Harmonic sums, Mellin transforms and Integrals

1 Introduction

• Feynman-diagram integrals are difficult to evaluate, while Mellin moments can simplify them at the cost of symbolic sums over harmonic series.Knowing all positive even moments permits reconstruction of the complete structure functions.Source: §1 · p. 2
• The framework supplies consistent computer-oriented notation and algorithms for sums, Mellin transforms, values at infinity, and inverse Mellin transformations.Most procedures are implemented in FORM, with additional appendices listing sums not directly handled by the general algorithms.Source: §1 · p. 2

2 Notations

• Harmonic-series notation tracks index signs, weights, extended weights, and products, including alternating sums and denominators sharing the series argument.The weight counts absolute index values, while extended weight adds denominator powers tied to the argument.Source: §2 · p. 3
• The natural basis represents each fixed-weight harmonic-series element as a single higher harmonic series, making summation formulae simpler.Products can be converted to this basis when needed, although alternative bases may be preferable for expressing divergent objects at infinity.Source: §2 · p. 4
• The notation and algorithms must handle symbolic denominators and exceptional cases such as equal parameters, where naive partial fractions produce invalid 1/(b-a) terms.FORM uses delta and theta functions, especially under integer-parameter assumptions, to manage these cases.Source: §2 · p. 4

3 Synchronization

• Synchronization is needed because combined objects may have different arguments, including factorials and complementary arguments such as i and n−i.Negative leading indices require additional sign care, while symbolic argument differences are outside the simplest synchronization tricks.Source: §3 · pp. 5–6
• Argument synchronization recursively brings S-functions, denominators, and factorials closer together, often combining factorials into binomial coefficients.For integer argument differences, partial fractioning produces terminating recursions; products with matched arguments are then reduced to the natural basis.Source: §3 · pp. 5–6

4 Mellin Transforms

• The Mellin-transform algorithm expands endpoint factors, separates logarithmic powers, integrates by parts, and repeats decomposition until only integrable powers or nested sums remain.It applies to combinations involving rational endpoint factors, logarithms, and polylogarithms, but requires functions to break down without fractional powers.Source: §4 · pp. 7–8
• Products of S-functions with identical arguments are converted to the natural basis so finite-limit and infinite-limit nested sums can be handled after the integration steps.The resulting expression contains harmonic series whose arguments depend on the Mellin parameter and harmonic series evaluated at infinity.Source: §4 · p. 8
• Logarithmic divergences are rewritten in powers of the single basic divergent sum S1(∞), after which cancelling ln L terms allow the limit L →∞ to be taken.The divergences are soft and behave as powers of ln L up to terms of order 1/L1.Source: §4 · p. 8

• The target sums are related to Euler-Zagier sums, whose evaluation is needed because Mellin-transform results contain harmonic series both at finite moments and at infinity.Source: §5 · p. 8
• The method first rewrites sums into products of lower-weight harmonic series, leaving a controlled number of series at the weight being evaluated.Source: §5 · p. 9
• Doubling identities for positive-index series provide additional relations, including the adjustment S1(2∞) → S1(∞) + ln(2) for divergent terms.Source: §5 · p. 9
• A diagonal-triangle identity replaces products of sums when the complementary upper-right region vanishes, yielding powerful equations unless both leading indices equal one.Source: §5 · p. 9
• The weight n + 1 computation constructs shuffle and theorem-based equations, substitutes lower-weight values, eliminates unknowns, and retains unavoidable sums as independent variables.Source: §5 · pp. 9–10
• The resulting FORM implementation reaches weight 7 in under six hours on a Pentium-II-300, while selected constants remain as numerical inputs when no reduction is available.Source: §5 · p. 10
• Most non-zeta constants are expected not to occur in massless Feynman diagrams, whereas Li4(1/2) does occur in massive three-loop electron g−2 corrections.Source: §5 · p. 10

