Harmonic Polylogarithms

• Polylogarithms become insufficient for calculations with more loops or several dimensionful scales, motivating broader function classes.Source: §1 · p. 2
• Harmonic polylogarithms are introduced through recursive integrals as a generalization of Nielsen’s polylogarithms and as a subset of multidimensional polylogarithms.Source: §1 · p. 2
• The paper studies products, endpoint behavior, expansions, argument identities, special values, numerical evaluation, and Mellin transforms of the hpl’s.Source: §1 · p. 2
• The Mellin transforms of harmonic polylogarithms give harmonic sums with a one-to-one correspondence, yielding an inverse Mellin-transform algorithm.Source: §1 · p. 2
• The algorithms are implemented in FORM, and the resulting procedures are available from the second author.Source: §1 · p. 2

Definitions

• Harmonic polylogarithms of weight w are indexed by a w-component vector and defined recursively using the leftmost index and associated rational kernels.Source: §2 · p. 3
• At weight 2, the nine harmonic polylogarithms can be expressed using logarithms and Euler’s dilogarithm.Source: §2 · p. 4
• Harmonic polylogarithms resemble Nielsen’s polylogarithms through weight 3 but are no longer expressible through Nielsen’s functions of the same weight from weight 4 onward.Source: §2 · p. 5
• Nonzero-index hpl’s vanish at x = 0, while behavior at x = 1 is finite or logarithmic according to the leading indices.Source: §2 · p. 5
• A compact index notation removes trailing zeroes by increasing the absolute value of the preceding index, connecting the notation to Nielsen polylogarithms and harmonic sums.Source: §2 · p. 5
• Special values at simple arguments are connected to Euler/Zagier sums and can be reduced to a small set of constants such as Riemann ζ-functions and ln 2.Source: §2 · p. 6

Identities between functions of the same argument

• Integration-by-parts identities relate products and reversed index sequences, and can reduce the number of highest-weight hpl’s.Source: §3 · p. 7
• Products of hpl’s become sums of higher-weight hpl’s formed by all order-preserving mergers of their index vectors.Source: §3 · pp. 7–8
• Product identities extract powers of ln(x) associated with trailing zero indices and powers of ln(1−x) associated with leading one indices.Source: §3 · pp. 8–9
• Combining both procedures expresses hpl’s with leading 1’s and trailing 0’s through logarithmic factors and irreducible hpl’s finite at x = 0 and x = 1.Source: §3 · pp. 9–10
• The reduction to minimal sets uses product and integration-by-parts relations without modifying the indices appearing in the original hpl.Source: §3 · p. 10
• Table 1 reports the sizes of the full, irreducible, and minimal bases.Source: §3 · p. 10, Table 1 · p. 10

Power-series structure

• Trailing zeroes in an H-function generate powers of ln(x), so the proper expansion combines powers of x with powers of ln(x).Repeated product identities remove trailing zeroes and reduce the remaining functions to those expandable in x.Source: §4 · pp. 11–12

Nested-sum representation

• The nested-sum conventions differ for finite upper limits: the paper follows a convention that avoids zero values when the number of nonzero indices exceeds the summation limit.The alternative definition is favored for sums to infinity but is less convenient at finite n.Source: §4 · p. 11
• Changing the summation notation introduces a leading term with the same index field and correction terms with fewer, possibly combined, indices.For k nonzero indices there are 2^k−1−1 correction terms.Source: §4 · p. 12
• H-functions without trailing zeroes have expansions whose coefficients are represented by Z-sums or S-sums, making them special cases of multidimensional polylogarithms with b_i = ±1/x.H-functions with trailing zeroes remain necessary for Feynman-diagram calculations because logarithms occur there.Source: §4 · p. 12

Alternative expansions

• Dividing the expansion by 1+x or 1−x yields alternative sums that are convenient in some cases but break down at x = 1 or x = −1.The failure point depends on which form is used.Source: §4 · p. 13

Evaluation at special arguments

• At x = 1, H-function values are related to S-sums at infinity, and conversion to H-functions at x = 1/2 provides a practical numerical method at higher weights.Exact expressions in a small set of constants remain feasible through roughly weights 7–9, whereas x = 1/2 expansions are sufficiently fast for most numerical applications.Source: §4 · p. 13

Product algebra

• Triangle-sum manipulations reduce inner sums to single S-functions, which can then be resummed into expressions containing single H-functions.This connects the series expansions of products of H-functions to the harmonic-sum algebra.Source: §5 · p. 14

Algebraic duality

• H-function product formulas correspond to harmonic-sum algebra, with general H-function rules dual to special triangle rules and x = 1 rules dual to general shuffle rules.H-functions at x = 1 can be directly expressed in terms of harmonic sums at infinity.Source: §5 · p. 14

Divergences and regularization

• For divergent objects at x = 1, triangle relations give incorrect subleading terms, so regularization must be handled consistently rather than by directly applying the usual relation.The issue arises when both objects in the triangle sum are divergent.Source: §5 · p. 14
• The consistent solution regularizes in S-space by replacing sums to infinity with sums to a large finite M, cancelling divergences, and then taking M →∞.The resulting shuffle-based relations at x = 1 support a properly defined Mellin transform.Source: §5 · p. 15

Values at x = 1

• Determining a minimal independent set of H-function values at x = 1 becomes formidable at large weight because many objects and relations must be solved simultaneously.The paper notes that the different algebraic relations used at x ≠ 1 and x = 1 can make expressions complicated even when they are algebraically reducible.Source: §5 · pp. 13, 15

