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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: 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 …
6: 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).
7: 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 . …
8: 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 ) ;
9: 1.3 Determinants
The cofactor A j k of a j k is … For real-valued a j k , … where ω 1 , ω 2 , , ω n are the n th roots of unity (1.11.21). … If 𝐷 n [ a j , k ] tends to a limit L as n , then we say that the infinite determinant 𝐷 [ a j , k ] converges and 𝐷 [ a j , k ] = L . … Here δ j , k is the Kronecker delta. …
10: 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. …