The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable quantum field theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré–iterated integrals, including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of N is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument x = 1, respectively, for the cyclotomic harmonic sums at N → ∞, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle algebras and three multiple argument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight

$\sf{w=1,2}$
w=1,2 sums up to cyclotomy
$\sf{l=20}$
l=20
. This paper is dedicated to Martinus Veltman on the occasion of his 80th birthday.

1.
G.
't Hooft
and
M. J. G.
Veltman
,
Nucl. Phys. B
44
,
189
(
1972
).
2.
G.
't Hooft
and
M. J. G.
Veltman
,
Nucl. Phys. B
153
,
365
(
1979
);
G.
't Hooft
and
M. J. G.
Veltman
,
Diagrammar, NATO Adv. Study Inst. Ser., Ser. B
4
,
177
(
1974
);
G.
Passarino
and
M. J. G.
Veltman
,
Nucl. Phys. B
160
,
151
(
1979
).
3.
P.
Appell
,
Sur Les Fonctions Hypérgéometriques de Plusieurs Variables
(
Gauthier-Villars
,
Paris
,
1925
);
P.
Appell
and
J.
Kampé de Fériet
,
Fonctions Hypérgéometriques; Polynômes d'Hermite
(
Gauthier-Villars
,
Paris
,
1926
);
W. N.
Bailey
,
Generalized Hypergeometric Series
(
Cambridge University Press
,
Cambridge, England
,
1935
);
Higher Transcendental Functions
,
Bateman Manuscript Project
Vol.
I
, edited by
A.
Erdélyi
(
McGraw-Hill
,
New York
,
1953
);
L. J.
Slater
,
Generalized Hypergeometric Functions
(
Cambridge University Press
,
Cambridge, England
,
1966
);
H.
Exton
,
Multiple Hypergeometric Functions and Applications
(
Ellis Horwood Limited
,
Chichester
,
1976
);
H.
Exton
,
Handbook of Hypergeometric Integrals
(
Ellis Horwood Limited
,
Chichester
,
1978
).
4.
C. G.
Bollini
and
J. J.
Giambiagi
,
Nuovo Cimento B
12
,
20
(
1972
);
J. F.
Ashmore
,
Nuovo Cimento B
4
,
289
(
1972
);
G. M.
Cicuta
and
E.
Montaldi
,
Nuovo Cimento B
4
,
329
(
1972
).
5.
A. M.
Legendre
,
Mém. Inst. Fr.
10
,
416
(
1809
);
S. D.
Poisson
,
Mém. Inst. Fr.
257
,
57
(
1811
);
C. F.
Gauss
,
Comment. Gotting. Bd.
2
,
34
(
1812
).
6.
J.
Blümlein
,
Comput. Phys. Commun.
180
,
2218
(
2009
);
e-print [arXiv:0901.3106 [hep-ph]].
7.
J.
Blümlein
and
S.
Kurth
,
Phys. Rev. D
60
,
014018
(
1999
);
8.
J. A. M.
Vermaseren
,
Int. J. Mod. Phys. A
14
,
2037
(
1999
);
9.
M. E.
Hoffman
,
J. Algebra
194
,
477
(
1997
);
M. E.
Hoffman
,
Nucl. Phys. B (Proc. Suppl.)
135
,
215
(
2004
);
10.
L.
Euler
,
Novi Commentarii academiae scientiarum imperialis Petropolitanae
(
1775
), Vol.
20
, p.
140
, reprinted in Opera Omnia Ser I, (
B.G. Teubner
,
Berlin
, 1927), Vol. 15, p. 217;
D.
Zagier
, in
First European Congress of Mathematics
(
Paris
,
1992
), Vol.
II
;
D.
Zagier
,
Prog. Math.
120
,
497
(Birkhäuseere, Basel,
1994
).
11.
D. J.
Broadhurst
, e-print [hep-th/9604128];
D. J.
Broadhurst
and
D.
Kreimer
,
Phys. Lett. B
393
,
403
(
1997
);
e-print [hep-th/9609128];
J. M.
Borwein
,
D. M.
Bradley
,
D. J.
Broadhurst
, and
P.
Lisonek
,
Trans. Am. Math. Soc.
353
,
907
(
2001
);
J.
Blümlein
,
D. J.
Broadhurst
, and
J. A. M.
Vermaseren
,
Comput. Phys. Commun.
181
,
582
(
2010
);
e-print [arXiv:0907.2557 [math-ph]], and references therein.
12.
J. A. M.
Vermaseren
,
A.
Vogt
, and
S.
Moch
,
Nucl. Phys. B
724
,
3
(
2005
);
13.
