allows to deal with nested sums such as harmonic sums, S-sums, cyclotomic sums
and cyclotmic S-sums as well as iterated integrals such as harmonic
polylogarithms, multiple polylogarithms and cyclotomic polylogarithms in an algorithmic fashion.
The package can calculte the Mellin transformation of the iterated integrals in terms of the nested sums and it can
compute integral representations of the nested sums.
The package can be used to compute algebraic and structural relations between the nested sums as well as between the
the iterated integrals and connected to it the package can find relations between the nested sums at infinity and the iterated integrals at one.
In addition the package provides algorithms to represent expressions involving the nested sums and iterated integrals in terms of basis representations.
Moreover, the package allows to compute (asymptotic) expansions of the nested sums and iterated integrals and it contains an algorithm which rewrites certain types
of nested sums into expressions in terms of cyclotomic S-sums.
Registration and Legal Notices
The source code for this package is password protected. To get the password
send an email to
It will be given for free to all researchers and non-commercial users.
Copyright © 1999–2019 The RISC Combinatorics Group, Austria — all rights reserved.
Commercial use of the software is prohibited without prior written permission.
A Note on Encoded Files
This package contains one or more Mathematica input files which are encoded. Those files
cannot be read or modified directly as plain text, but can be loaded into
Mathematica just like any normal input file (i.e., with
There is no need (and also no way) to decode them by using additional software
or a special key.
If loading an encoded file causes a syntax error, open it with a
text editor and remove any blank lines at the beginning (for some
reason your Mac could have inserted them silently...).
The package is contained in the Mathematica input file
After loading the package, type HarmonicSumsFunctionList
to get a description of the available commands.
Thanks to the tremendous effort of Abilio De Freitas
all the available commands of HarmonicSums
now have usage messages. These messages are summarized in the notebook UsageMessages.nb
The following precomputed tables are available in order to extend the functionality of the package and are used by the functions ReduceToBasis
Right now you are using Version 1.0 released on August 19, 2019.
This version was tested with Mathematica versions 11.3 and 12.0.
Please report any bugs, comments and requests for further tables to
Discovering and Proving Infinite Pochhammer Sum Identities
The notebook PochhammerSums.nb
accompanies the article
J. Ablinger. Discovering and Proving Infinite Pochhammer Sum Identities. arXiv:1902.11001 [math.CO].
Proving two conjectural series for ζ(7) and discovering more series for ζ(7).
The notebook ProofOfZeta7Identities.nb
and the relation table
accompany the article
J. Ablinger. Proving two conjectural series for ζ(7) and discovering more series for ζ(7). arXiv:1908.06631 [math.CO].
The theoretical background of the algorithms implemented in
HarmonicSums is described in
- Computer Algebra Algorithms for Special Functions in Particle Physics (PhD Thesis).
RISC, Johannes Kepler University, April 2012.
- A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics (Diploma Thesis).
RISC, Johannes Kepler University, February 2009.
The PhD thesis also contains a chapter about how to use the package.
We ask you to quote the following block of papers using the package HarmonicSums:
J. Ablinger. A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics. Johannes Kepler University. Diploma Thesis. February 2009. [arXiv:1011.1176 [math-ph]].
J. Ablinger. Computer Algebra Algorithms for Special Functions in Particle Physics. Johannes Kepler University. PhD Thesis. April 2012.
J. Ablinger. Computing the Inverse Mellin Transform of Holonomic Sequences using Kovacic's Algorithm. PoS RADCOR2017, 069, 2017. [arXiv:1801.01039 [cs.SC]].
J. Ablinger. Inverse Mellin Transform of Holonomic Sequences. Johannes Kepler University. PoS LL 2016, 067, 2016. [arXiv:1606.02845 [cs.SC]].
J. Ablinger. The package HarmonicSums: Computer Algebra and Analytic aspects of Nested Sums. Loops and Legs in Quantum Field Theory - LL 2014. [arXiv:1407.6180 [cs.SC]].
J. Ablinger. Discovering and Proving Infinite Pochhammer Sum Identities. [arXiv:1902.11001 [math.CO]].
J. Ablinger, J. Blümlein and C. Schneider. Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms. [arXiv:1302.0378 [math-ph]].
J. Ablinger, J. Blümlein and C. Schneider. Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials. J. Math. Phys. 52 (2011) 102301. [arXiv:1105.6063 [math-ph]].
J. Blümlein. Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5. Comput. Phys. Commun. 180 (2009) 2218. [arXiv:0901.3106 [hep-ph]].
E. Remiddi and J. A. M. Vermaseren. Harmonic polylogarithms. Int. J. Mod. Phys. A 15 (2000) 725. [hep-ph/9905237].
J. A. M. Vermaseren. Harmonic sums, Mellin transforms and integrals. Int. J. Mod. Phys. A 14 (1999) 2037. [hep-ph/9806280].