• Inverse Mellin reconstruction can be performed by choosing x-space functions whose Mellin transforms span the relevant harmonic-series space and solving the resulting linear relations.Source: §6 · p. 11
• A complete natural-basis reconstruction requires 3w−1 candidate functions after symmetry reduction, with 3, 8, 18, 48, and 116 new functions at weights 3 through 7.Source: §6 · p. 11
• The formulas assume suitable endpoint behavior, especially F(1)=0 when needed; otherwise additional terms are required to cancel divergences.Source: §6 · p. 11
• The construction starts from weight-one functions and builds higher weights through composed integrations such as F+−0++0(x), with Mellin transformation used for systematic checking.Source: §6 · p. 13
• Weight-three inversions introduce dilogarithms, while weight-four inversions use trilogarithms with arguments including x, −x, (1+x)/2, and several rational transformations.Source: §6 · p. 13
• The recursive formulas isolate a single weight w+1 harmonic series on the right-hand side, avoiding large equation systems as the weight increases.Source: §6 · p. 13

• The program evaluates a difficult integral by treating it as a Mellin transformation, reducing its accompanying sum to the same harmonic-sum class handled elsewhere.Source: §7 · p. 13
• The resulting expression contains only S-functions at infinity of maximal weight 5, allowing substitution from the prepared tables.Source: §7 · p. 14
• The technique extends beyond the displayed example to whole classes of integrals considerably beyond those previously treated in reference.Source: §7 · p. 14
• For deep-inelastic-scattering diagrams, symbolic Mellin moments convert the remaining calculation into harmonic-series summation, motivating the SUMMER program's development.Source: §7 · p. 14
• At two loops, the program is general, whereas three-loop calculations were initially available only for a small set of fixed moments and were being extended symbolically.Source: §7 · p. 14

• The algorithms provide a foundation for sums arising in calculations such as deep-inelastic-scattering Feynman diagrams.Source: §8 · p. 15
• They analytically evaluate whole classes of integrals, solve Mellin transforms for broad function categories, and provide a numerical approach to inverse Mellin transforms.Source: §8 · p. 15
• Most algorithms and tables were implemented in FORM version 3 and made available through the author's homepage.Source: §8 · p. 15

A Conversion to the Basis

• A recursion converts products of S-functions with the same argument into single higher S-functions, starting from S(1) and S(2).The construction terminates when one factor has no remaining indices and can be replaced by 1.Source: A Conversion to the Basis · p. 15
• The basis-conversion algorithm is directly constructed, implemented efficiently in FORM, and tested by exact rational evaluations for arbitrary functions and arguments.A weight-10 run generated 1683 terms and used 85104 bytes on a Pentium Pro 200.Source: A Conversion to the Basis · pp. 15–16
• The FORM implementation represents intermediate products with SS and uses signed-index pseudo-addition to produce the final S-function representation.The procedure was tested on products including five-index positive and negative harmonic series.Source: A Conversion to the Basis · pp. 15–16

B Conjugations

• For positive-index S-functions, conjugation maps a natural-basis element to one lower-weight S-function supplemented by powers of its argument to preserve extended weight.The proof uses induction over weight, linear independence, and the fact that applying conjugation twice returns the original function.Source: B Conjugations · p. 16
• Associate functions reduce conjugate-finding to a constructive recursion that can be implemented in FORM.A function divided by n^m is first rewritten through its associate function, whose result directly determines the needed conjugate.Source: B Conjugations · p. 17

C Sums involving n −i

• Partial fractioning reduces denominator powers to k′ ≤ k or m′ ≤ m, while reversing the summation direction evaluates the resulting i−j sums.This makes the innermost sum progressively simpler and defines a useful recursion.Source: C Sums involving n −i · p. 18
• The sumnmii algorithm implements this recursion for sums involving n−i, including cases with negative m or factors (−1)^i.A test run generated 478 terms and produced 208 output terms in 0.28 seconds.Source: C Sums involving n −i · p. 18
• A related recursion for G-functions lowers the weight through partial fractioning and handles the k = 0 case separately before reconstructing G(k,n).Changing the summation direction can introduce (−1)^n and therefore harmonic functions with negative indices.Source: C Sums involving n −i · p. 19
• The simpler sumnm example evaluates to a linear combination of single harmonic functions with negative and positive indices.The reported output contains terms such as S(R(-3),n), S(R(-2,1),n), and S(R(3),n).Source: C Sums involving n −i · p. 19

