This material has been published in J. Math. Soc. Japan 58 (2006), 183-210, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by the Mathematical Society of Japan. This material may not be copied or reposted without explicit permission.

Christian Krattenthaler and Tanguy Rivoal

How can we escape Thomae's relations?

(26 pages)

Abstract. In 1879, Thomae discussed the relations between two generic hypergeometric 3F2-series with argument 1. It is well-known since then that, in combination with the trivial ones which come from permutations of the parameters of the hypergeometric series, Thomae had found a set of 120 relations. More recently, Rhin and Viola asked the following question (in a different, but equivalent language of integrals): If there exists a linear dependence relation over Q between two convergent 3F2-series with argument 1, with integral parameters, and whose values are irrational numbers, is this relation a specialisation of one of the 120 Thomae relations? A few years later, Sato answered this question in the negative, by giving six examples of relations which cannot be explained by Thomae's relations. We show that Sato's counter-examples can be naturally embedded into two families of infinitely many 3F2-relations, both parametrised by three independent parameters. Moreover, we find two more infinite families of the same nature. The families, which do not seem to have been recorded before, come from certain 3F2-transformation formulae and contiguous relations. We also explain in detail the relationship between the integrals of Rhin and Viola and 3F2-series.

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