On functional equations for Nielsen polylogarithms

Danylo Radchenko; Herbert Gangl; Steven Charlton

REVIEW 3 major objections 4 minor 43 references

On functional equations for Nielsen polylogarithms

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The weight-5 Nielsen polylogarithm $S_{3,2}$ satisfies the dilogarithm five-term relation modulo $\mathrm{Li}_5$ and products, with explicit $\mathrm{Li}_5$ corrections.

desk verdict A solid new symbol-level depth reduction for S3,2 that deserves refereeing, with the analytic lift and several evaluations conditional on an unpublished cleaning theorem. read the letter →

arxiv 1908.04770 v1 pith:HFEVDAPM submitted 2019-08-13 math.NT math.KT

Steven Charlton , Herbert Gangl , Danylo Radchenko This is my paper · ORCID

classification math.NTmath.KT MSC 11G5533E2039B32

keywords Nielsenpolylogarithmsfunctionalequationsfive-termrelationmod-productssymboldepthreductionsingle-valuedspecialvaluespolylogarithmladders

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that the weight-5 Nielsen polylogarithm $S_{3,2}$ behaves like the classical dilogarithm $\mathrm{Li}_2$ in a precise sense: whenever a combination of arguments satisfies the five-term relation for $\mathrm{Li}_2$, the same combination applied to $S_{3,2}$ vanishes modulo $\mathrm{Li}_5$ and products of lower-weight functions. The main theorem gives the explicit $\mathrm{Li}_5$ corrections through a five-variable identity involving three higher cross-ratios. This confirms a depth-reduction prediction in the motivic framework for weight 5 and implies that every rational functional equation for $\mathrm{Li}_2$, including the distribution relations, produces a corresponding relation for $S_{3,2}$ with algorithmically determined $\mathrm{Li}_5$ terms. The paper also derives functional equations and evaluations for Nielsen polylogarithms up to weight 8, including reductions of $S_{4,2}$, $S_{5,3}$, and $S_{6,2}$, and gives a general family of higher-weight depth reductions that yields Nielsen ladders in every weight.

What carries the argument

The mod-products symbol $\mathrm{Symb}^{/A_1}$ carries the argument: it is an invariant of iterated integrals that records only coproduct components not coming from products, so an identity written in it is exactly a statement modulo products and lower-weight functions. The paper computes $\mathrm{Symb}^{/A_1}(S_{n,p}(z))$ explicitly and uses recursion to reduce weights. To pass from symbol identities to genuine functions, the paper invokes the clean single-valued construction from an unpublished source, which combines a cleaning map that completes product terms with a single-valued map; the main theorem of that source asserts that $\sum_i \lambda_i \hat f_i = \text{constant}$ whenever $\mathrm{Symb}^{/A_1}(\sum_i\lambda_i f_i)=0$. The proof of Theorem 14 also uses the action of the symmetric group $S_5$ on cross-ratio variables: the relevant tensor space $\bigwedge^2 V \otimes \operatorname{Sym}^3(V)$ splits into a 4-dimensional and a 6-dimensional irreducible representation, and the alternating sum is shown to vanish by explicit polynomial identities inside these components.

What would settle it

Evaluate the clean single-valued identity of Corollary 15 at a randomly chosen tuple $(x_1,\dots,x_5)$ of distinct complex numbers with non-zero imaginary parts and check the left-hand side to 100 decimal places; any non-zero value would disprove the analytic claim. Alternatively, if the unpublished cleaning theorem of [9] is false, the identity may fail only at the analytic lift while the symbol identity (10) still holds, and a direct analytic proof of (10) would settle the distinction.

