It is shown that a set of six series representations, and correspondingly a set of six 3F2(1)'s is necessary and sufficient to account for the 72 symmetries of the 3j-coefficient.

A note on the symmetries of the 3j-coefficient

An explicit form of the five Regge symmetries of the 3j coefficient is given. It is shown that a set of six 3F2(1) hypergeometric functions is necessary and sufficient to account for the 72 symmetries of the 3j coefficient, each accounting for 12 symmetries. The eight‐element group, recently discussed by Lockwood, accounts for only eight symmetries of the 3j coefficient.

Symmetries of the 3j coefficient

The equivalence between the two sets of 4F3(1) S for the 6j coefficient is explained through two sets of equalities. It is shown that the 144 symmetries can be partitioned into 12 sets of 12 each, depending on the number of terms in the series representation for the 6j coefficient. The 12‐element group, recently discussed by Lockwood, is shown to account for only 12 symmetries of the 6j coefficient.

Symmetries of the 6j coefficient

A set of six canonical parameters is derived for the 6j coefficient in a way analogous to that used previously for the 3j coefficient. The procedure involves implicitly a twelve‐to‐one homomorphism which reduces the known 144‐element symmetry group to a 12‐element group. The symmetries are completely and elegantly described by only five of the six parameters. A further property of the canonical parameters is their complete independence.

Canonical parameters of the 6j coefficient

Expressions for the 3j and the 6j symbol involving generalized hypergeometric series, valid for all admissible values of the j's and the re's, are derived and it is shown that, for a symbol of either type, all such expressions are deducible from two basic forms by simple substitutions.

Canonical forms of the generalized hypergeometric series for the 3j and the 6j symbol

The 3j coefficient is expressed as a function of five new parameters which have unique properties. They are completely independent, satisfy simple validity criteria, and display the symmetry properties of the function in a particularly transparent manner. By means of the new parameters, the known 72‐element symmetry group is reduced to an eight‐element group, and the absolute symmetries are separated in a clear way from those which contain a phase factor.

Loren A. Lockwood

Papers

Canonical parameters of the 3j coefficient

Canonical parameters of the 6j coefficient