# §34.2 Definition: Symbol

The quantities in the symbol are called angular momenta. Either all of them are nonnegative integers, or one is a nonnegative integer and the other two are half-odd positive integers. They must form the sides of a triangle (possibly degenerate). They therefore satisfy the triangle conditions

34.2.1

where is any permutation of . The corresponding projective quantum numbers are given by

34.2.2,

and satisfy

34.2.3

See Figure 34.2.1 for a schematic representation.

Figure 34.2.1: Angular momenta and projective quantum numbers , .

If either of the conditions (34.2.1) or (34.2.3) is not satisfied, then the symbol is zero. When both conditions are satisfied the symbol can be expressed as the finite sum

34.2.4

where

34.2.5

and the summation is over all nonnegative integers such that the arguments in the factorials are nonnegative.

Equivalently,

where is defined as in §16.2.

For alternative expressions for the symbol, written either as a finite sum or as other terminating generalized hypergeometric series of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).