Sigma

A Mathematica Package for Discovering and Proving Multi-Sum Identities

Authors

Software URL

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, within the Computer Algebra and Applications group at RISC.

Registration and Legal Notices

The source code for this package is password protected. To get the password send an email to Carsten Schneider. It will be given for free to all researchers and non-commercial users. Copyright © 1999–2025 The RISC Computer Algebra and Applications 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

Sigma.m (encoded)

For a demo see the built in help of Sigma.

Literature

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]

For further literature click [here].

Bugs

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