§34.5 Basic Properties: Symbol

§34.5(i) Special Cases

In the following equations it is assumed that the triangle inequalities are satisfied and that is again defined by (34.3.4).

If any lower argument in a symbol is 0, , or 1, then the symbol has a simple algebraic form. Examples are provided by:

34.5.1
34.5.2
34.5.3
34.5.4
34.5.5
34.5.6
34.5.7

§34.5(ii) Symmetry

The 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, for example,

34.5.8

Next,

34.5.9
34.5.10

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.

§34.5(iii) Recursion Relations

In the following equation it is assumed that the triangle conditions are satisfied.

where

For further recursion relations see Varshalovich et al. (1988, §9.6) and Edmonds (1974, pp. 98–99).

§34.5(v) Generating Functions

For generating functions for the symbol see Biedenharn and van Dam (1965, p. 255, eq. (4.18)).

§34.5(vi) Sums

34.5.15
34.5.16

Equations (34.5.15) and (34.5.16) are the sum rules. They constitute addition theorems for the symbol.

34.5.23

Equation (34.5.23) can be regarded as an alternative definition of the symbol.

For other sums see Ginocchio (1991).