In the following equations it is assumed that the triangle inequalities are
satisfied and that J is again defined by (34.3.4).
If any lower argument in a 6j symbol is 0, 12, or 1, then the 6j
symbol has a simple algebraic form. Examples are provided by:
The 6j symbol is invariant under interchange of any two columns and also
under interchange of the upper and lower arguments in each of any two columns,
Equations (34.5.9) and (34.5.10) are called
Regge symmetries. Additional symmetries are obtained by applying
(34.5.8) to (34.5.9) and (34.5.10). See
Srinivasa Rao and Rajeswari (1993, pp. 102–103) and references given there.
In the following equation it is assumed that the triangle conditions are
For further recursion relations see Varshalovich et al. (1988, §9.6) and
Edmonds (1974, pp. 98–99).
For generating functions for the 6j symbol see
Biedenharn and van Dam (1965, p. 255, eq. (4.18)).
Equations (34.5.15) and (34.5.16) are the sum rules.
They constitute addition theorems for the 6j symbol.
Equation (34.5.23) can be regarded as an alternative definition of
the 6j symbol.
For other sums see Ginocchio (1991).