 
Summary: Advances in Applied Mathematics 30 (2003) 424441
www.elsevier.com/locate/aam
When does Zeilberger's algorithm succeed?
S.A. Abramov 1
Russian Academy of Sciences, Dorodnicyn Computing Centre, Vavilova 40,
119991, Moscow GSP1, Russia
Received 30 May 2002; accepted 20 June 2002
Abstract
A terminating condition of the wellknown Zeilberger's algorithm for a given hypergeometric
term T (n,k) is presented. It is shown that the only information on T (n,k) that one needs in order
to determine in advance whether this algorithm will succeed is the rational function T (n,k + 1)/
T (n,k).
2003 Elsevier Science (USA). All rights reserved.
1. Introduction
Let K be a field of characteristic 0. A hypergeometric term (or simply a term) T (k) in
k over K satisfies a linear recurrence of the form
f (k)T (k + 1) + g(k)T (k) = 0, (1)
f,g K[k]\{0}, the variable k is integervalued. The certificate Ck(T ) of the term T (k) is
the rational function T (k + 1)/T (k) = g(k)/f (k). A term T (n,k) in two integervalued
variables over K satisfies the recurrences