S.
Moch
,
P.
Uwer
, and
S.
Weinzierl
,
J. Math. Phys.
43
,
3363
(
2002
);
e-print [hep-ph/0110083].
14.
J.
Ablinger
,
J.
Blümlein
, and
C.
Schneider
, (unpublished).
15.
J.
Ablinger
,
I.
Bierenbaum
,
J.
Blümlein
,
A.
Hasselhuhn
,
S.
Klein
,
C.
Schneider
, and
F.
Wissbrock
,
Nucl. Phys. B (Proc. Suppl.)
205–206
,
242
(
2010
);
e-print [arXiv:1007.0375 [hep-ph]].
16.
J.
Ablinger
,
J.
Blümlein
,
S.
Klein
,
C.
Schneider
, and
F.
Wissbrock
,
Nucl. Phys. B
844
,
26
(
2011
);
[PubMed]
e-print [arXiv:1008.3347 [hep-ph]].
17.
C.
Schneider
,
J. Symb. Comput.
43
,
611
(
2008
);
C.
Schneider
,
Ann. Comb.
9
,
75
(
2005
);
C.
Schneider
,
J. Differ. Equations
11
,
799
(
2005
);
C.
Schneider
,
Ann. Comb.
14
(
4
),
533
(
2010
);
e-print [arXiv:0808.2596];
in
Proceedings of the Conference on Motives, Quantum Field Theory, and Pseudodifferential Operators, Clay Mathematics Institute, Boston University, Cambridge, MA, 2–13 June, 2008
;
Clay Mathematics Proceedings
, edited by
A.
Carey
,
D.
Ellwood
,
S.
Paycha
, and
S.
Rosenberg
(
Clay Mathematical Institute
,
Boston, MA
,
2010
), Vol. 12, p.
285
;
C.
Schneider
,
Sém. Lothar. Combin.
56
,
1
(
2007
), Article B56b, Habilitationsschrift JKU Linz (2007), and references therein;
J.
Ablinger
,
J.
Blümlein
,
S.
Klein
, and
C.
Schneider
,
Nucl. Phys. B (Proc. Suppl.)B
,
110
(
2010
);
e-print [arXiv:1006.4797 [math-ph]].
18.
See, e.g.,
J.
Zhao
, e-print arXiv:0707.1459 [mathNT];
J.
Zhao
,
Acad. Sci., Paris, C. R.
346
,
1029
(
2008
);
J.
Zhao
,
Doc. Math.
15
,
1
(
2010
);
J.
Zhao
,
J. Algebra Appl.
9
,
327
(
2010
);
e-print [arXiv:0904.0888[mathNT]]; [cf. also footnote 8 of Ref. 29].
19.
A. B.
Goncharov
,
Math. Res. Lett.
5
,
497
(
1998
).
20.
D. J.
Broadhurst
,
Eur. Phys. J. C
8
,
311
(
1999
);
21.
M.
Bigotte
,
G.
Jacob
,
N. E.
Oussous
, and
M.
Petitot
,
Theor. Comput. Sci
273
,
271
(
2002
).
22.
S.
Weinzierl
,
J. Math. Phys.
45
2656
(
2004
);
e-print [hep-ph/0402131];
M. Y.
Kalmykov
,
B. F. L.
Ward
, and
S.
Yost
,
J. High Energy Phys.
0702
,
040
(
2007
);
e-print [hep-th/0612240];
M. Y.
Kalmykov
,
J. High Energy Phys.
0604
,
056
(
2006
);
e-print [hep-th/0602028];
M. Y.
Kalmykov
,
B. F. L.
Ward
, and
S. A.
Yost
,
J. High Energy Phys.
0710
,
048
(
2007
);
e-print [arXiv:0707.3654 [hep-th]];
T.
Huber
and
D.
Maitre
,
Comput. Phys. Commun.
178
,
755
(
2008
);
e-print [arXiv:0708.2443 [hep-ph]];
M. Y.
Kalmykov
and
B. A.
Kniehl
,
Nucl. Phys. B
809
,
365
(
2009
);
e-print [arXiv:0807.0567 [hep-th]].
23.
S.
Lang
,
Algebra
, 3rd ed. (
Springer
,
New York
,
2002
).
24.
L.
Euler
,
Novi Commentarii academiae scientiarum imperialis Petropolitanae
(
1760
), Vol.
8
, p.
74
;
M.
Takase
, “
Euler's Theory of Numbers
,” in
Euler Reconsidered
, edited by
R.
Baker
(
Kedrick
,
Heber City, UT
,
2007
), p.
377
;
25.
H.
Poincaré
,
Acta Math.
