Ha Q. Le

A criterion for the applicability of Zeilberger's algorithm to rational functions

We consider the applicability (or terminating condition) of the well-known Zeilbergers algorithm and give the complete solution to this problem for the case where the original hypergeometric term F(n,k) is a rational function. We specify a class of identifies Σk=0nF(n,k)= 0, F(n,k) ∈ C(n,k), that cannot be proven by Zeilbergers algorithm. Additionally, we give examples showing that the set of hypergeometric terms on which Zeilbergers algorithm terminates is a proper subset of the set of all hypergeometric terms, but a super-set of the set of proper terms.

Applicability of Zeilberger’s Algorithm to Rational Functions

We consider the applicability (or terminating condition) of the well-known Zeilberger’s algorithm and give the complete solution to this problem for the case where the original hypergeometric term F(n, k) is a rational function. We specify a class of identities \(\sum\nolimits_{k = 0}^n {F\left( {n,k} \right) = 0} \) \(F\left( {n,k} \right) \in \mathbb{C}\left( {n,k} \right)\) that cannot be proven by Zeilberger’s algorithm. Additionally we give examples showing that the set of hypergeometric terms for which Zeilberger’s algorithm terminates is a proper subset of the set of all hypergeometric terms, but a super-set of the set of proper terms.

On the q-Analogue of Zeilberger's Algorithm to Rational Functions

We consider the applicability (or terminating conditions) of the q-analogue of Zeilbergers algorithm and give the complete solution to this problem for the case when the original q-hypergeometric term is a rational function.

Client-server communication standards for mathematical computation

This paper presents a niotlel for a \Vorld Wide 1Veb comput.at,ioual wrwr which uses a Computer .Ugebra System i1S its undrrlying cnginc, and which collllrlurlicat,es wit,h its clients using three enwrging standards: OpenMath. h~lnt.hML and VRML. An ilnplerllelltatioli of t.he server and ibs applications to education and rcscxch arc dcscrilml.

Rational canonical forms and efficient representations of hypergeometric terms

We propose four multiplicative canonical forms that exhibit the shift structure of a given rational function. These forms in particular allow one to represent a hypergeometric term efficiently. Each of these representations is optimal in some sense.

A direct algorithm to construct the minimal Z-pairs for rational functions

In this paper, we present a direct algorithm to construct the minimal Z-pairs for rational functions. We describe a Maple implementation of the algorithm and show timing comparisons between this algorithm and other related algorithms. We also summarize an analogous algorithm for the q-difference case.

On the order of the recurrence produced by the method of creative telescoping

We present an algorithm which computes a non-trivial lower bound for the order of the minimal telescoper for a given hypergeometric term. The combination of this algorithm and techniques from indefinite summation leads to an efficiency improvement in Zeilbergers algorithm. We also describe a Maple implementation, and conduct experiments which show the improvement that it makes in the construction of the telescopers.

Computing the minimal telescoper for sums of hypergeometric terms

Let <i>T</i> (<i>n, k</i>) be a hypergeometric term of <i>n</i> and <i>k.</i> We present in this paper an algorithm to construct the minimal telescoper for <i>U</i> (<i>n, k</i>) = ∑<inf><i>m=b</i></inf>^<i>n</i>-1 <i>T</i> (<i>m, k</i>), <i>b</i> ε ℤ, if it exists. We show a Maple implementation of this method and discuss the problem of finding closed forms of definite sums of <i>U</i> (<i>n, k</i>).