Watch

Extended reading notes

Core claim

The central discovery is equation (10) of Theorem 14: after antisymmetrisation over the five variables, $$11 S_{3,2}(\operatorname{cr}(x_1,x_2,x_3,x_4)) + \mathrm{Li}_5(15[r_1]-9[r_2]+[r_3]) = 0$$ in the mod-products symbol $\mathrm{Symb}^{/A_1}$; here $\operatorname{cr}$ is the cross-ratio and $r_1,r_2,r_3$ are certain higher ratios built from pairwise differences $x_i-x_j$. Since the symbol kills products, this says exactly that $S_{3,2}$ of the five-term relation is a combination of $\mathrm{Li}_5$'s modulo products. The authors lift this symbol identity to a clean single-valued identity (Corollary 15), and from it deduce that distribution relations for $S_{3,2}$ hold modulo $\mathrm{Li}_5$ with explicit corrections, that any $\mathrm{Li}_2$ evaluation reachable through the five-term relation upgrades to an $S_{3,2}$ evaluation, and that alternating the identity over a sixth point yields a known nontrivial $\mathrm{Li}_5$ functional equation. Beyond weight 5, the paper reduces $S_{3,3}$ to $S_{4,2}$ and $\mathrm{Li}_6$, $S_{4,3}$ to $S_{5,2}$ and $\mathrm{Li}_7$, and $S_{5,3}$ to $S_{6,2}$ and $\mathrm{Li}_8$ on algebraic families, and proposes numerically verified evaluations such as $S_{4,2}(-1)$ and $S_{6,2}(-1)$ in terms of classical polylogarithms.

Load-bearing premise

The analytic consequences rely on an unpublished result that a clean single-valued polylogarithm identity automatically upgrades an algebraic invariant identity to a genuine function identity; if that result is incomplete, the analytic corollaries fail even though the invariant identity (10) remains.

Editorial extensions

If this is right

Every rational functional equation for $\mathrm{Li}_2$, not just the five-term relation, yields a corresponding relation for $S_{3,2}$ modulo $\mathrm{Li}_5$ and products with explicitly computable corrections; in particular the distribution relations for $S_{3,2}$ hold in this sense (Corollary 19).
Known $\mathrm{Li}_2$ evaluations accessible through the five-term relation, including the golden-ratio values, Lewin's $\sqrt{2}-1$ ladder, and the $1/3$ ladder, lift to $S_{3,2}$ evaluations with explicit $\mathrm{Li}_5$ terms.
In weight 6, $S_{3,3}$ reduces to $S_{4,2}$ and $\mathrm{Li}_6$, and $S_{4,2}$ evaluated on algebraic $\mathrm{Li}_3$ functional equations reduces to $\mathrm{Li}_6$; this gives reductions of $S_{3,3}$ at roots of unity and conditional evaluations of $S_{4,2}(-1)$, $S_{4,2}(1/2)$, and $S_{3,3}(-1)$ in classical polylogarithms.
In weight 7, $S_{4,3}$ reduces to $S_{5,2}$ and $\mathrm{Li}_7$; in weight 8, $S_{5,3}$ evaluated on algebraic $\mathrm{Li}_2$ equations reduces to $S_{6,2}$ and $\mathrm{Li}_8$, and a numerically verified candidate evaluates $S_{6,2}(-1)$ through $\mathrm{Li}_8$, $\zeta(3,5)$, and products of logs.
The depth-reduction theorem of Section 10 holds for all $m\ge 1$: $S_{2m,m}$, $S_{2m-1,m}$, and $S_{2m-2,m}$ satisfy explicit reductions or two- and three-term relations, so Nielsen ladders exist in arbitrary weight (Corollary 41).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not spell this out, but the same mechanism should produce explicit $\mathrm{Li}_{n+p}$ corrections whenever an analogous motivic cobracket vanishes, so the recursions of Theorem 40 could be used to guess higher-weight identities algebraically.
Because the clean lift relies on an unpublished theorem, a natural test is to compute the clean single-valued lift of a known mod-products symbol identity independently; a failure there would cut the analytic consequences while leaving the symbol identity (10) intact.
The conjectural special values imply that all alternating multiple zeta values up to weight 6 reduce to classical polylogarithms; proving the undetermined $\zeta(6)$ coefficient in the $S_{4,2}(-1)$ reduction would complete that reduction.
The same higher-ratio and $S_5$-module identities could be transferred to other depth-2 polylogarithms or to their single-valued variants, giving explicit functional equations useful in amplitude computations.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The real news here is Theorem 14: a purely algebraic proof that the weight-5 Nielsen polylogarithm S3,2, evaluated on the dilogarithm five-term relation, reduces modulo products to an explicit combination of Li5 terms. That is a new, concrete corroboration of Goncharov's depth-reduction prediction, and the proof is genuine and self-contained at the symbol level. The explicit ratios r1, r2, r3 are new, and the representation-theoretic computation inside S5 is clean and convincing. The same framework also gives a nice general family in Theorem 40. No fitted parameters, no circularity: the argument really does what it says.