4
,
201
(
1884
);
J. A.
Lappo-Danilevsky
,
Mémoirs sur la Théorie des Systèmes Différentielles Linéaires
(
Chelsea
,
New York
,
1953
);
K. T.
Chen
,
Trans. A.M.S.
156
(
3
),
359
(
1971
).
26.
E.
Remiddi
and
J. A. M.
Vermaseren
,
Int. J. Mod. Phys. A
15
,
725
(
2000
);
27.
J.
Ablinger
, code harmonicsums (unpublished).
28.
J.
Blümlein
,
Comput. Phys. Commun.
159
,
19
(
2004
);
29.
J.
Blümlein
,
S.
Klein
,
C.
Schneider
, and
F.
Stan
, e-print arXiv:1011.2656 [cs.SC].
30.
J.
Blümlein
, “
Structural Relations of Harmonic Sums and Mellin Transforms at Weight w=6
,” in
Proceedings of the Conference on Motives, Quantum Field Theory, and Pseudodifferential Operators, Clay Mathematics Institute, Boston University, Cambridge, MA, June 2–13, 2008; Clay Mathematics Proceedings
edited by
A.
Carey
,
D.
Ellwood
,
S.
Paycha
, and
S.
Rosenberg
(
2010
), Vol.
12
, p.
167
;
e-print [arXiv:0901.0837 [math-ph]].
31.
J.
Ablinger
,
J.
Blümlein
, and
C.
Schneider
(unpublished).
32.
J.
Blümlein
,
Comput. Phys. Commun.
133
,
76
(
2000
);
S. I.
Alekhin
and
J.
Blümlein
,
Phys. Lett. B
594
,
299
(
2004
);
J.
Blümlein
and
S. O.
Moch
,
Phys. Lett. B
614
,
53
(
2005
);
A. V.
Kotikov
and
V. N.
Velizhanin
, e-print [hep-ph/0501274];
S.
Albino
,
Phys. Lett. B
674
,
41
(
2009
);
e-print [arXiv:0902.2148 [hep-ph]].
33.
T.
Gehrmann
and
E.
Remiddi
,
Comput. Phys. Commun.
141
,
296
(
2001
);
34.
J.
Vollinga
and
S.
Weinzierl
,
Comput. Phys. Commun.
167
,
177
(
2005
);
35.
T.
Gehrmann
and
E.
Remiddi
,
Comput. Phys. Commun.
144
,
200
(
2002
);
R.
Bonciani
,
G.
Degrassi
, and
A.
Vicini
, e-print arXiv:1007.1891 [hep-ph].
36.
N.
Nielsen
,
Handbuch der Theorie der Gammafunktion
(
Teubner
,
Leipzig
,
1906
); reprinted by (
Chelsea
,
Bronx, New York
,
1965
).
37.
J.
Stirling
,
Methodus differentialis sive tractatus de summatione et interpolatione serierum infinitarum
(
Impensis Ric. Manby
,
London
,
1730
), p.
27
.
38.
K. G.
Chetyrkin
and
M.
Steinhauser
,
Nucl. Phys. B
573
,
617
(
2000
);
e-print [hep-ph/9911434].
39.
A. I.
Davydychev
and
M. Y.
Kalmykov
,
Nucl. Phys. B
605
,
266
(
2001
);
e-print [hep-th/0012189].
40.
L. G.
Almeida
and
C.
Sturm
,
Phys. Rev.
D82
,
054017
(
2010
);
e-print [arXiv:1004.4613 [hep-ph]];
J. A.
Gracey
, e-print arXiv:1104.5382 [hep-ph].
41.
C.
Reutenauer
,
Free Lie Algebras
(
Calendron
,
Oxford
,
1993
).
42.
E.
Witt
,
J. Reine Angew. Math.
177
,
152
(
1937
);
E.
Witt
,
Math. Zeitschr.
64
,
195
(
1956
).
43.
A. F.
Möbius
,
J. Reine Angew. Math.
9
,
105
(
1832
);
G. W.
Hardy
and
E. M.
Wright
,
An Introduction to the Theory of Numbers
, 5th ed. (
Calendron
,
Oxford
,
1979
).
44.
E.
Catalan
,
Mem. Acad. Imp. Sci. Saint-Pétersbourg, Ser.
7
,
31
(
1883
);
45.
A.
Migotti
,
Sitzungsber Math.-Naturwiss. Kl Kaiser. Akad. Wiss. Wien
87
,
7
(
1883
);
P.
Erdös
,
Bull. Am. Math. Soc.
52
,
179
(
1946
);
M.
