# §34.4 Definition: Symbol

The symbol is defined by the following double sum of products of symbols:

34.4.1

where the summation is taken over all admissible values of the ’s and ’s for each of the four symbols; compare (34.2.2) and (34.2.3).

Except in degenerate cases the combination of the triangle inequalities for the four symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths ; see Figure 34.4.1.

Figure 34.4.1: Tetrahedron corresponding to symbol.

The symbol can be expressed as the finite sum

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

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, §§9.2.1, 9.2.3).