We consider finite iterated generalized harmonic sums weighted by the binomial

$\binom{2k}{k}$
2kk in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator insertions starting at 3-loop order in the coupling constant and extends the classes of the nested harmonic, generalized harmonic, and cyclotomic sums. The binomially weighted sums are associated by the Mellin transform to iterated integrals over square-root valued alphabets. The values of the sums for N → ∞ and the iterated integrals at x = 1 lead to new constants, extending the set of special numbers given by the multiple zeta values, the cyclotomic zeta values and special constants which emerge in the limit N → ∞ of generalized harmonic sums. We develop algorithms to obtain the Mellin representations of these sums in a systematic way. They are of importance for the derivation of the asymptotic expansion of these sums and their analytic continuation to
$N \in \mathbb {C}$
NC
. The associated convolution relations are derived for real parameters and can therefore be used in a wider context, as, e.g., for multi-scale processes. We also derive algorithms to transform iterated integrals over root-valued alphabets into binomial sums. Using generating functions we study a few aspects of infinite (inverse) binomial sums.

1.
S. S.
Schweber
,
QED and the Men Who Made it: Dyson, Feynman, Schwinger, and Tomonaga
(
Princeton University Press
,
Princeton, NJ
,
1994
), and references therein.
2.
S. L.
Glashow
, “
Partial symmetries of weak interactions
,”
Nucl. Phys.
22
,
579
588
(
1961
).
3.
S.
Weinberg
, “
A model of leptons
,”
Phys. Rev. Lett.
19
,
1264
1266
(
1967
).
4.
G.
't Hooft
, “
Renormalization of massless Yang-Mills fields
,”
Nucl. Phys. B
33
,
173
199
(
1971
).
5.
G.
't Hooft
, “
Renormalizable Lagrangians for massive Yang-Mills fields
,”
Nucl. Phys. B
35
,
167
188
(
1971
).
6.
G.
't Hooft
and
M. J. G.
Veltman
, “
Regularization and Renormalization of gauge fields
,”
Nucl. Phys. B
44
,
189
213
(
1972
).
7.
G.
't Hooft
and
M. J. G.
Veltman
, “
Scalar one loop integrals
,”
Nucl. Phys. B
153
,
365
401
(
1979
).
8.
D. J.
Gross
and
F.
Wilczek
, “
Ultraviolet behavior of nonabelian gauge theories
,”
Phys. Rev. Lett.
30
,
1343
1345
(
1973
).
9.
H. D.
Politzer
, “
Reliable perturbative results for strong interactions?
,”
Phys. Rev. Lett.
30
,
1346
1349
(
1973
).
10.
H.
Fritzsch
,
M.
Gell-Mann
, and
H.
Leutwyler
, “
Advantages of the color octet gluon picture
,”
Phys. Lett. B
47
,
365
368
(
1973
).
11.
G.
't Hooft
and
M. J. G.
Veltman
, “
Diagrammar
,”
NATO Adv. Study Inst. Ser. B Phys.
4
,
177
322
(
1974
).
12.
M. J. G.
Veltman
,
Diagrammatica: The Path to Feynman Diagrams
,
Cambridge Lecture Notes in Physics
Vol.
4
(
Cambridge University Press
,
Cambridge
,
1994
).
13.
J.
Ablinger
and
J.
Blümlein
, “
Harmonic sums, polylogarithms, special numbers, and their generalizations
,”
Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions
,
Texts & Monographs in Symbolic Computation
, edited by
C.
Schneider
and
J.
Blümlein
(
Springer
,
Wien
,
2013
), pp.
1
32
; e-print arXiv:1304.7071 [math-ph].
14.
J.
Ablinger
,
J.
Blümlein
, and
C.
Schneider
, “
Generalized harmonic, cyclotomic, and binomial sums, their polylogarithms and special numbers
,”
J. Phys. Conf. Ser.
523
,
012060
(
2014
); e-print arXiv:1310.5645 [math-ph].
15.
J. M.
Borwein
,
D. M.
Bradley
,
D. J.
Broadhurst
, and
P.
Lisonek
, “
Special values of multiple polylogarithms
,”
Trans. Am. Math. Soc.
