The HarmonicSums package by
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 © 1999–2012 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
and is accompanied by the Mathematica notebook:
demo.nb (for Mathematica version 8.0 and 9.0)
The following precomputed tables are available in order to extend the functionality of the package and are used by the functions ReduceToBasis and ReduceConstants:
Right now you are using Version 1.0 released on January 5, 2013.
This version is compatible with Mathematica versions from 5.2 to 9.0.
Please report any bugs, comments and requests for further tables to
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 [bib]
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, J. Blümlein and C. Schneider. Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms. arXiv:1212.xxxx [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].