The summation package 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
, 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
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...).
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.
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
For further literature click
Sigma is developed for Mathematica 9 and higher versions and might not run properly on
earlier versions. Please report any bugs