Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals

• High-order expansions of transcendental functions are non-trivial, especially in higher-order radiative corrections and multi-scale loop calculations.Source: §1 · p. 2
• The paper defines Z-sums, studies their algebraic properties, and gives computer-implementable algorithms for expanding functions around integer indices to arbitrary order in ε.Source: §1 · p. 2
• Nested sums provide a framework for generalized hypergeometric, Appell, and related function expansions, as well as higher-loop multi-scale integrals.Source: §1 · p. 2
• The C-topology with one leg offshell is reduced to multiple nested sums for arbitrary propagator powers and dimensions, supporting two-loop e+e−→3 jets calculations.Source: §1 · p. 2

2 Definition and properties of nested sums

• Z-sums and S-sums are nested-sum definitions whose depth and weight characterize their structure, with different upper summation boundaries.Source: §2 · p. 3
• Z-sums and S-sums can be converted into one another recursively, allowing properties to be expressed in whichever notation is more natural.Source: §2 · p. 3
• Products of Z-sums with a common upper limit reduce recursively to single Z-sums, providing an algorithmic multiplication rule based on partitioning summation regions.Source: §2 · p. 4, Figure 1 · p. 4
• The Z-sums form a quasi-shuffle algebra, conserve weight under multiplication, and satisfy a Hopf-algebra structure.Source: §2 · p. 4

2.1 Special cases

• Z-sums interpolate between Goncharov multiple polylogarithms and Euler-Zagier sums, while special parameter choices recover multiple ζ-values, harmonic sums, and related polylogarithms.Source: §2.1 · pp. 5–6

3 Algorithms

• The framework first expands Gamma-function factors, synchronizes subsums, applies partial fractioning, and converts products or offsets into standard nested-sum forms.Source: §3 · pp. 7–8
• The four algorithm types cover sums involving Z-sums, products of Z- or S-sums, binomials, and paired terms depending on i and n−i.Source: §3 · pp. 7–8
• The recursions are designed for implementation in computer-algebra systems including GiNaC, FORM, Mathematica, and Maple.Source: §3 · p. 8

3.1 Algorithm A

• Algorithm A recursively reduces Gamma-weighted sums to Z-sums and yields multiple polylogarithms or harmonic polylogarithms when the upper limit is infinite.Source: §3.1 · p. 9

3.2 Algorithm B

• Algorithm B handles paired finite sums through partial fractioning and recursion that terminates by reducing the subsum depth, but its upper limit cannot be infinite.Source: §3.2 · p. 10

3.3 Algorithm C

• Algorithm C uses S-sum transformations, raising and lowering operators, and controlled limits to reduce sums to finite Z-sums together with multiple polylogarithms.Source: §3.3 · pp. 10–13

3.4 Algorithm D

• Algorithm D recursively lowers the weight of sums involving both S(i,...) and S(n−i,...), producing finite Z-sums plus multiple polylogarithms rather than finite Z-sums alone.Source: §3.4 · pp. 14–15

4 Applications

• The algorithms expand higher transcendental functions around integer indices and express the results through nested sums suitable for computer implementation.The expansion parameter occurs in Pochhammer symbols.Source: §4 · p. 16

4.1 Generalized hypergeometric functions

• Generalized hypergeometric functions are rewritten as transcendental sums and expanded with examples that agree with known results.The rewriting uses Pochhammer symbols.Source: §4.1 · p. 16

4.2 Appell functions

• The two Appell functions are reduced to nested sums of types B and D, enabling expansions with the corresponding algorithms.The presented algorithm outputs linear combinations of polylogarithms rather than their products.Source: §4.2 · pp. 17–18

4.3 Kampé de Fériet function

• The Kampé de Fériet function is reduced to a nested sum of type C and expanded using algorithms A and C.Source: §4.3 · p. 18

4.4 The C-topology

• The C-topology with one massive external leg, arbitrary propagator powers, and arbitrary dimensions is reduced through a one-loop triangle and a remaining parameter integration.The resulting expression becomes a type-D sum after changing summation indices.Source: §4.4 · pp. 18–19
• For specific integer propagator powers and dimensions, the C-topology expression can be expanded in ε and agrees with Gehrmann and Remiddi in two previously known cases.The earlier results covered m = 2 and propagator-power sets (1,1,1,1,1) and (1,1,1,1,2).Source: §4.4 · p. 20

