bijectively through the Mellin transform and its inverse.The inverse construction solves for linear combinations of Mellin transforms representing a given S-sum.
- Inverse Mellin transform: The inverse-transform procedure orders S-sums by weight and depth to identify the most complicated terms during reduction.A sum is more complicated when it is larger under the ordering by weight, then depth.
8 Differentiation of S-Sums
The section defines analytic differentiation of S-sums by extending even and odd analytic continuations, then derives two Mellin-based strategies and illustrates them with explicit generalized-polylogarithm expressions.
- Analytic continuation: Even and odd S-sums admit unique analytic continuations within the class of C-type functions, enabling differentiation with respect to the summation index.The differentiated expressions remain integral representations of S-sum expressions.
- Mellin-transform strategy: One differentiation strategy computes a Mellin transform, differentiates the analytically continued integrals, and applies the inverse Mellin transform to return to S-sums.With current technologies, this approach is restricted to ¯S-sums.
- Parity cases: The methods apply to even forms S_a(b;2n), odd forms S_a(b;2n+1), and ordinary S-sums whose outputs encode both parity cases.Substituting n→2n or n→2n+1 yields the corresponding even or odd differentiation.
- Generalization: The authors state that the proposed strategy works in general for S-sums after reducing the relevant integrals to Mellin transforms of generalized polylogarithms.The derivations establish finiteness and generalized-polylogarithm representations for the required integral terms.
- Example: For S_2(2;n), differentiation produces a combination of S_3(2;n), S_2(2;n), and generalized polylogarithm constants.The displayed result includes H_0(2), H_0,0,−1(1), H_0,0,1(1), and H_0,1,−1(1).
- Direct strategy: A second strategy differentiates the iterated-integral representation directly and then rewrites the resulting expressions in terms of S-sums and finite S-sums at infinity.The construction uses generalized polylogarithms and their Mellin transforms to complete the conversion.
9 Relations between S-Sums
The section develops algebraic, differential, duplication, and basis-reduction relations for S-sums, using their quasi-shuffle structure to organize dependent sums.
- Quasi-Shuffle or Stuffle Relations: S-sums inherit a quasi-shuffle algebra, equivalently forming a free polynomial algebra on Lyndon words over the indexed alphabet.This structure follows from transforming between S-sums and Z-sums and supports systematic relation generation.
- Quasi-Shuffle or Stuffle Relations: The quasi-shuffle relations determine algebraic dependencies and allow selected families of S-sums to be represented by smaller basis sets.For depth 2 on a four-letter alphabet, 1/4 q = 6 basis sums are identified, with the other 10 sums expressed through them.
- Differential Relations: Differentiation with respect to the upper summation limit produces additional S-sum relations and reduces the depth-2 basis to 3 sums in the example.The section gives an explicit derivative identity for S2(2; n) involving S3(2; n), products, and generalized harmonic polylogarithm values.
- Duplication Relations: Duplication relations are derived for S-sums evaluated at 2n by summing over all 2^m sign combinations, but they do not always reduce the basis.In the example, further reduction would require introducing new sums of the same depth and weight.
- Examples for Specific Index Sets: For depth 3, algebraic relations express 19 sums through 9 basis sums, while a related depth-3 family expresses 18 sums through 6 basis sums.These reductions may introduce additional lower-depth S-sums governed by quasi-shuffle relations.
- Examples for Specific Index Sets: For the specified weight-2 alphabet, algebraic and differential relations express 38 sums through 17 basis sums, whose algebraic independence is verified as sequences.The resulting S-sums form a polynomial subring of the ring of sequences.
10 Relations between S-Sums at Infinity
The paper develops relations among convergent infinite S-sums using stuffle, duplication, shuffle, and generalized-polylogarithm duality relations. These relations reduce expressions to smaller bases of constants for specified alphabets.
- Stuffle relations: Convergent S-sum stuffle relations remain valid at infinity and provide one class of algebraic reductions.
- Duplication relations: Duplication relations also remain valid for sums finite at infinity because the limits at n and 2n agree.
- Shuffle relations: Generalized-polylogarithm shuffle relations are converted into relations among convergent infinite S-sums.
- Duality relations: Duality and argument transformations of generalized polylogarithms generate additional relations among infinite S-sums.
- Basis reduction: The reduction strategy extends alphabets temporarily, then retains relations that return to the original finite index set.
- Basis reduction: PSLQ searches up to 2000 digits found no further relations for the reported constants, which define basis constants beyond multiple zeta values.
11 Asymptotic Expansion of S-Sums
This section develops asymptotic-expansion algorithms for S-sums by translating them into Mellin transforms of generalized polylogarithms and repeatedly integrating by parts. The complete treatment covers S-sums with weights bi in [−1,1], while extensions handle selected larger-weight cases.
- Expansion framework: Asymptotic expansions are represented through coefficient series with a remainder term omitted after the desired order.
- Scope: The general expansion problem is not completely solved, but the presented methods are sufficient for current QCD calculations.
- Generalized polylogarithms: Repeated integration by parts computes expansions of Mellin transforms involving generalized polylogarithms.
- S-sum algorithm: The resulting algorithm computes asymptotic expansions for S_a1,...,ak(b1,...,bk;n) with bi in [−1,1] and bi ≠ 0.
- S-sum algorithm: The algorithm recursively transforms difficult S-sums into less complicated sums and terminates through decreasing complexity.
- Extended sums: An extension covers selected S̄-sums with |b1| > 1 by using suitable integral representations, recursion, and repeated integration by parts.
12 Application from Quantum Chromodynamics
The QCD applications examine when generalized harmonic sums arise in massless and massive higher-loop calculations. Their contributions may cancel in final results, but they remain nonvanishing for several massive three-loop integral families.
- Massless Wilson coefficients: Massless three-loop Wilson-coefficient calculations can contain generalized sums in intermediate expressions, yet all such sums cancel in the final result.
- Massive Wilson coefficients: For a massive three-loop Wilson-coefficient contribution, the highest generated generalized-sum weight was w = 5, while all generalized sums canceled in the final result.
- Massive Wilson coefficients: Combining sums diagram by diagram reduces the number of nested sums but can produce fewer sums with much larger summands.
- Massive operator matrix elements: Generalized sums do not always vanish in massive three-loop operator matrix elements, contributing up to weight w = 4 for ladder diagrams with six massive propagators and Benz-type diagrams.
- Convergence: Physical expressions are restricted here to convergent combinations because individual diagram terms can diverge even when their combinations converge.
- Analytic representation: Generalized harmonic sums connect to generalized harmonic polylogarithms through Mellin transforms and yield special constants and basis relations at n → ∞.
13 Appendix: Available commands of the package HarmonicSums.m
The appendix documents HarmonicSums.m commands for defining, transforming, expanding, reducing, and converting harmonic, generalized, cyclotomic, S-, and Z-sums. It also describes basis computations and available relation tables.
- Definitions: The package defines S-sums with positive integer indices, nonzero real weights, finite upper limit n, and generalized polylogarithms with real indices.
- Reduction: ReduceToBasis and ReduceSums reduce polylogarithms and sums to algebraically independent basis elements using stored or dynamically computed tables.
- Transforms: TransformToSSums converts indefinite nested sums to harmonic, S-, or cyclotomic sums whenever possible.
- Transforms: Mellin and inverse Mellin commands transform expressions between integral representations and nested-sum forms.
- Series and differentiation: HarmonicSumsSeries computes series expansions of sums and associated polylogarithms about finite points or infinity.
- Conversions: HToSinf and SinfToH convert generalized polylogarithms at 1 and infinite S-sums in both directions, while HToS and SToH invert power-series representations.