This material has been published in
J. Math. Soc. Japan 58 (2006), 183-210,
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Christian Krattenthaler and
How can we escape Thomae's relations?
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
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
A few years later, Sato answered this question in the negative,
by giving six examples of relations which cannot be explained by
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
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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