The soft spot is exactly where the reader's stress-test lands. The advertised analytic five-term identity (Corollary 15) and the special values that flow from it depend on the unpublished reference [9], which supplies the theorem that clean single-valued functions lift mod-products symbol identities to analytic identities up to a constant. The present paper gives neither a statement of the theorem's hypotheses nor a proof. If that lifting theorem is false or has hidden conditions, the analytic conclusions do not follow from the symbol identity. This is a real gap in the chain from algebraic identity to analytic functional equation. It does not damage Theorem 14 itself, which is the core new result.

The weight 6 and 8 evaluations (e.g. S4,2(-1), S6,2(-1)) are honest: they are marked with '?' and verified numerically to high precision, and the coproduct analysis explains most coefficients. Those should not be read as proven theorems, and the paper is careful about that. The abstract is slightly stronger than what is rigorously shown, since it says functional equations are derived without flagging the dependence on [9] and the numerical parts.

Net assessment: this is a serious paper with a substantial new algebraic result, written by people who know the subject. The main theorem should be published. The authors need to either make [9] available with a complete proof, or restrict the stated five-term conclusion to the symbol level and keep the analytic lift as conditional. A referee should verify Theorem 14's computation carefully and push for clarity on the lifting hypothesis.

I would send this to a referee. The core result earns the time.

Referee Report

3 major / 4 minor

Summary. This paper studies Nielsen polylogarithms S_{n,p}. Its main algebraic result, Theorem 14, gives an explicit mod-products symbol identity showing that S_{3,2} evaluated on the dilogarithm five-term relation reduces to a specific combination of Li_5 terms, thereby corroborating a depth-reduction prediction of Goncharov. The paper also establishes a basis for Nielsen symbols in each weight (Theorem 7), derives several depth reductions and functional equations in weights 6–8, including ladders and special values, and proves a general family of depth reductions (Theorem 40). Many of the analytic and clean single-valued statements are obtained by applying a lifting theorem from the unpublished reference [9], and several special-value identities are marked as numerical ('?') only.

Significance. If the algebraic result Theorem 14 is taken on its own, it is a substantial and largely self-contained computation with explicit, checkable steps, and it provides strong evidence for the expected depth reduction of S_{3,2} modulo Li_5. The representation-theoretic S5 arguments and the explicit form of the Li_5 terms are valuable. However, the abstract advertises analytic functional equations for Nielsen polylogarithms, and those analytic statements currently depend on an unpublished lifting theorem that is neither stated nor proved in the manuscript. The numerical identities are honestly marked, but they are not proven. Consequently, the central algebraic contribution is sound while the advertised analytic consequences are conditional on external material.

major comments (3)

[Section 4.1, Corollary 15] The paper's advertised analytic functional equations rely on the assertion that clean single-valued functions automatically lift any mod-products symbol identity to an analytic identity, citing 'the main result in [9]'. Since [9] is unpublished and no statement of hypotheses or proof is given, the analytic versions of Theorem 14 (Corollary 15 and all results derived from it, including the S_{3,2} evaluations and ladders in Section 6.4) are not established within this manuscript. This is a load-bearing gap between the symbol-level Theorem 14 and the abstract's claim of deriving new functional equations for Nielsen polylogarithms. The authors should either include a self-contained statement and proof of the lifting theorem, or explicitly reformulate the main claims as symbol-level and mark the analytic lift as conditional on [9].
[Section 2.2 and Section 4.1] The symbol calculus in Section 2.2 deliberately ignores signs by working modulo 2-torsion, and this convention is used repeatedly in the proof of Theorem 14 (e.g., 'we can ignore signs in the tensor factors'). It is not automatic that a mod-products identity obtained in this quotient lifts to an analytic identity for the clean single-valued functions, whose values are not taken modulo 2-torsion. Please verify explicitly that the sign choices in (10) are compatible with the lifting theorem, and that the antisymmetry argument in the proof of Corollary 15, which sets the constant to zero, does not depend on a sign ambiguity introduced by the quotient.
[Remark 16] Remark 16 states that 'with some work, one can in fact give an analytic version of Corollary 15' but provides no statement, proof, or reference. Since the analytic version is precisely what the abstract promises and is needed to make the paper's headline results unconditional, this remark should either be expanded into a full argument or removed, with the paper's claims adjusted accordingly.

