Short Description

The HarmonicSums package by Jakob Ablinger 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 Peter Paule. 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 <<"file" or Get["file"]). 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

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 and ReduceConstants: 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 Jakob Ablinger.

Discovering and Proving Infinite Pochhammer Sum Identities

The notebook PochhammerSums.nb accompanies the article

Proving two conjectural series for ζ(7) and discovering more series for ζ(7).

The notebook ProofOfZeta7Identities.nb and the relation table HLogCyclo3tow7.m accompany the article


The theoretical background of the algorithms implemented in HarmonicSums is described in 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: