The Abel Lemma On Summation By Parts And Classical Hypergeometric Series

By means of the Abel lemma on summation by parts, this paper investigates system-atically computation of classical hypergeometric series. Several terminating hypergeomet-ric summation formulae are established. Numerous new contiguous and exotic relationsfor ₃F₂(1)-series are derived, which permit us to construct further counter examples toRhin-Viola conjecture.1. Recently, Chu (2006) has proposed the Abel lemma on summation by parts as a newcomputation method to investigate classical hypergeometric series. This approachis further explored in this paper to derive several terminating summation identities,including those well-known ones due to Dougall, Whipple, Gasper, Chu, Gessel andStanton, which have originally been discovered through series rearrangements, com-binatorial inversions, hypergeometric transformations and symbolic calculus on com-puter algebra.2. The Abel lemma on summation by parts is systematically employed to study recur-rence relations of classical hypergeometric ₃F₂ (1)-series. We show four typical patternsof contiguous relations, from which ten contiguous two-term relations and eighteenexotic three-term relations are obtained. They contain all the main theorems andpropositions due to Krattenthaler and Rivoal (2006) as very special cases.3. Based on the contiguous two-term relations on ₃F₂(1)-series found in this paper, threeclasses of infinitely counter-examples to Rhin-Viola conjecture are further constructed.One of these classes generalizes substantially the corresponding result due to Krat-tenthaler and Rivoal (2006) with an extra free parameter.