minor comments (4)

[Abstract] The phrase 'viewed modulo Li5 and products' should specify that the reduction is at the level of mod-products symbols, to match the precise statement of Theorem 14.
[Section 3.3, Theorem 7] The proof invokes 'standard evaluations' for the determinants of A'_M, B'_M, and C'_M; please include a reference or a short derivation so the basis claim is fully self-contained.
[Equations (19), (20), Proposition 34, and related items] Several identities are explicitly marked with '?' and are verified only numerically to high precision. These should be placed in a clearly labeled 'numerical evidence' or 'conjectures' section, rather than being presented as propositions or results in the main text, to avoid confusion with the proven statements.
[Throughout] There are numerous typographical and formatting issues in the arXiv text, such as irregular spacing in the title and an extra closing parenthesis in the heading of Appendix B. A careful proofreading pass is needed.

Circularity Check

1 steps flagged · score 4.0 of 10

Analytic functional equations and special values inherit an unproved, same-author cleaning theorem [9]; the symbol-level Theorem 14 is independently derived.

self citation load bearing [Section 4.1 ('Cleaning procedure') and its use in Corollary 15, Section 6.3]
"The main result in [9] is that the clean single-valued functions ˆfi automatically lift a mod-products symbol identity Symb /A1 (∑ i λifi) = 0 to an analytic identity ∑ i λiˆfi = constant. ... Proof. By the antisymmetry, the constant in the clean single-valued identity must be 0."

The advertised analytic identity of Corollary 15 is obtained by applying the lifting theorem of the unpublished reference [9], whose authors include two authors of this paper. The paper supplies neither a proof nor a detailed statement of the hypotheses of this theorem. Consequently the step from the symbol-level identity (10) to the analytic clean single-valued identity is not derived in this paper; it reduces to a load-bearing self-citation. All subsequent clean single-valued evaluations and ladders inherit this dependence. This does not infect Theorem 14 itself, which is proved by direct computation, but it makes the analytic half of the paper rely on an unverified same-author citation.

full rationale

The central algebraic result, Theorem 14, is derived within the paper: the mod-products symbol of the left-hand side of (10) is computed explicitly, decomposed under the S5-representation V4 ⊕ V6, and shown to vanish by polynomial identities. No fitted constants or renaming of inputs occur there. The supporting Propositions 10, 11, 12, 21, 23, 25, 29, and 36 are either proved in the text or quoted as auxiliary results from publicly available theses that do not assert the target reduction. The numerical evaluations marked with '?'= are explicitly labeled conjectural and so are not disguised predictions. However, the paper's analytic functional equations for Nielsen polylogarithms, including Corollary 15 and the special-value ladders, depend on the cleaning and single-valued lifting theorem from the unpublished same-author preprint [9]. That theorem is load-bearing for those analytic statements and is not proved or stated with hypotheses in the present paper. This is a genuine self-citation dependency rather than an equivalence-by-construction, so the score is moderate rather than high.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The main theorem rests on the motivic symbol calculus plus the unpublished clean single-valued lifting theorem. The latter is the main external dependency. The LLL-found coefficients in the conjectural evaluations, e.g., equation (19) and Proposition 34, are not used in the proof of the main theorem, so they are not listed as fitted parameters. No free parameters are fit to data in the central derivation.