353
,
907
941
(
2001
); e-print arXiv:math/9910045 [math-ca].
16.
J.
Blümlein
,
D. J.
Broadhurst
, and
J. A. M.
Vermaseren
, “
The multiple Zeta value data mine
,”
Comput. Phys. Commun.
181
,
582
625
(
2010
); e-print arXiv:0907.2557 [math-ph], and references therein.
17.
S.
Laporta
and
E.
Remiddi
, “
The analytical value of the electron (g − 2) at order α3 in QED
,”
Phys. Lett. B
379
,
283
291
(
1996
); e-print arXiv:hep-ph/9602417.
18.
T.
van Ritbergen
,
J. A. M.
Vermaseren
, and
S. A.
Larin
, “
The four loop β-function in quantum chromodynamics
,”
Phys. Lett. B
400
,
379
384
(
1997
); e-print arXiv:hep-ph/9701390.
19.
M.
Czakon
, “
The four-loop QCD β-function and anomalous dimensions
,”
Nucl. Phys. B
710
,
485
498
(
2005
); e-print arXiv:hep-ph/0411261.
20.
P. A.
Baikov
,
K. G.
Chetyrkin
,
J. H.
Kühn
, and
J.
Rittinger
, “
Complete
${\cal O}(\alpha _s^4)$
O(αs4)
QCD corrections to hadronic Z-decays
,”
Phys. Rev. Lett.
108
,
222003
(
2012
); e-print arXiv:1201.5804 [hep-ph].
21.
J. A. M.
Vermaseren
, “
Harmonic sums, Mellin transforms and integrals
,”
Int. J. Mod. Phys. A
14
,
2037
2076
(
1999
); e-print arXiv:hep-ph/9806280.
22.
J.
Blümlein
and
S.
Kurth
, “
Harmonic sums and Mellin transforms up to two loop order
,”
Phys. Rev. D
60
,
014018
(
1999
); e-print arXiv:hep-ph/9810241.
23.
E.
Remiddi
and
J. A. M.
Vermaseren
, “
Harmonic polylogarithms
,”
Int. J. Mod. Phys. A
15
,
725
754
(
2000
); e-print arXiv:hep-ph/9905237.
24.
J.
Ablinger
,
J.
Blümlein
, and
C.
Schneider
, “
Harmonic sums and polylogarithms generated by cyclotomic polynomials
,”
J. Math. Phys.
52
,
102301
(
2011
); e-print arXiv:1105.6063 [math-ph].
25.
S.
Moch
,
P.
Uwer
, and
S.
Weinzierl
, “
Nested sums, expansion of transcendental functions and multiscale multiloop integrals
,”
J. Math. Phys.
43
,
3363
3386
(
2002
); e-print arXiv:hep-ph/0110083.
26.
J.
Ablinger
,
J.
Blümlein
, and
C.
Schneider
, “
Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms
,”
J. Math. Phys.
54
,
082301
(
2013
); e-print arXiv:1302.0378 [math-ph].
27.
C.
Reutenauer
,
Free Lie Algebras
(
Oxford University Press
,
1993
).
28.
M.
Hoffman
, “
Quasi-shuffle products
,”
J. Algebraic Combin.
11
,
49
68
(
2000
); e-print arXiv:math/9907173 [math.QA].
29.
J.
Blümlein
, “
Algebraic relations between harmonic sums and associated quantities
,”
Comput. Phys. Commun.
159
,
19
54
(
2004
); e-print arXiv:hep-ph/0311046.
30.
J.
Blümlein
, “
Structural relations of harmonic sums and Mellin transforms up to weight w = 5
,”
Comput. Phys. Commun.
180
,
2218
2249
(
2009
); e-print arXiv:0901.3106 [hep-ph].
31.
J.
Blümlein
, “
Structural relations of harmonic sums and Mellin transforms at weight w = 6
,” in
Motives, Quantum Field Theory, and Pseudodifferential Operators
,
Clay Mathematics Proceedings
, Vol.
12
, edited by
A.
Carey
,
D.
Ellwood
,
S.
Paycha
, and
S.
Rosenberg
(
American Mathematical Society
,
2010
), pp.
167
186
; arXiv:0901.0837 [math-ph].
32.
