New Counterexamples To The Rhin-Viola Conjecture

In the thesis, we construct some new counterexamples to the conjecture of Rhin-Violaby using contiguous relations given by Chu-Wang on hypergeometric series ₃F₂(1). These newcounterexamples have two variable parameters, so they are more general than Chu-Wang's. Themain contents of this thesis are summarized as follows:The first chapter is a survey on history and development of the hypergeometric series.In Chapter 2, we give the modified Abel lemma on summation by parts. By using thislemma we introduce how to get four patterns of three-terms relations and three important contiguous relations, and also give how to use Abel's method to prove these four important relationsgiven by Krattenthaler and Rivoal.In Chapter 3, we introduce the conjecture of Rhin-Viola, Thomae's relations, six counterexamples and counterexamples given by Chu-Wang which contain only one variable parameter.Using these three important contiguous relations, we construct new counterexamples to theconjecture of Rhin-Viola and give proof.