D Some sums to infinity

• Sums to infinity are reduced recursively by introducing an auxiliary sum over m, whose final difference can then be summed.The resulting expressions may contain S-functions evaluated at infinity, which are treated separately.Source: D Some sums to infinity · p. 20
• A second class of infinite sums uses an analogous auxiliary function and recursion, allowing the sums to be solved.The last term is simpler than the original, which enables the recursive reduction.Source: D Some sums to infinity · p. 20

E Miscellaneous Sums

• The miscellaneous section states that its algorithms cover selected sums to arbitrary complexity but do not represent complete classes, and lack proofs.They were checked only for rather large parameter values.Source: E Miscellaneous Sums · p. 21
• The Gamma-function expansion supplies harmonic-series combinations, which can be rewritten as products of single-sum harmonic series.This provides another representation for the special harmonic series used in the algorithms.Source: E Miscellaneous Sums · p. 21
• The section derives product expansions for higher harmonic and U-functions, with coefficients related to permutation-cycle structure.Examples include expansions for S1,1,1,1(n) and U1,1,1,1(n,m).Source: E Miscellaneous Sums · p. 21
• Additional identities organize nested sums by weight, index signs, and the number of nested sums, while a conjugation identity can halve the required work or cause cancellations.The construction also introduces U0(n,m)=1 and connects harmonic-series expansions to the Gamma function.Source: E Miscellaneous Sums · pp. 21–22

F Summation tables

• Many evaluated sums fall outside the general algorithms, are solvable only at fixed weight, and were not readily available in the literature.These results are presented in tables, including formulas later found unnecessary for the final program.Source: F Summation tables · pp. 22–23
• The tables also include literature formulas, some derived formally and others obtained by guessing and testing across many values.Sums already handled by earlier procedures are omitted to avoid unnecessarily lengthy tables.Source: F Summation tables · p. 23
• The tabulated identities generally assume integer parameters i,j,k,l,m,n ≥ 0, although some formulas extend to noninteger values.The extension is stated only for some cases rather than for the tables as a whole.Source: F Summation tables · p. 23
• Auxiliary functions, including Δ and Δ′, are introduced for manipulating triangular two-dimensional summation regions that do not touch the origin.These functions support recurring summation structures in the tabulated formulas.Source: F Summation tables · p. 23

F.1 Sums without (−1)j

• F.1 presents low-complexity and miscellaneous sums without (−1)j, then develops partial-integration-like identities used to derive subsequent formulas.These identities are described as central to deriving the next equations and as useful for the later sums.Source: §F.1 · pp. 24–26

F.2 Sums with (−1)j

• F.2 treats sums containing (−1)j separately because they give completely different results from corresponding non-alternating sums.The section includes basic, mixed, factorial-numerator, and multiple-binomial-coefficient formulas.Source: §F.2 · pp. 26, 27, 29, 30
• The alternating-sum formulas cover extra pieces with arguments j, j + 1, j + 2, and j + n, alongside corresponding harmonic-sum expressions.Examples include formulas involving S1,1,1(j), S1,1(n+j) − S2(n+j), and S2(n+j).Source: §F.2 · pp. 27–29
• Several alternating identities also handle sums with two factorials in the numerator and more than one binomial coefficient.The appendix points to general formulas and related appendices for these cases.Source: §F.2 · pp. 27, 30

F.3 Partial sums

• Partial sums over only part of the binomial-coefficient range are useful for derivations but difficult to obtain, limiting how many are presented.The limitation concerns the availability of partial-sum formulas rather than their usefulness.Source: §F.3 · p. 30