Summer 2016: Special Functions & WZ Theory (AARMS Summer School)


This course is part of the 2016 AARMS Summer School at Dalhousie University.

Instructor Armin Straub
Lecture Week 1: class meets MTWRF, 11-12:30pm   (MTF in Dunn 302, W in Dunn 135, R in Dunn 301B)
Week 2: class meets MTWRF, 9-10:30am   (MTWF in Dunn 135, R in Dunn 301B)
Week 3: class meets MTWRF, 11-12:30pm   (MTF in Dunn 302, W in Dunn 135, R in Dunn 301B)
Week 4: class meets TWRF, 9-10:30am   (TW in Rowe 1016, RF in Dunn 301B)
Syllabus syllabus.pdf


Each of the problems comes with a number of experience points (XP's) you gain by doing it (mostly correctly).

The goal is to reach a minimum of 42 experience points by the end of the course.

Many more XP's can be collected, and this is meant to allow you to focus on the problems you find most interesting and skip some that may be less relevant to you.

DatePDFXP'sdue by
7=2+3+1+1Wednesday, July 20
10=2+1+2+2+3Wednesday, July 20
10=1+1+2+2+2+2Wednesday, July 20
3=1+2 (bonus: 7=2+2+2+1)Wednesday, July 27
8=1+2+5 (bonus: 4)Wednesday, July 27
10=2+1+1+1+5 (bonus: 2)Wednesday, July 27
12=2+1+2+3+1+1+2Wednesday, July 27
7=2+3+2Wednesday, July 27
07/21problems09.pdf4=4 (bonus: 9=3+3+3)Tuesday, August 2
07/25problems10.pdf8=2+2+2+2Tuesday, August 2
07/27problems11.pdf7=1+2+2+2Tuesday, August 2
07/29problems12.pdf5=2+3 (bonus: 3)optional
91currently achievable XPs (aim for at least 42 XP)


As part of this course, we will explore the open-source free computer algebra system Sage.

If you don't have access to Sage yet, please create an account at This free cloud service does not require you to install anything, and you can access your files and computations from any computer as long as you have internet.

Introduction to Sage

Here are the notebooks that we explored in lab:

If you are logged into your cloud account, you can play around with these files (which initially show up as read-only) by copying them to one of your personal projects (click on the light blue button with an i and select Copy...).

LaTeX and Sage

The superb package sagetex allows you to include Sage computations into your LaTeX file. Below is the example file we discussed in lab:

See for more information on sagetex. More examples are included with the sagetex package; see, for instance, the resulting document:

WZ algorithms for Sage

Following the description in the excellent book A=B by Marko Petkovsek, Herbert S. Wilf and Doron Zeilberger, I have implemented Celine's method, Gosper's algorithm and Zeilberger's algorithm in Sage, so we can play with these and prove identities:

The code requires the ore_algebra package by Manuel Kauers, Maximilian Jaroschek, and Fredrik Johansson to be installed. This package is already installed in the SageMathCloud.

Note that this is a simple minded implementation, not optimized for speed and with some need for polishing. Please report any issues you encounter.

Currently known issues:

  • When running code for the first time, there is several instances of a DeprecationWarning. These warnings are annoying (make them disappear by running the code again) but should not impact computations. As far as I know, none of these are due to issues in
  • The current version of the ore_algebra package does not apply operators properly if several variables are involved. Hence, the function zeilberger_verify usually fails. Let me know if you would like a patch that fixes that particular issue.

Recommended literature

  • Herbert S. Wilf. Generatingfunctionology. Academic Press, 1990. freely available online
  • Marko Petkovsek, Herbert S. Wilf and Doron Zeilberger. A=B. A. K. Peters, Ltd., 1st edition, 1996. freely available online
  • George E. Andrews, Richard Askey, and Ranjan Roy. Special Functions. Cambridge University Press, 1999.
  • Manuel Kauers and Peter Paule. The Concrete Tetrahedron. Springer-Verlag, 2011.
Last change: 2016/08/07
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