• Nested-sum algorithms suitable for computer implementation expand a class of transcendental functions and evaluate integrals in high-energy physics.Source: §5 · p. 20
• The two-loop C-topology is evaluated for arbitrary propagator powers and arbitrary dimensions.Source: §5 · p. 20
• Nested sums satisfy a Hopf algebra connected with Kreimer’s Hopf algebra.Source: §5 · p. 20

A The Hopf algebra of Z-sums

• The quasi-shuffle structure guarantees that Z-sums also form a Hopf algebra.The paper invokes a general theorem connecting quasi-shuffle algebras with Hopf algebras.Source: A The Hopf algebra of Z-sums · p. 20
• Z-sums form a quasi-shuffle algebra because their multiplication matches the recursive quasi-shuffle product.The letter product combines degrees and arguments, while the algebra product is recursively defined.Source: A The Hopf algebra of Z-sums · pp. 20–21
• Z-sums can be represented as rooted trees without sidebranchings, with the outermost sum corresponding to the root.Nested sums with multiple subsums produce rooted trees with sidebranchings.Source: A The Hopf algebra of Z-sums · p. 21
• The coalgebra is formulated directly through admissible cuts of rooted trees, identifying one root-containing subtree and the remaining forest.An admissible cut places at most one cut on every path from a vertex to the root.Source: A The Hopf algebra of Z-sums · pp. 21–22
• The coproduct and antipode are defined from admissible cuts, with explicit examples given for depth-one and depth-two Z-sums.The antipode reverses and combines indices in the depth-two example.Source: A The Hopf algebra of Z-sums · p. 22
• The Z-sum coalgebra is identical to the coalgebra in the Hopf algebra of Kreimer and Connes, although their algebra products differ.In the Kreimer–Connes algebra, the product of two rooted trees need not remain a rooted tree.Source: A The Hopf algebra of Z-sums · p. 22

B Review of Goncharov’s multiple polylogarithms

• Goncharov’s multiple polylogarithms are reviewed as the specialization used to express the paper’s final results.The paper notes that these functions are important specializations of nested sums.Source: B Review of Goncharov’s multiple… · p. 22
• The review is based largely on Borwein et al. and relates their notation to Goncharov’s notation.The section also recalls Goncharov’s introduction of the functions.Source: B Review of Goncharov’s multiple… · p. 22

B.1 Integral representations

• The section introduces notation for iterated integrals and gives an integral representation for multiple polylogarithms.The representation is also written in the notation of Borwein et al.Source: §B.1 · p. 23
• Changing variables according to t → 1−t yields a dual integral representation.The variables b_j are related to the x_j parameters through the paper’s preceding notation.Source: §B.1 · p. 23
• Multiple polylogarithms also admit a depth-dimensional integral representation in addition to the weight-dimensional representations.The section explicitly distinguishes these two dimensional viewpoints.Source: §B.1 · p. 23

B.2 The shuffle algebra

• Multiple polylogarithms can be denoted by a single string instead of separate x_j and m_j parameters.This notation is used to rewrite the iterated-integral representation.Source: §B.2 · p. 24
• The iterated-integral representation induces a shuffle product based on permutations preserving the relative order within each input string.The product conserves the total weight.Source: §B.2 · p. 24
• The shuffle structure provides a second algebraic description alongside the Z-sum Hopf-algebra structure.The paper states that Z-sums form a Hopf algebra and refers to Appendix A for further details.Source: §B.2 · p. 24

• Multiple polylogarithms contain classical polylogarithms as their depth-one case.The section begins by identifying depth-one multiple polylogarithms with classical polylogarithms.Source: §B.3 · p. 24
• Nielsen’s generalized polylogarithms are related to multiple polylogarithms through an explicit correspondence.Source: §B.3 · pp. 24–25
• Harmonic polylogarithms of Remiddi and Vermaseren are related to multiple polylogarithms for positive indices and are defined recursively.The recursive definition is presented through an integral involving f_0(t) and lower-depth harmonic polylogarithms.Source: §B.3 · p. 25
• Two-dimensional harmonic polylogarithms were introduced by Gehrmann and Remiddi by extending the defining fractions.Their integral representation shows that 2dHPL belong to Goncharov’s multiple polylogarithms.Source: §B.3 · p. 25