J. A. M.
Vermaseren
,
A.
Vogt
, and
S.-O.
Moch
, “
The third-order QCD corrections to deep-inelastic scattering by photon exchange
,”
Nucl. Phys. B
724
,
3
182
(
2005
); e-print arXiv:hep-ph/0504242.
33.
J.
Ablinger
,
J.
Blümlein
,
S.
Klein
,
C.
Schneider
, and
F.
Wißbrock
, “
The
$O(\alpha _s^3)$
O(αs3)
massive operator matrix elements of O(Nf) for the structure function F2(x, Q2) and transversity
,”
Nucl. Phys. B
844
,
26
54
(
2011
); e-print arXiv:1008.3347 [hep-ph].
34.
J.
Blümlein
,
A.
Hasselhuhn
,
S.
Klein
, and
C.
Schneider
, “
The
$O(\alpha _s^3 N_f T_F^2 C_{A,F})$
O(αs3NfTF2CA,F)
contributions to the gluonic massive operator matrix elements
,”
Nucl. Phys. B
866
,
196
211
(
2013
); e-print arXiv:1205.4184 [hep-ph].
35.
J.
Ablinger
,
J.
Blümlein
,
A.
Hasselhuhn
,
S.
Klein
,
C.
Schneider
, and
F.
Wißbrock
, “
Massive 3-loop ladder diagrams for quarkonic local operator matrix elements
,”
Nucl. Phys. B
864
,
52
84
(
2012
); e-print arXiv:1206.2252 [hep-ph].
36.
J.
Ablinger
,
J.
Blümlein
,
A.
De Freitas
,
A.
Hasselhuhn
,
A.
von Manteuffel
,
M.
Round
,
C.
Schneider
, and
F.
Wißbrock
, “
The transition matrix element Agq(N) of the variable flavor number scheme at
$O(\alpha _s^3)$
O(αs3)
,”
Nucl. Phys. B
882
,
263
288
(
2014
); e-print arXiv:1402.0359 [hep-ph].
37.
J.
Ablinger
,
A.
Behring
,
J.
Blümlein
,
A.
De Freitas
,
A.
Hasselhuhn
,
A.
von Manteuffel
,
M.
Round
,
C.
Schneider
, and
F.
Wißbrock
, “
The 3-loop non-singlet heavy flavor contributions and anomalous dimensions for the structure function F2(x, Q2) and transversity
,”
Nucl. Phys. B
886
,
733
823
(
2014
); e-print arXiv:1406.4654 [hep-ph].
38.
J.
Ablinger
,
A.
Behring
,
J.
Blümlein
,
A.
De Freitas
,
A.
von Manteuffel
, and
C.
Schneider
, “
The 3-loop pure singlet heavy flavor contributions to the structure function F2(x, Q2) and the anomalous dimension
,”
Nucl. Phys. B
(in press); preprint arXiv:1407.7832 [hep-ph].
39.
M.
Buza
,
Y.
Matiounine
,
J.
Smith
,
R.
Migneron
, and
W. L.
van Neerven
, “
Heavy quark coefficient functions at asymptotic values Q2 ≫ m2
,”
Nucl. Phys. B
472
,
611
658
(
1996
); e-print arXiv:hep-ph/9601302.
40.
M.
Buza
,
Y.
Matiounine
,
J.
Smith
, and
W. L.
van Neerven
, “
$O(\alpha _s^2)$
O(αs2)
corrections to polarized heavy flavor production at Q2 ≫ m2
,”
Nucl. Phys. B
485
,
420
456
(
1997
); e-print arXiv:hep-ph/9608342.
41.
M.
Buza
,
Y.
Matiounine
,
J.
Smith
, and
W. L.
van Neerven
, “
Charm electroproduction viewed in the variable flavor number scheme versus fixed order perturbation theory
,”
Eur. Phys. J. C
1
,
301
320
(
1998
); e-print arXiv:hep-ph/9612398.
42.
M.
Buza
and
W. L.
van Neerven
, “
$O(\alpha _s^2)$
O(αs2)
contributions to charm production in charged current deep inelastic lepton-hadron scattering
,”
Nucl. Phys. B
500
,
301
324
(
1997
); e-print arXiv:hep-ph/9702242.