assumptions (4)

domain assumption Goncharov's motivic Hopf algebra of iterated integrals and the mod-products symbol recursion are valid (Section 2, based on [22] and [13]).
The main computations take place in this symbolic framework; the paper does not prove the framework itself.
domain assumption The clean single-valued map and the cleaning map R from [9] behave as stated (Section 4.1).
Reference [9] is listed as 'in preparation'; the analytic lift of Corollary 15 depends on it.
standard math Classical polylogarithm functional equations used as inputs (Section 5) are correct, namely the algebraic families for Li2, Li3, Li4 from [14] and [8].
These are established in the cited theses and used as lemmas for the S3,2, S4,2, and S5,2 reductions.
standard math Standard determinant evaluations for binomial matrices in Theorem 7 are correct.
Used to prove the basis of mod-products symbols; the determinants are quoted from 'standard evaluations' without derivation.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On functional equations for Nielsen polylogarithms." pith.science (2026). https://pith.science/paper/HFEVDAPM

@misc{pith2026190804770,
  author       = {Pith},
  title        = {Pith review of: On functional equations for Nielsen polylogarithms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HFEVDAPM}},
  note         = {Machine review of arXiv:1908.04770}
}

abstract

We derive new functional equations for Nielsen polylogarithms. We show that, when viewed modulo $\mathrm{Li}_5$ and products of lower weight functions, the weight $5$ Nielsen polylogarithm $S_{3,2}$ satisfies the dilogarithm five-term relation. We also give some functional equations and evaluations for Nielsen polylogarithms in weights up to 8, and general families of identities in higher weight.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 41 canonical work pages

[9]

Charlton, C

S. Charlton, C. Duhr, F. Dulat, and H. Gangl. Clean single -valued multiple polylogarithms. In preparation

work page
[1]

Bl¨ umlein, D

J. Bl¨ umlein, D. J. Broadhurst, and J. A. M. Vermaseren. T he multiple zeta value data mine. Comput. Phys. Comm. , 181(3):582–625, 2010

work page 2010
[2]

J. M. Borwein, D. M. Bradley, and D. J. Broadhurst. Evalua tions of k-fold Euler/Zagier sums: a compendium of results for arbitrary k. Electron. J. Combin. , 4(2):Research Paper 5, approx. 21, 1997. The Wilf Festschr ift (Philadelphia, PA, 1996)

work page 1997
[3]

J. M. Borwein and A. Straub. Mahler measures, short walks and log-sine integrals. Theoret. Comput. Sci. , 479:4–21, 2013

work page 2013
[4]

J. M. Borwein and A. Straub. Relations for Nielsen polylo garithms. J. Approx. Theory , 193:74–88, 2015

work page 2015
[5]

F. C. S. Brown. Representation theory of polylogarithms . Unpublished notes

work page
[6]

F. C. S. Brown. Polylogarithmes multiples uniformes en u ne variable. C. R. Math. Acad. Sci. Paris , 338(7):527–532, 2004

work page 2004
[7]

F. C. S. Brown. On the decomposition of motivic multiple z eta values. In Galois-Teichm¨ uller theory and arithmetic geometry, volume 63 of Adv. Stud. Pure Math. , pages 31–58. Math. Soc. Japan, Tokyo, 2012

work page 2012

Show all 43 references

[8]

Charlton

S. Charlton. Identities arising from coproducts on multiple zeta values and multiple polylogarithms . PhD thesis, Durham University, 2016

2016
[10]

K. T. Chen. Iterated path integrals. Bull. Amer. Math. Soc. , 83(5):831–879, 1977

1977
[11]

A. I. Davydychev and M. Yu. Kalmykov. New results for the ε-expansion of certain one-, two- and three-loop Feynman diagrams. Nuclear Phys. B , 605(1-3):266–318, 2001

2001
[12]

Del Duca, L

V. Del Duca, L. J. Dixon, C. Duhr, and J. Pennington. The B FKL equation, Mueller-Navelet jets and single-valued harmonic polylogarithms. J. High Energy Phys. , (2):086, front matter+28, 2014

