### Research Experience for Undergraduates (REU) Seminar

Title: An Introduction to WZ Theory

Speaker: **Andrew Sills**, Rutgers University

Date: July 6, 2004 1 - 2 pm

Location: DIMACS Center, CoRE Bldg, Room 431, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

A geometric series is a series a(0) + a(1) + a(2) + ... in which the
ratio a(k+1)/a(k) of consective terms is constant for all k=0,1,2,3,...
In contrast, a hypergeometric series is a series in which the ratio of
consective terms is not constant, but rather a rational function.

Hypergeometric series were first studied by Gauss, and over the centuries
many identities involving hypergeomtric series were discovered.
In the course of studying such series, it became standard to look for,
by ad hoc methods, recurrence relations satisfied by the series.

A major advance was made in the 1940's by Sister Mary Celine Fasenmyer, who
discovered an ALGORITHM for finding a recurrence relation satisfied by
a given hypergeometric term. In the early 1990's Herbert Wilf (U. Penn)
and Doron Zeilberger (now at Rutgers) discovered a faster algorithm
which accomplishes the goal of Sister Celine's algorithm, and further
allows a computer to rigorously prove (not just test a large number of
cases, but actually PROVE) hypergeometric identities completely
automatically.