43.
I.
Bierenbaum
,
J.
Blümlein
, and
S.
Klein
, “
Two-loop massive operator matrix elements and unpolarized heavy flavor production at asymptotic values Q2 ≫ m2
,”
Nucl. Phys. B
780
,
40
75
(
2007
); e-print arXiv:hep-ph/0703285.
44.
I.
Bierenbaum
,
J.
Blümlein
, and
S.
Klein
, “
Two-loop massive operator matrix elements for polarized and unpolarized deep-inelastic scattering
,” e-print arXiv:0706.2738 [hep-ph].
45.
I.
Bierenbaum
,
J.
Blümlein
,
S.
Klein
, and
C.
Schneider
, “
Two-loop massive operator matrix elements for unpolarized heavy flavor production to O(ɛ)
,”
Nucl. Phys. B
803
,
1
41
(
2008
); e-print arXiv:0803.0273 [hep-ph].
46.
I.
Bierenbaum
,
J.
Blümlein
, and
S.
Klein
, “
The gluonic operator matrix elements at
$O(\alpha _s^2)$
O(αs2)
for DIS heavy flavor production
,”
Phys. Lett. B
672
,
401
406
(
2009
); e-print arXiv:0901.0669 [hep-ph].
47.
J.
Blümlein
,
A.
Hasselhuhn
, and
T.
Pfoh
, “
The
$O(\alpha _s^2)$
O(αs2)
heavy quark corrections to charged current deep-inelastic scattering at large virtualities
,”
Nucl. Phys. B
881
,
1
41
(
2014
); e-print arXiv:1401.4352 [hep-ph].
48.
A.
Behring
,
I.
Bierenbaum
,
J.
Blümlein
,
A.
De Freitas
,
S.
Klein
, and
F.
Wißbrock
, “
The logarithmic contributions to the
$O(\alpha _s^3)$
O(αs3)
asymptotic massive Wilson coefficients and operator matrix elements in deeply inelastic scattering
,”
Eur. Phys. J. C
74
,
3033
(
2014
); e-print arXiv:1403.6356 [hep-ph].
49.
J.
Ablinger
,
J.
Blümlein
,
A.
De Freitas
,
A.
Hasselhuhn
,
A.
von Manteuffel
,
M.
Round
, and
C.
Schneider
, “
The
$O(\alpha _s^3 T_F^2)$
O(αs3TF2)
contributions to the gluonic operator matrix element
,”
Nucl. Phys. B
885
,
280
317
(
2014
); e-print arXiv:1405.4259 [hep-ph].
50.
J.
Ablinger
,
J.
Blümlein
,
C.
Raab
,
C.
Schneider
, and
F.
Wißbrock
, “
Calculating massive 3-loop graphs for operator matrix elements by the method of hyperlogarithms
,”
Nucl. Phys. B
885
,
409
447
(
2014
); e-print arXiv:1403.1137 [hep-ph].
51.
U.
Aglietti
and
R.
Bonciani
, “
Master integrals with 2 and 3 massive propagators for the 2 loop electroweak form-factor - planar case
,”
Nucl. Phys. B
698
,
277
318
(
2004
); e-print arXiv:hep-ph/0401193.
52.
J.
Fleischer
,
A. V.
Kotikov
, and
O. L.
Veretin
, “
Analytic two loop results for self-energy type and vertex type diagrams with one nonzero mass
,”
Nucl. Phys. B
547
,
343
374
(
1999
); e-print arXiv:hep-ph/9808242.
53.
C.
Schneider
, “
Simplifying multiple sums in difference fields
,” in
Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions
,
Texts & Monographs in Symbolic Computation
, edited by
C.
Schneider
and
J.
Blümlein
(
Springer
,
Wien
,
2013
), pp.
325
360
; e-print arXiv:1304.4134 [cs.SC].
54.
J. M.
Borwein
,
D. J.
Broadhurst
, and
J.
Kamnitzer
, “
Central binomial sums, multiple Clausen values, and zeta values
,”
Exp. Math.
10
,
25
34
(
2001
).
55.
R.
Osburn
and
C.
Schneider
, “
Gaussian hypergeometric series and extensions of supercongruences
,”
Math. Comput.
