##### This material has been published in
J. Math. Soc. Japan **58** (2006), 183-210,
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## 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 _{3}*F*_{2}-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 _{3}*F*_{2}-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 _{3}*F*_{2}-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 _{3}*F*_{2}-transformation
formulae and contiguous relations.
We also explain in detail the relationship between the integrals of
Rhin and Viola and _{3}*F*_{2}-series.

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