Zeilberger's Algorithm And Its Applications

This thesis is mainly concerned with some new developments on Zeilberger's algorithm, including the terminating condition of the q-analogue of Zeilberger's algorithm (or q-Zeilberger algorithm, for short), application to prove nonterminating basic hypergeometric identities, and Zeilberger's algorithm to double summation case.In Chapter 1, we give a background on Zeilberger's algorithm, and introduce some notations and basic definitions that are used throughout the entire thesis.In Chapter 2, we consider the open problem when the q-Zeilberger algorithm terminates. Le found a solution to this problem when the given bivariate q-hypergeometric term is a rational function in q, qⁿ,q^k. We solve the problem for the general case by giving a characterization of bivariate q-hypergeometric terms for which the q-Zeilberger algorithm terminates. First, for a given bivariate q-hypergeometric term T(n,k), using the method given by Abramov-Petkovsek, we present an algorithm q-decomp which provides an additive decomposition of T(n,k) such that T(n,k) = (K - 1)T₁(n,k) +T₂(n,k). Meanwhile, we prove that the bivariate q-hypergeometric term T₂(n, k) given by g-decomp has some nice properties. Then we study the structure of bivariate q-hypergeometric term. Based on the properties of the structure and divisibility considerations, we conclude that for the above T₂(n,k), g-Zeilberger algorithm terminates if and only if T₂(n,k) is a g-proper hypergeometric term. Thus, we yield the main result of this chapter, which presents a criterion of g-Zeilberger algorithm for T(n, k). Last we give an algorithm to determine whether a bivariate g-hypergeometric term T(n, k) has a telescoped recurrence.In Chapter 3, we present a systematic method for proving nonterminating basic hypergeometric identities. Assume that f(a,···, c) = ∑_k t_k(a,···, c) is a nonterminating basic hypergeometric series, where k is the summation index. By introducing an additional parameter n, i.e., by setting some parameters a,···,c...