78
,
275
292
(
2009
).
56.
O. M.
Ogreid
and
P.
Osland
,
J. Comput. Appl. Math.
98
,
245
271
(
1998
); e-print arXiv:hep-th/9801168.
57.
A. I.
Davydychev
and
M. Y.
Kalmykov
,
Nucl. Phys. B
605
,
266
318
(
2001
); e-print arXiv:hep-th/0012189.
58.
M. Y.
Kalmykov
and
O.
Veretin
, “
Single scale diagrams and multiple binomial sums
,”
Phys. Lett. B
483
,
315
323
(
2000
); e-print arXiv:hep-th/0004010.
59.
F.
Jegerlehner
,
M. Y.
Kalmykov
, and
O.
Veretin
, “
$\overline{\rm MS}$
MS ¯
versus pole masses of gauge bosons. 2. Two loop electroweak fermion corrections
,”
Nucl. Phys. B
658
,
49
112
(
2003
); e-print arXiv:hep-ph/0212319.
60.
A. I.
Davydychev
and
M. Y.
Kalmykov
, “
Massive Feynman diagrams and inverse binomial sums
,”
Nucl. Phys. B
699
,
3
64
(
2004
); e-print arXiv:hep-th/0303162.
61.
S.
Weinzierl
, “
Expansion around half integer values, binomial sums and inverse binomial sums
,”
J. Math. Phys.
45
,
2656
2673
(
2004
); e-print arXiv:hep-ph/0402131.
62.
M. Y.
Kalmykov
,
B. F. L.
Ward
, and
S. A.
Yost
, “
Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ɛ-expansion of generalized hypergeometric functions with one half-integer value of parameter
,”
JHEP
0710
,
048
(
2007
); e-print arXiv:0707.3654 [hep-th].
63.
H.
Mellin
, “
Über die fundamentale Wichtigkeit des Satzes von Cauchy für die Theorien der Gamma- und hypergeometrischen Funktionen
,”
Acta Soc. Fennicae
21
,
1
115
(
1896
).
64.
H.
Mellin
, “
Über den Zusammenhang zwischen den linearen Differential- und Differenzen- gleichungen
,”
Acta Math.
25
,
139
164
(
1902
).
65.
E. E.
Kummer
, “
Ueber die Transcendenten, welche aus wiederholten Integrationen rationaler Formeln entstehen
,”
J. Reine Angew. Math.
21
,
74
90
(
1840
).
66.
H.
Poincaré
, “
Sur les groupes des équations linéaires
,”
Acta Math.
4
,
201
312
(
1884
).
67.
J.
Blümlein
, “
Analytic continuation of Mellin transforms up to two loop order
,”
Comput. Phys. Commun.
133
,
76
104
(
2000
); e-print arXiv:hep-ph/0003100.
68.
J.
Blümlein
and
S.
Moch
, “
Analytic continuation of the harmonic sums for the 3-loop anomalous dimensions
,”
Phys. Lett. B
614
,
53
61
(
2005
); e-print arXiv:hep-ph/0503188.
69.
N.
Nielsen
,
Handbuch der Theorie der Gammafunktion
(
B. G. Teubner
,
Leipzig
,
1906
).
70.
E.
Landau
, “
Über die Grundlagen der Theorie der Fakultätenreihen
,”
Sitzber. d. Bayerische Akad. d. Wissenschaften
XXXVI
(
Heft I
),
151
218
(
1906
).
71.
J.
Stirling
,
Methodus differentialis sive tractatus de summatione et interpolatione serierum infinitarum
, (
Gul. Bowyer
,
London
,
1730
), p.
27
.
72.
R.
Ree
, “
Lie elements and an algebra associated with shuffles
,”
Ann. Math.
68
,
210
220
(
1958
).
73.
M.
Deneufchâtel
,
G. H. E.
Duchamp
,
V.
Hoang Ngoc Minh
, and
A. I.
Solomon
, “
Independence of hyperlogarithms over function fields via algebraic combinatorics
,” in
Proceedings of CAI
(
CAI
,
Linz, Austria
,
2011
),
127
139
; e-print arXiv:1101.4497.
74.
R. H.
Risch
, “
The problem of integration in finite terms
,”
Trans. Am. Math. Soc.
