Ira Gessel, Brandeis University, U.S.A.
e-mail: ira@cs.brandeis.edu

On the WZ Method

Abstract

The WZ method of Wilf and Zeilberger is an efficient way of using a computer to prove hypergeometric summation formulas. However, there are many other interesting aspects of the method, and  I will discuss two of them.

First I will describe how in some cases there is a combinatorial interpretation of the WZ method. In its simplest form this combinatorial interpretation is known to probabilists as the "Banach matchbox problem." A more interesting q-version of the same basic idea was studied by Kadell in 1987.

Next I will discuss "WZ forms," a concept due to Zeilberger which gives a convenient framework for applying the WZ method to summation formulas with more than one free parameter. Every summation formula to which the WZ method applies (and there are few to which it does not apply) has associated to it a WZ form.  A "change of variables formula" for WZ forms, due in essence to Gosper in the late 1970's, can be used to unify the WZ forms for nearly all known summation formulas. However, there remain some interesting open questions on equivalence of WZ forms.