The summation package Sigma

Short Description

Sigma is a Mathematica package that can handle multisums in terms of indefinite nested sums and products. The summation principles of Sigma are: telescoping, creative telescoping and recurrence solving. The underlying machinery of Sigma is based on difference field theory. The package has been developed by Carsten Schneider, a member of the RISC Combinatorics group.

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 Sigma package consists of the file For a demo see the built in help of Sigma.


For a detailed description and a collection of non-trivial examples we refer, e.g., to the article
C. Schneider, Symbolic Summation Assists Combinatorics, Sem.Lothar.Combin. 56, pp.1-36. 2007. Article B56b. [pdf]
C. Schneider. Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation. To appear in: Anti-Differentiation and the Calculation of Feynman Amplitudes, J. Blümlein and C. Schneider (ed.), Texts and Monographs in Symbolic Computuation, 2021. arXiv:2102.01471 [cs.SC].

For further literature click [here].


Sigma is developed for Mathematica 9 and higher versions and might not run properly on earlier versions. Please report any bugs to Carsten Schneider.