139
,
167
189
(
1969
).
75.
B. M.
Trager
, “
Integration of simple radical extensions
,” in
Proceedings of EUROSAM'79
(
EUROSAM'79
,
Marseille, France
,
1979
), pp.
408
414
.
76.
R.
Bonciani
,
G.
Degrassi
, and
A.
Vicini
, “
On the generalized Harmonic polylogarithms of one complex variable
,”
Comput. Phys. Commun.
182
,
1253
1264
(
2011
); e-print arXiv:1007.1891 [hep-ph].
77.
J.
Hadamard
,
Lectures on Cauchy's Problem in Linear Partial Differential Equations
(
Dover Publications
,
New York
,
1923
).
78.
L.
Schwartz
,
Théorie des distributions
(
Hermann
,
Paris
,
1950
), Vols.
1 and 2
.
79.
C.
Schneider
, “
Symbolic summation assists combinatorics
,”
Sém. Lothar. Combin.
56
,
1
36
(
2007
).
80.
J.
Ablinger
, “
A computer algebra toolbox for harmonic sums related to particle physics
,” e-print arXiv:1011.1176 [math-ph].
81.
J.
Ablinger
, “
Computer algebra algorithms for special functions in particle physics
,” e-print arXiv:1305.0687 [math-ph].
82.
J.
Blümlein
,
A.
Hasselhuhn
, and
C.
Schneider
, “
Evaluation of multi-sums for large scale problems
,”
PoS RADCOR
2011
,
032
; e-print arXiv:1202.4303 [math-ph].
83.
J.
Ablinger
,
J.
Blümlein
,
S.
Klein
, and
C.
Schneider
, “
Modern summation methods and the computation of 2- and 3-loop Feynman diagrams
,”
Nucl. Phys. Proc. Suppl.
205–206
,
110
(
2010
); e-print arXiv:1006.4797 [math-ph].
84.
C.
Schneider
, “
Modern summation methods for loop integrals in quantum field theory: The packages sigma, EvaluateMultiSums and SumProduction
,”
J. Phys. Conf. Ser.
523
,
012037
(
2014
); e-print arXiv:1310.0160 [cs.SC].
85.
C. B.
Koutschan
, HolonomicFunctions (User's Guide), Technical Report No. 10-01 in RISC Report Series,
Johannes Kepler Universität Linz
, Austria,
2010
; see http://www.risc.uni-linz.ac.at/publications/download/risc_3934/hf.pdf.
86.
C. G.
Raab
, “
Definite integration in differential fields
,” Ph.D. thesis, (
Johannes Kepler Universität Linz
, Austria,
2012
).
87.
W.
Decker
,
G.-M.
Greuel
,
G.
Pfister
, and
H.
Schönemann
, Singular 3-1-6 — A computer algebra system for polynomial computations, (
2012
) http://www.singular.uni-kl.de.
88.
M.
Bronstein
,
Symbolic Integration I – Transcendental Functions
, 2nd ed. (
Springer
,
Berlin
,
2005
).
89.
Ø.
Ore
, “
The theory of non-commutative polynomials
,”
Ann. Math.
34
,
480
508
(
1933
).
90.
G. E. T.
Almkvist
and
D.
Zeilberger
, “
The method of differentiating under the integral sign
,”
J. Symbolic Comput.
10
,
571
591
(
1990
).
91.
F.
Chyzak
, “
An extension of Zeilberger's fast algorithm to general holonomic functions
,”
Discrete Math.
217
,
115
134
(
2000
).
92.
C. B.
Koutschan
, “
A fast approach to creative telescoping
,”
Math. Comput. Sci.
4
,
259
266
(
2010
).
93.
S. A.
Abramov
and
M.
Petkovšek
, “
D'Alembertian solutions of linear differential and difference equations
,” in
Proc. ISSAC'94
, edited by
J.
von zur Gathen
(
ACM Press
,
1994
), pp.
169
174
.
94.
Ch.
Hermite
, “
Sur la réduction des intégrales hyperelliptiques aux fonctions de première, de seconde et de troisième espèce
,”
Bull. Sci. Math. Astron.
(2e série)
7
,
36
42
(
1883
).
