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1: 34.6 Definition: 9 j Symbol
§34.6 Definition: 9 j Symbol
The 9 j symbol may be defined either in terms of 3 j symbols or equivalently in terms of 6 j symbols:
34.6.1 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = all  m r s ( j 11 j 12 j 13 m 11 m 12 m 13 ) ( j 21 j 22 j 23 m 21 m 22 m 23 ) ( j 31 j 32 j 33 m 31 m 32 m 33 ) ( j 11 j 21 j 31 m 11 m 21 m 31 ) ( j 12 j 22 j 32 m 12 m 22 m 32 ) ( j 13 j 23 j 33 m 13 m 23 m 33 ) ,
34.6.2 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = j ( 1 ) 2 j ( 2 j + 1 ) { j 11 j 21 j 31 j 32 j 33 j } { j 12 j 22 j 32 j 21 j j 23 } { j 13 j 23 j 33 j j 11 j 12 } .
The 9 j symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …
2: 34.2 Definition: 3 j Symbol
§34.2 Definition: 3 j Symbol
The quantities j 1 , j 2 , j 3 in the 3 j symbol are called angular momenta. …They therefore satisfy the triangle conditions …The corresponding projective quantum numbers m 1 , m 2 , m 3 are given by … When both conditions are satisfied the 3 j symbol can be expressed as the finite sum …
3: 34.4 Definition: 6 j Symbol
§34.4 Definition: 6 j Symbol
The 6 j symbol is defined by the following double sum of products of 3 j symbols: …where the summation is taken over all admissible values of the m ’s and m ’s for each of the four 3 j symbols; compare (34.2.2) and (34.2.3). … The 6 j symbol can be expressed as the finite sum … where F 3 4 is defined as in §16.2. …
4: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
18.29.1 ( b c , b d , c d ; q ) n ( Q n ( e i θ ; a , b , c , d q ) + Q n ( e i θ ; a , b , c , d q ) ) ,
18.29.2 Q n ( z ; a , b , c , d q ) z n ( a z 1 , b z 1 , c z 1 , d z 1 ; q ) ( z 2 , b c , b d , c d ; q ) , n ; z , a , b , c , d , q fixed.
5: 20 Theta Functions
Chapter 20 Theta Functions
6: 34.11 Higher-Order 3 n j Symbols
§34.11 Higher-Order 3 n j Symbols
For information on 12 j , 15 j ,…, symbols, see Varshalovich et al. (1988, §10.12) and Yutsis et al. (1962, pp. 62–65 and 122–153).
7: 27.9 Quadratic Characters
§27.9 Quadratic Characters
For an odd prime p , the Legendre symbol ( n | p ) is defined as follows. If p divides n , then the value of ( n | p ) is 0 . … If an odd integer P has prime factorization P = r = 1 ν ( n ) p r a r , then the Jacobi symbol ( n | P ) is defined by ( n | P ) = r = 1 ν ( n ) ( n | p r ) a r , with ( n | 1 ) = 1 . The Jacobi symbol ( n | P ) is a Dirichlet character (mod P ). …
8: 34.14 Tables
§34.14 Tables
Some selected 9 j symbols are also given. Other tabulations for 3 j symbols are listed on pp.  11-12; for 6 j symbols on pp.  16-17; for 9 j symbols on p. …
9: 34.10 Zeros
§34.10 Zeros
In a 3 j symbol, if the three angular momenta j 1 , j 2 , j 3 do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the 3 j symbol is zero. Similarly the 6 j symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four 3 j symbols in the summation. …However, the 3 j and 6 j symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. Such zeros are called nontrivial zeros. …
10: 34.12 Physical Applications
§34.12 Physical Applications
The angular momentum coupling coefficients ( 3 j , 6 j , and 9 j symbols) are essential in the fields of nuclear, atomic, and molecular physics. … 3 j , 6 j , and 9 j symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).