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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: 19.2 Definitions
§19.2(iv) A Related Function: R C ( x , y )
Formulas involving Π ( ϕ , α 2 , k ) that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using R C ( x , y ) . … When x and y are positive, R C ( x , y ) is an inverse circular function if x < y and an inverse hyperbolic function (or logarithm) if x > y : …For the special cases of R C ( x , x ) and R C ( 0 , y ) see (19.6.15). …
19.2.21 R C ( x , y ) = 0 1 ( v 2 x + ( 1 v 2 ) y ) 1 / 2 d v ,
3: 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).
4: 26.5 Lattice Paths: Catalan Numbers
C ( n ) is the Catalan number. …(Sixty-six equivalent definitions of C ( n ) are given in Stanley (1999, pp. 219–229).) …
26.5.3 C ( n + 1 ) = k = 0 n C ( k ) C ( n k ) ,
26.5.4 C ( n + 1 ) = 2 ( 2 n + 1 ) n + 2 C ( n ) ,
26.5.7 lim n C ( n + 1 ) C ( n ) = 4 .
5: 9 Airy and Related Functions
Chapter 9 Airy and Related Functions
6: 1.12 Continued Fractions
C n is called the n th approximant or convergent to C . … Define … A contraction of a continued fraction C is a continued fraction C whose convergents { C n } form a subsequence of the convergents { C n } of C . Conversely, C is called an extension of C . … Then the convergents C n satisfy …
7: 34 3j, 6j, 9j Symbols
Chapter 34 3 j , 6 j , 9 j Symbols
8: 34.14 Tables
§34.14 Tables
Tables of exact values of the squares of the 3 j and 6 j symbols in which all parameters are 8 are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of 3 j , 6 j , and 9 j symbols on pp. … Some selected 9 j symbols are also given. … 16-17; for 9 j symbols on p. …  310–332, and for the 9 j symbols on pp. …
9: 16.26 Approximations
For discussions of the approximation of generalized hypergeometric functions and the Meijer G -function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).
10: 34.9 Graphical Method
§34.9 Graphical Method
For specific examples of the graphical method of representing sums involving the 3 j , 6 j , and 9 j symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).