  For the hypergeometric function of unit argument 3F2(1) we prove the
existence and uniqueness of three-term relations with arbitrary integer shifts.
We show that not only the original 3F2(1) function but also other five
functions related to it satisfy one and the same three-term relation. This fact
is referred to as simultaneousness. The uniqueness and simultaneousness provide
three-term relations with a group symmetry of order 72.
