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1: 19.2 Definitions
where p j is a polynomial in t while ρ and σ are rational functions of t . … Here a , b , p are real parameters, and k c and x are real or complex variables, with p 0 , k c 0 . … If 1 < k 1 / sin ϕ , then k c is pure imaginary. …
§19.2(iv) A Related Function: R C ( x , y )
For the special cases of R C ( x , x ) and R C ( 0 , y ) see (19.6.15). …
2: 34.6 Definition: 9 j Symbol
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 } .
3: 34.7 Basic Properties: 9 j Symbol
34.7.1 { j 11 j 12 j 13 j 21 j 22 j 13 j 31 j 31 0 } = ( 1 ) j 12 + j 21 + j 13 + j 31 ( ( 2 j 13 + 1 ) ( 2 j 31 + 1 ) ) 1 2 { j 11 j 12 j 13 j 22 j 21 j 31 } .
34.7.2 j 12 j 34 ( 2 j 12 + 1 ) ( 2 j 34 + 1 ) ( 2 j 13 + 1 ) ( 2 j 24 + 1 ) { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } = δ j 13 , j 13 δ j 24 , j 24 .
34.7.3 j 13 j 24 ( 1 ) 2 j 2 + j 24 + j 23 j 34 ( 2 j 13 + 1 ) ( 2 j 24 + 1 ) { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } { j 1 j 3 j 13 j 4 j 2 j 24 j 14 j 23 j } = { j 1 j 2 j 12 j 4 j 3 j 34 j 14 j 23 j } .
34.7.4 ( j 13 j 23 j 33 m 13 m 23 m 33 ) { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = m r 1 , m r 2 , r = 1 , 2 , 3 ( 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 ) .
34.7.5 j ( 2 j + 1 ) { j 11 j 12 j j 21 j 22 j 23 j 31 j 32 j 33 } { j 11 j 12 j j 23 j 33 j } = ( 1 ) 2 j { j 21 j 22 j 23 j 12 j j 32 } { j 31 j 32 j 33 j j 11 j 21 } .
4: 34.5 Basic Properties: 6 j Symbol
34.5.9 { j 1 j 2 j 3 l 1 l 2 l 3 } = { j 1 1 2 ( j 2 + l 2 + j 3 l 3 ) 1 2 ( j 2 l 2 + j 3 + l 3 ) l 1 1 2 ( j 2 + l 2 j 3 + l 3 ) 1 2 ( j 2 + l 2 + j 3 + l 3 ) } ,
34.5.10 { j 1 j 2 j 3 l 1 l 2 l 3 } = { 1 2 ( j 2 + l 2 + j 3 l 3 ) 1 2 ( j 1 l 1 + j 3 + l 3 ) 1 2 ( j 1 + l 1 + j 2 l 2 ) 1 2 ( j 2 + l 2 j 3 + l 3 ) 1 2 ( j 1 + l 1 + j 3 + l 3 ) 1 2 ( j 1 + l 1 j 2 + l 2 ) } .
34.5.11 ( 2 j 1 + 1 ) ( ( J 3 + J 2 J 1 ) ( L 3 + L 2 J 1 ) 2 ( J 3 L 3 + J 2 L 2 J 1 L 1 ) ) { j 1 j 2 j 3 l 1 l 2 l 3 } = j 1 E ( j 1 + 1 ) { j 1 + 1 j 2 j 3 l 1 l 2 l 3 } + ( j 1 + 1 ) E ( j 1 ) { j 1 1 j 2 j 3 l 1 l 2 l 3 } ,
34.5.16 ( 1 ) j 1 + j 2 + j 3 + j 1 + j 2 + l 1 + l 2 { j 1 j 2 j 3 l 1 l 2 l 3 } { j 1 j 2 j 3 l 1 l 2 l 3 } = j ( 1 ) l 3 + l 3 + j ( 2 j + 1 ) { j 1 j 1 j j 2 j 2 j 3 } { l 3 l 3 j j 1 j 1 l 2 } { l 3 l 3 j j 2 j 2 l 1 } .
34.5.23 ( j 1 j 2 j 3 m 1 m 2 m 3 ) { j 1 j 2 j 3 l 1 l 2 l 3 } = m 1 m 2 m 3 ( 1 ) l 1 + l 2 + l 3 + m 1 + m 2 + m 3 ( j 1 l 2 l 3 m 1 m 2 m 3 ) ( l 1 j 2 l 3 m 1 m 2 m 3 ) ( l 1 l 2 j 3 m 1 m 2 m 3 ) .
