# §34.3 Basic Properties: Symbol

## §34.3(i) Special Cases

When any one of is equal to , or 1, the symbol has a simple algebraic form. Examples are provided by

34.3.1
34.3.2,
34.3.3.

For these and other results, and also cases in which any one of is or 2, see Edmonds (1974, pp. 125–127).

Next define

34.3.4

Then assuming the triangle conditions are satisfied

Lastly,

34.3.7

Again it is assumed that in (34.3.7) the triangle conditions are satisfied.

## §34.3(ii) Symmetry

Even permutations of columns of a symbol leave it unchanged; odd permutations of columns produce a phase factor , for example,

34.3.8
34.3.9

Next,

34.3.10
34.3.11
34.3.12

Equations (34.3.11) and (34.3.12) are called Regge symmetries. Additional symmetries are obtained by applying (34.3.8)–(34.3.10) to (34.3.11)) and (34.3.12). See Srinivasa Rao and Rajeswari (1993, pp. 44–47) and references given there.

## §34.3(iii) Recursion Relations

In the following three equations it is assumed that the triangle conditions are satisfied by each symbol.

34.3.13
34.3.14
34.3.15

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).

## §34.3(iv) Orthogonality

34.3.16
34.3.18

In the summations (34.3.16)–(34.3.18) the summation variables range over all values that satisfy the conditions given in (34.2.1)–(34.2.3). Similar conventions apply to all subsequent summations in this chapter.

## §34.3(v) Generating Functions

For generating functions for the symbol see Biedenharn and van Dam (1965, p. 245, Eq. (3.42) and p. 247, Eq. (3.55)).

## §34.3(vi) Sums

For sums of products of symbols, see Varshalovich et al. (1988, pp. 259–262).

## §34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

For the polynomials see §18.3, and for the functions and see §14.30.

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)).