Endo
,
Comment. Math. Univ. St. Pauli
23
,
121
(
1974
);
R.
Thangadurai
, “
On the coefficients of cyclotomic polynomials
,” in
Cyclotomic Fields and Related Topics
(
Bhaskaracharya
,
Pune
,
2000
), p.
311
.
46.
E.
Landau
,
Sitzungsber. Math.-Naturw. Kl. Bayerische Akad. Wiss. München
36
,
151
(
1906
).
47.
C. F.
Gauss
,
Comment. Gotting.
2
,
33
(
1812
);
J. L. W. V.
Jensen
,
Nyt Tidsskr Math.
2B
,
33
(
1891
).
48.
C. F.
Gauss
,
Discquisitiones Arithmeticae
(
Leipzig/Gerhard Fleischer
,
1801
), pp.
365
,
366
;
É.
Galois
, (uvre mathématiques publiées en 1846 dans le Journal de Liouville), Tome XI
381
444
(
1846
).
49.
See http://mathworld.wolfram.com/TrigonometryAnglesPi5.html and similar pages for trigonometric functions of cyclotomic angles.
50.
Ch.
Hermite
,
Acad. Sci., Paris, C. R.
77
,
18
(
1873
);
F.
Lindemann
,
Math. Ann.
20
,
213
(
1882
);
K.
Weierstrass
,
Sitzungsber. K. Preuss. Akad. Wiss.
2
,
1067
(
1885
);
A.
Baker
,
Transcendental Number Theory
(
Cambridge University Press
,
Cambridge, England
,
1975
), and references therein.
51.
H. M.
Srivastave
and
J.
Choi
,
Series Associated with the Zeta and Related Functions
(
Kluwer Academic Publishers
,
Dordrecht
,
2001
).
52.
A.
Hurwitz
,
Z Math. Phys.
27
,
86
(
1882
);
T. M.
Apostol
,
Introduction to Analytic Number Theory
(
Springer
,
Berlin
,
1976
), Chap. 12.
53.
M. W.
Coffey
,
J. Math. Phys.
49
,
043510
(
2008
);
D.
Cvijovic
,
J. Math. Phys.
50
,
023515
(
2009
).
54.
S.
Takakazu
,
Katsuyo Sampo
(
Syotsudo
,
Edo
,
1712
).
55.
J.
Bernoulli
,
Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis
(
Brüder Thurneysen
,
Basel
,
1713
).
56.
J.
Miller
and
V. S.
Adamchik
,
J. Comput. Appl. Math.
100
,
201
(
1998
).
57.
K. S.
Kölbig
,
J. Comput. Appl. Math.
75
,
43
(
1996
).
58.
L.
Saalschütz
,
Vorlesungen über die Bernoullischen Zahlen
(
Springer
,
Berlin
,
1893
).
59.
N.
Nielsen
,
Traité élémentaire des nombres de Bernoulli
(
Gauthier-Villars
,
Paris
,
1923
).
60.
L.
Euler
,
Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum
II (
Typographeo Petro Galeatii
,
Ticini
,
1787
), Chap. VIII, pp.
224
226
, 1st issue (Academiae Imperialis Scientiarum, St. Petersburg,
1755
).
61.
H. F.
Scherk
, “
Erste Abhandlung - Betrifft die Bernoullischen und die Eulerschen Zahlen
,” in
Mathematische Abhandlungen
(
G. Reimer
,
Berlin
,
1825
).
62.
M.
Lerch
,
Enseign. Math.
5
,
450
(
1903
);
A.
Laurinčikas
and
R.
Garunkštis
,
The Lerch Zeta-function
(
Kluwer Academic Publishers
,
Dordrecht
,
2010
), and references therein.
63.
W.
Spence
,
An Essay on Logarithmic Transcendents
(
John Murray, London and Archibald Constable and Company
,
Edinburgh
,
1809
).
64.
L.
Lewin
,
Dilogarithms and Associated Functions
(
Macdonald
,
London
,
1958
);
Polylogarithms and Associated Functions
(
North-Holland
,
New York
,
1981
);
D.
Zagier
, in
Frontiers in Number Theory, Physics, and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization
, edited by
P.
Cartier
 et al (
Springer
,
Berlin
,
2006
), Vol.
2
, p.
3
.
65.
R.
Clausen
,
J. Reine Angew. Math.
8
,
298
(
1832
).
66.
D.
Cvijović
and
J.
Klinkowski
,
Math. Comput.
68
,
1623
(
1999
);
D.
Cvijović
,
J. Math. Anal. Appl.
332
,
1056
(
2007
).
67.
A. M.
Legendre
,
Exercices de calcul intégral, tome
1
,
247
(
1811
).