2014
[13]

C. Duhr, H. Gangl, and J. R. Rhodes. From polygons and sym bols to polylogarithmic functions. J. High Energy Phys. , (10):075, front matter + 77, 2012

2012
[14]

H. Gangl. Funktionalgleichungen von Polylogarithmen . PhD thesis, Bonn University, 1995. 34 CHARLTON, GANGL, AND RADCHENKO

1995
[15]

H. Gangl. Functional equations for higher logarithms. Selecta Math. (N.S.) , 9(3):361–377, 2003

2003
[16]

H. Gangl. Functional equations and ladders for polylog arithms. Commun. Number Theory Phys. , 7(3):397–410, 2013

2013
[17]

H. Gangl. Multiple polylogarithms in weight 4. arXiv preprint arXiv:1609.05557 , 2016

2016 arXiv
[18]

Gehrmann and E

T. Gehrmann and E. Remiddi. Numerical evaluation of har monic polylogarithms. Comput. Phys. Comm. , 141(2):296– 312, 2001

2001
[19]

A. B. Goncharov. Polylogarithms and motivic Galois gro ups. In Motives (Seattle, WA, 1991) , volume 55 of Proc. Sympos. Pure Math. , pages 43–96. Amer. Math. Soc., Providence, RI, 1994

1991
[20]

A. B. Goncharov. Geometry of conﬁgurations, polylogar ithms, and motivic cohomology. Adv. Math. , 114(2):197–318, 1995

1995
[21]

A. B. Goncharov. Multiple polylogarithms and mixed Tat e motives, 2001. arXiv:math.AG/0103059v4

2001
[22]

A. B. Goncharov. Galois symmetries of fundamental grou poids and noncommutative geometry. Duke Math. J. , 128(2):209–284, 2005

2005
[23]

A. B. Goncharov and D. Rudenko. Motivic correlators, cl uster varieties and zagier’s conjecture on zeta (f, 4). arXiv preprint arXiv:1803.08585 , 2018

2018 arXiv
[24]

PARI/GP version 2.9.4, 2018

The PARI Group. PARI/GP version 2.9.4, 2018. http://pari.math.u-bordeaux.fr/

2018
[25]

Kellerhals

R. Kellerhals. Volumes in hyperbolic 5-space. Geom. Funct. Anal. , 5(4):640–667, 1995

1995
[26]

A. N. Kirillov. Dilogarithm identities. Progr. Theoret. Phys. Suppl., 118:61–142, 1995. Quantum ﬁeld theory, integrable models and beyond (Kyoto, 1994)

1995
[27]

K. S. K¨ olbig. Nielsen’s generalized polylogarithms. SIAM J. Math. Anal. , 17(5):1232–1258, 1986

1986
[28]

K. S. K¨ olbig, J. A. Mignaco, and E. Remiddi. On Nielsen’ s generalized polylogarithms and their numerical calculat ion. Nordisk Tidskr. Informationsbehandling (BIT) , 10:38–73, 1970

1970
[29]

L. Lewin. The dilogarithm in algebraic ﬁelds. J. Austral. Math. Soc. (Series A) 33 (1982), 302–330

1982
[30]

L. Lewin. Polylogarithms and associated functions . North-Holland Publishing Co., New York-Amsterdam, 1981

1981
[31]

L. Lewin. Structural properties of polylogarithms . Number 37. American Mathematical Soc., 1991

1991
[32]

N. Nielsen. Der Eulersche Dilogarithmus und seine Vera llgemeinerungen. Nova Acta Leopoldina , 90:123–211, 1909

1909
[33]

Radchenko

D. Radchenko. Higher cross-ratios and geometric functional equations fo r polylogarithms. PhD thesis, Bonn University, 2016

2016
[34]

Remiddi and J

E. Remiddi and J. A. M. Vermaseren. Harmonic polylogari thms. Internat. J. Modern Phys. A , 15(5):725–754, 2000

2000
[35]

Shang, Q

N. Shang, Q. Feng, and H. Qin. Some new transformation pr operties of the Nielsen generalized polylogarithm. Int. J. Math. Math. Sci. , pages Art. ID 210890, 10, 2014