5: 34.4 Definition: 6 j Symbol
34.4.1 { j 1 j 2 j 3 l 1 l 2 l 3 } = m r m s ( 1 ) l 1 + m 1 + l 2 + m 2 + l 3 + m 3 ( j 1 j 2 j 3 m 1 m 2 m 3 ) ( j 1 l 2 l 3 m 1 m 2 m 3 ) ( l 1 j 2 l 3 m 1 m 2 m 3 ) ( l 1 l 2 j 3 m 1 m 2 m 3 ) ,
Except in degenerate cases the combination of the triangle inequalities for the four 3 j symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths j 1 , j 2 , j 3 , l 1 , l 2 , l 3 ; see Figure 34.4.1. …
34.4.2 { j 1 j 2 j 3 l 1 l 2 l 3 } = Δ ( j 1 j 2 j 3 ) Δ ( j 1 l 2 l 3 ) Δ ( l 1 j 2 l 3 ) Δ ( l 1 l 2 j 3 ) s ( 1 ) s ( s + 1 ) ! ( s j 1 j 2 j 3 ) ! ( s j 1 l 2 l 3 ) ! ( s l 1 j 2 l 3 ) ! ( s l 1 l 2 j 3 ) ! 1 ( j 1 + j 2 + l 1 + l 2 s ) ! ( j 2 + j 3 + l 2 + l 3 s ) ! ( j 3 + j 1 + l 3 + l 1 s ) ! ,
where F 3 4 is defined as in §16.2. For alternative expressions for the 6 j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F 3 4 of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
6: 26.16 Multiset Permutations
Let S = { 1 a 1 , 2 a 2 , , n a n } be the multiset that has a j copies of j , 1 j n . 𝔖 S denotes the set of permutations of S for all distinct orderings of the a 1 + a 2 + + a n integers. The number of elements in 𝔖 S is the multinomial coefficient (§26.4) ( a 1 + a 2 + + a n a 1 , a 2 , , a n ) . … The q -multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by …and again with S = { 1 a 1 , 2 a 2 , , n a n } we have …
7: 34.3 Basic Properties: 3 j Symbol
When any one of j 1 , j 2 , j 3 is equal to 0 , 1 2 , or 1 , the 3 j symbol has a simple algebraic form. …For these and other results, and also cases in which any one of j 1 , j 2 , j 3 is 3 2 or 2 , see Edmonds (1974, pp. 125–127). … Even permutations of columns of a 3 j symbol leave it unchanged; odd permutations of columns produce a phase factor ( 1 ) j 1 + j 2 + j 3 , for example,
34.3.8 ( j 1 j 2 j 3 m 1 m 2 m 3 ) = ( j 2 j 3 j 1 m 2 m 3 m 1 ) = ( j 3 j 1 j 2 m 3 m 1 m 2 ) ,
For the polynomials P l see §18.3, and for the function Y l , m see §14.30. …
8: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
( n n 1 , n 2 , , n k ) is the number of ways of placing n = n 1 + n 2 + + n k distinct objects into k labeled boxes so that there are n j objects in the j th box. … These are given by the following equations in which a 1 , a 2 , , a n are nonnegative integers such that … M 1 is the multinominal coefficient (26.4.2): …For each n all possible values of a 1 , a 2 , , a n are covered. … where the summation is over all nonnegative integers n 1 , n 2 , , n k such that n 1 + n 2 + + n k = n . …
9: 34.1 Special Notation
( j 1 j 2 j 3 m 1 m 2 m 3 ) ,
{ j 1 j 2 j 3 l 1 l 2 l 3 } ,
{ j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } .
An often used alternative to the 3 j symbol is the Clebsch–Gordan coefficient
34.1.1 ( j 1 m 1 j 2 m 2 | j 1 j 2 j 3 m 3 ) = ( 1 ) j 1 j 2 + m 3 ( 2 j 3 + 1 ) 1 2 ( j 1 j 2 j 3 m 1 m 2 m 3 ) ;
10: 16.10 Expansions in Series of F q p Functions
§16.10 Expansions in Series of F q p Functions
16.10.1 F q + s p + r ( a 1 , , a p , c 1 , , c r b 1 , , b q , d 1 , , d s ; z ζ ) = k = 0 ( 𝐚 ) k ( α ) k ( β ) k ( z ) k ( 𝐛 ) k ( γ + k ) k k ! F q + 1 p + 2 ( α + k , β + k , a 1 + k , , a p + k γ + 2 k + 1 , b 1 + k , , b q + k ; z ) F s + 2 r + 2 ( k , γ + k , c 1 , , c r α , β , d 1 , , d s ; ζ ) .
Expansions of the form n = 1 ( ± 1 ) n F p + 1 p ( 𝐚 ; 𝐛 ; n 2 z 2 ) are discussed in Miller (1997), and further series of generalized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, §§5.10.2 and 5.11), and Prudnikov et al. (1990, §§5.3, 6.8–6.9).