68.
See Ref. 60, p. 499 (edition of 1755) and Ref. 59.
69.
B. C.
Berndt
,
Ramanujan's Notebooks
, Part I (
Springer
,
Berlin
,
1985
).
70.
R.
Sitaramachandrarao
,
J. Number Theory
25
,
1
(
1987
).
71.
N.
Nielsen
, “
Der Eulersche Dilogarithmus und seine Verallgemeinerungen
,” in
Nova Acta Leopoldina
, (
Halle
,
1909
), Vol.
XC
, Nr. 3;
K. S.
Kölbig
,
SIAM J. Math. Anal.
16
,
1232
(
1986
).
72.
P. F.
Jordan
,
Bull. Am. Math. Soc.
79
,
681
(
1973
).
73.
M. S.
Milgram
, e-print arXiv:math/0406338v2.
74.
O. M.
Ogreid
and
P.
Osland
,
J. Comput. Appl. Math.
98
,
245
(
1998
);
e-print [hep-th/9801168];
M. W.
Coffey
,
J. Comput. Appl. Math.
183
,
84
(
2005
);
e-print [math-ph/0505051].
75.
B. J.
Laurenzi
, e-print arXiv:1010.6229 [math-ph].
76.
P.
Deligne
,
Publ. Math., Inst. Hautes Etud. Sci.
112
,
101
141
(
2010
);
77.
P.
Deligne
and
A. B.
Gocharov
, Groupes fondamentaux motivique de Tate mixte, IAS preprint, see http://www.math.ias.edu/people/faculty/deligne/preprints.
78.
N. J. A.
Sloane
, The On-Line Encyclopedia of Integer Sequences; see http://oeis.org/.
79.
H. R. P.
Ferguson
and
D. H.
Bailey
, RNR Technical Report RNR-91-032, July 14,
1992
.
80.
F.
Vieta
,
Opera mathematica
,
1579
; reprinted (
Officina Bonaventurae & Abrahami Elzeviriorum
,
Leiden
,
1646
).
81.
H.
Lugowski
and
J.
Weinert
,
Grundzüge der Algebra
(
Teubner
,
Leipzig
,
1960
), Vol.
III
.
82.
G.
Racinet
,
C. R. Math. Acad. Sci.
333
(
1
),
5
(
2001
);
83.
F. W.
Newman
,
Cambridge Dublin Math. J.
II
,
77
100
(
1847
);
F. W.
Newman
,
Cambridge Dublin Math. J.
II
,
172
191
(
1847
).
84.
A. N.
Kirillov
,
Prog. Theor. Phys. Suppl.
118
,
61
(
1995
);
e-print [hep-th/9408113].
85.
M. Y.
Kalmykov
and
B. A.
Kniehl
,
Nucl. Phys. Proc. Suppl.
205–206
,
129
(
2010
);
e-print [arXiv:1007.2373 [math-ph]].
86.
T.
Nagell
,
Introduction to Number Theory
(
Wiley
,
New York
,
1951
).
87.
Complex representations are related to the so-called colored harmonic sums
$S_{b,\vec{a}}(p, \vec{r}; N) = \sum _{k=1}^N \frac{p^k}{k^b} S_{\vec{a}}(\vec{r}; k)$
Sb,a(p,r;N)=k=1NpkkbSa(r;k)
with
$b, a_i \in {\mathbb N}_+, p, r_i \in \cup _{l = 2}^M \lbrace \exp [2\pi i (n/l)], n \in \lbrace 1,\ldots,l-1\rbrace \rbrace$
b,aiN+,p,ril=2M{exp[2πi(n/l)],n{1,...,l1}}
, cf. Refs. 18 and 19.
88.
Note that we defined here the second letter by 1/(x − 1) which differs in sign from the corresponding letter in Ref. 26. Numerical implementations were given in Refs. 33 and 34. A few extensions of iterated integrals introduced in Ref. 26 based on linear denominator functions of different kind, which are used in quantum-field theoretic calculations, were made in Refs. 34 and 35.
89.
For the relations given in this section we mostly present the results, giving for a few cases the proofs in Appendix  B. The other proofs proceed in a similar manner.
90.
See Ref. 49 for special values of the trigonometric functions occurring in (5.6).
91.
Special examples were also considered in Ref. 53.
92.
We corrected typos in Eq. (10) of Ref. 66.
93.
For similar sums see Ref. 74.
94.
Relations between colored nested infinite harmonic sums have been investigated also in Refs. 76 and 77 recently.
95.
We would like to thank D. Broadhurst for communicating this relation to us.
You do not currently have access to this content.