When any one of is equal to , or , the symbol has a simple algebraic form. Examples are provided by
For these and other results, and also cases in which any one of is or , see Edmonds (1974, pp. 125–127).
Then assuming the triangle conditions are satisfied
Again it is assumed that in (34.3.7) the triangle conditions are satisfied.
Even permutations of columns of a symbol leave it unchanged; odd permutations of columns produce a phase factor , for example,
In the following three equations it is assumed that the triangle conditions are satisfied by each symbol.
For these and other recursion relations see Varshalovich et al. (1988, §8.6). See also Micu (1968), Louck (1958), Schulten and Gordon (1975a), Srinivasa Rao and Rajeswari (1993, pp. 220–225), and Luscombe and Luban (1998).
For generating functions for the symbol see Biedenharn and van Dam (1965, p. 245, Eq. (3.42) and p. 247, Eq. (3.55)).
For sums of products of symbols, see Varshalovich et al. (1988, pp. 259–262).
Equations (34.3.19)–(34.3.22) are particular cases of more general results that relate rotation matrices to symbols, for which see Edmonds (1974, Chapter 4). The left- and right-hand sides of (34.3.22) are known, respectively, as Gaunt’s integral and the Gaunt coefficient (Gaunt (1929)).