General transformation framework

• H-functions of weight w at transformed arguments can be expressed homogeneously using H-functions of the related argument and constants at special arguments.The constructions proceed recursively by induction on the weight.Source: §6 · pp. 15, 19

x^2→x

• For x^2→x identities, excluding −1 indices in H-functions of x^2 allows recursive reduction, including Li2(x^2)=2Li2(x)+2Li2(−x).The harmonic-polylogarithm identities are wider than the corresponding Nielsen-polylogarithm relations.Source: §6 · pp. 15–16

x→1−x

• The x→1−x identities are derived recursively, with finite constants H(0,m⃗;1) entering when the leading index is 0.Leading index 1 requires reduction to a basis where it appears only through H(1;1−x).Source: §6 · pp. 16–17

x→1/x

• The x→1/x identities are established by induction after reducing leading-index-1 functions, yielding explicit weight-3 examples.Finite H-values at argument 1 serve as constants in the resulting expressions.Source: §6 · pp. 17–19

x→−x

• The x→−x transformation introduces the constant π with weight 1 and requires tracking the imaginary part of ln(x) under continuation.For example, H(1;−1) takes the real value −ln 2.Source: §6 · p. 18

x→(1−t)/(1+t)

• The transformation x→(1−t)/(1+t) applies to arbitrary index sets and has no counterpart among Nielsen’s polylogarithms.Its recursive proof substitutes lower-weight identities before the final integration; for example, H(−1,−1,1;x) becomes −H(0,−1,−1;t)+H(−1,−1,1;1).Source: §6 · pp. 19–20

Definition and structure

• G(m⃗w;x) is defined for index vectors whose first index is always 1, and each G-function is a homogeneous combination of H-functions of weight w.The paper establishes this representation inductively.Source: §7 · p. 20

Recursive derivation

• The recursive derivation differentiates G-functions with respect to x, integrates by parts in t, and substitutes previously established lower-weight identities.For a=−1, this procedure produces the stated derivative relation.Source: §7 · pp. 20–21

Example

• A weight-4 example expresses G(1,0,−1,1;x) as a combination of four H-functions with argument x.The displayed relation contains H(0,−1,0,1;x), H(0,−1,1,1;x), H(0,0,0,1;x), and H(0,0,1,1;x).Source: §7 · p. 21

Special values

• H-functions generalize the use of Nielsen polylogarithms for special arguments, providing a broader identity system for evaluating definite integrals and mathematical constants.The same integration-by-parts and change-of-variable strategy applies more systematically to H-functions.Source: §8 · p. 21

Validation and scope

• The redundant transformation relations reproduce the known table of weight-4 definite integrals, but their sufficiency for higher-weight tables remains untested.The paper explicitly reports verification only through weight 4.Source: §8 · p. 22

Transformation-based evaluation

• Values at x=1/2 can be related to values at x=1 through x→z/(1+z), while transformations also connect values at x=−1, x=2, and x=1.These conversions combine 1−x→x, x→1/x, x→(1−t)/(1+t), and G-to-H identities.Source: §8 · p. 22

Constant evaluation

• Two relations provide power series for ζ3, and related dependencies express s6=S−5,−1(∞) using H5,1(1/2), H6(1/2), and lower-weight constants.Analogous dependencies can be derived for higher-weight constants.Source: §8 · p. 23

Numerical evaluation

• For arbitrary x, series around x=0 converge rapidly at small x, while the transformation t=(1−x)/(1+x) improves convergence as x approaches 1.For larger values, x→1/x and sign-flip identities provide alternative evaluation regions.Source: §8 · p. 23

Values at x=1

• H-functions at x=1 can be rewritten as S-sums or Z-sums at infinity, but values with first index 1 diverge and must be handled through sums or regularized representations.For low weights, finite results can be reduced to a limited set of fundamental constants.Source: §8 · p. 23

Mellin transforms

• Mellin transforms are obtained at low weight directly and at higher weight by recursive partial integration, with H-functions at x=1 supplying the remaining values.Only the H-functions used in subtraction must be finite; other divergences can be regularized through S-sums.Source: §9 · pp. 24–25

Limitation

• For weights currently larger than 9, evaluating H-functions at x=1 or S-sums at infinity becomes rather hard because cancellations among divergent terms are difficult to expose.The algebra for x=1 differs from the general-x algebra, leaving many divergent intermediate H-functions.Source: §9 · p. 25

Inverse Mellin algorithm

• Inverse Mellin transforms are constructed by locating the most complicated S-sums, building corresponding H-functions, subtracting them, and iterating until no S-sums remain.The algorithm is stated to terminate properly.Source: §9 · pp. 25–26

Parity dependence

• Parity must be specified because the inverse Mellin transform uses either all even or all odd moments, fixing factors of (−1)^N.Terms containing 1/(1+x) change sign for odd N.Source: §9 · pp. 26–27

S-sum mapping

• The S-sum-to-H-function mapping determines the kernel from the parity of negative indices, copies the index field, and applies additional sign rules during recursive construction.The rules include signs from the last index, weight and depth, and each negative index.Source: §9 · p. 26

Higher-weight evaluation

• At higher weights, numerical evaluation remains possible, but substituting H-values at x=1 may fail and resulting formulas become much longer.An irreducible sum can remain in a nontrivial Mellin-transform example.Source: §9 · pp. 25, 27