2014
[36]

W ojtkowiak

Z. W ojtkowiak. The basic structure of polylogarithmic functional equations. In Structural properties of polylogarithms, volume 37 of Math. Surveys Monogr. , pages 205–231. Amer. Math. Soc., Providence, RI, 1991

1991
[37]

W ojtkowiak

Z. W ojtkowiak. Functional equations of iterated integ rals with regular singularities. Nagoya Math. J. , 142:145–159, 1996

1996
[38]

W ojtkowiak

Z. W ojtkowiak. Mixed Hodge structures and iterated int egrals. I. In Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998) , volume 3 of Int. Press Lect. Ser. , pages 121–208. Int. Press, Somerville, MA, 2002

1998
[39]

D. Zagier. Polylogarithms, Dedekind zeta functions an d the algebraic K-theory of ﬁelds. In Arithmetic algebraic geometry (Texel, 1989) , volume 89 of Progr. Math., pages 391–430. Birkh¨ auser Boston, Boston, MA, 1991

1989
[40]

D. Zagier. Special values and functional equations of p olylogarithms. In Structural properties of polylogarithms , vol- ume 37 of Math. Surveys Monogr. , pages 377–400. Amer. Math. Soc., Providence, RI, 1991

1991
[41]

D. Zagier. The dilogarithm function. In Frontiers in number theory, physics, and geometry. II , pages 3–65. Springer, Berlin, 2007

2007
[42]

Zagier and H

D. Zagier and H. Gangl. Classical and elliptic polyloga rithms and special values of L-series. In The arithmetic and geometry of algebraic cycles (Banﬀ, AB, 1998) , volume 548 of NATO Sci. Ser. C Math. Phys. Sci. , pages 561–615. Kluwer Acad. Publ., Dordrecht, 2000. Appendix A...

1998
[43]

for Li 2 (see [30, Equations 1.20 and 1.21], or [41, Section 1.1]): Li2(φ−2) = 2 5 ζ(2) − log2(φ) , Li2(φ−1) = 3 5 ζ(2) − log2(φ) , Li2(−φ−1) = − 2 5 ζ(2) + 1 2 log2(φ) , Li2(−φ) = − 3 5 ζ(2) − log2(φ) . Corresponding to these Li 2 evaluations, we have the following evaluation...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [9]

Charlton, C

S. Charlton, C. Duhr, F. Dulat, and H. Gangl. Clean single -valued multiple polylogarithms. In preparation

work page

[2] [1]

Bl¨ umlein, D

J. Bl¨ umlein, D. J. Broadhurst, and J. A. M. Vermaseren. T he multiple zeta value data mine. Comput. Phys. Comm. , 181(3):582–625, 2010

work page 2010

[3] [2]

J. M. Borwein, D. M. Bradley, and D. J. Broadhurst. Evalua tions of k-fold Euler/Zagier sums: a compendium of results for arbitrary k. Electron. J. Combin. , 4(2):Research Paper 5, approx. 21, 1997. The Wilf Festschr ift (Philadelphia, PA, 1996)

work page 1997

[4] [3]

J. M. Borwein and A. Straub. Mahler measures, short walks and log-sine integrals. Theoret. Comput. Sci. , 479:4–21, 2013

work page 2013

[5] [4]

J. M. Borwein and A. Straub. Relations for Nielsen polylo garithms. J. Approx. Theory , 193:74–88, 2015

work page 2015

[6] [5]

F. C. S. Brown. Representation theory of polylogarithms . Unpublished notes

work page

[7] [6]

F. C. S. Brown. Polylogarithmes multiples uniformes en u ne variable. C. R. Math. Acad. Sci. Paris , 338(7):527–532, 2004

work page 2004

[8] [7]

F. C. S. Brown. On the decomposition of motivic multiple z eta values. In Galois-Teichm¨ uller theory and arithmetic geometry, volume 63 of Adv. Stud. Pure Math. , pages 31–58. Math. Soc. Japan, Tokyo, 2012

work page 2012