  In 1879, Thomae discussed the relations between two generic hypergeometric
$_3F_2$-series with argument 1. It is well-known since then that there are 120
such relations (including the trivial ones which come from permutations of the
parameters of the hypergeometric series). 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 $\mathbf Q$ between two
convergent $_3F_2$-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 $_3F_2$-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 $_3F_2$-transformation formulae and
contiguous relations. We also explain in detail the relationship between the
integrals of Rhin and Viola and $_3F_2$-series.