95.
K. O.
Geddes
,
Ha Q.
Le
, and
Z.
Li
, “
Differential rational normal forms and a reduction algorithm for hyperexponential functions
,” in
Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC)
(
ISSAC
,
Santander, Spain
,
2004
), pp.
183
190
.
96.
Sh.
Chen
,
M.
Kauers
, and
M. F.
Singer
, “
Telescopers for rational and algebraic functions via residues
,” in
Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC)
(
ISSAC'12
,
Grenoble, France
,
2012
), pp.
130
137
.
97.
A.
Bostan
,
Sh.
Chen
,
F.
Chyzak
,
Z.
Li
, and
G.
Xin
, “
Hermite reduction and creative telescoping for hyperexponential functions
,” in
Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC)
(
ISSAC'13
,
Boston, MA
,
2013
), pp.
77
84
.
98.
L.
Lewin
,
Polylogarithms and Associated Functions
(
North-Holland
,
New York
,
1981
).
99.
M.
Kauers
and
P.
Paule
,
The Concrete Tetrahedron, Text and Monographs in Symbolic Computation
(
Springer
,
Wien
,
2011
).
100.
M.
Karr
, “
Summation in finite terms
,”
J. Assoc. Comput. Mach.
28
,
305
350
(
1981
).
101.
M.
Bronstein
, “
On solutions of linear ordinary difference equations in their coefficient field
,”
J. Symbolic Comput.
29
,
841
877
(
2000
).
102.
C.
Schneider
, “
Symbolic summation in difference fields
,” Ph.D. thesis (RISC, Johannes Kepler University, Linz,
2001
).
103.
C.
Schneider
, “
Solving parameterized linear difference equations in terms of indefinite nested sums and products
,”
J. Differ. Equ. Appl.
11
,
799
821
(
2005
).
104.
C.
Schneider
, “
Simplifying Sums in ΠΣ-Extensions
,”
J. Algebra Appl.
6
,
415
441
(
2007
).
105.
C.
Schneider
, “
A refined difference field theory for symbolic summation
,”
J. Symbolic Comput.
43
,
611
644
(
2008
); e-print arXiv:0808.2543.
106.
C.
Schneider
, “
Structural theorems for symbolic summation
,”
Appl. Algebra Eng. Commun. Comput.
21
,
1
32
(
2010
).
107.
C.
Schneider
, “
A symbolic summation approach to find optimal nested sum representations
,” in
Motives, Quantum Field Theory, and Pseudodifferential Operators
,
Clay Mathematics Proceedings
, Vol.
12
, edited by
A.
Carey
,
D.
Ellwood
,
S.
Paycha
, and
S.
Rosenberg
(
American Mathematical Society
,
2010
), pp.
285
308
; e-print arXiv:0808.2543.
108.
C.
Schneider
, “
Parameterized telescoping proves algebraic independence of sums
,”
Ann. Combust.
14
,
533
552
(
2010
) [arXiv:0808.2596].
109.
C.
Schneider
, “
Fast algorithms for refined parameterized telescoping in difference fields
,” in Lecture Notes in Computer Science (LNCS), edited by J. Guitierrez, J. Schicho, and M. Weimann (in press); preprint arXiv:1307.7887 [cs.SC] (
2013
).
110.
M.
Petkovšek
, “
Hypergeometric solutions of linear recurrences with polynomial coefficients
,”
J. Symbolic Comput.
14
,
243
264
(
1992
).
111.
P. A.
Hendriks
and
M. F.
Singer
, “
Solving difference equations in finite terms
,”
J. Symbolic Comput.
27
,
239
259
(
1999
).
112.
J.
Blümlein
,
M.
Kauers
,
S.
Klein
, and
C.
Schneider
, “
Determining the closed forms of the
$O(\alpha ^3_s)$
O(αs3)
anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra
,”
Comput. Phys. Commun.
180
,
2143
2165
(
2009
); e-print arXiv:0902.4091 [hep-ph].
113.
In Ref. 61 also infinite sums are considered, which result of the Γ-function around rational numbers.
114.
Due to the emerging hypergeometric weights at the side of the sums, corresponding quasi-shuffle relations are more involved and will be dealt with elsewhere.
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