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1: 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 } .
2: 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 } .
3: 28.16 Asymptotic Expansions for Large q
Let s = 2 m + 1 , m = 0 , 1 , 2 , , and ν be fixed with m < ν < m + 1 . …
28.16.1 λ ν ( h 2 ) 2 h 2 + 2 s h 1 8 ( s 2 + 1 ) 1 2 7 h ( s 3 + 3 s ) 1 2 12 h 2 ( 5 s 4 + 34 s 2 + 9 ) 1 2 17 h 3 ( 33 s 5 + 410 s 3 + 405 s ) 1 2 20 h 4 ( 63 s 6 + 1260 s 4 + 2943 s 2 + 486 ) 1 2 25 h 5 ( 527 s 7 + 15617 s 5 + 69001 s 3 + 41607 s ) + .
4: 13.28 Physical Applications
The reduced wave equation 2 w = k 2 w in paraboloidal coordinates, x = 2 ξ η cos ϕ , y = 2 ξ η sin ϕ , z = ξ η , can be solved via separation of variables w = f 1 ( ξ ) f 2 ( η ) e i p ϕ , where
f 1 ( ξ ) = ξ 1 2 V κ , 1 2 p ( 1 ) ( 2 i k ξ ) ,
f 2 ( η ) = η 1 2 V κ , 1 2 p ( 2 ) ( 2 i k η ) ,
and V κ , μ ( j ) ( z ) , j = 1 , 2 , denotes any pair of solutions of Whittaker’s equation (13.14.1). … See Chapter 33. …
5: 29.7 Asymptotic Expansions
p = 2 m + 1 ,
29.7.4 τ 1 = p 2 6 ( ( 1 + k 2 ) 2 ( p 2 + 3 ) 4 k 2 ( p 2 + 5 ) ) .
The same Poincaré expansion holds for b ν m + 1 ( k 2 ) , since …
29.7.7 τ 3 = p 2 14 ( ( 1 + k 2 ) 4 ( 33 p 4 + 410 p 2 + 405 ) 24 k 2 ( 1 + k 2 ) 2 ( 7 p 4 + 90 p 2 + 95 ) + 16 k 4 ( 9 p 4 + 130 p 2 + 173 ) ) ,
In Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions 𝐸𝑐 ν m ( z , k 2 ) and 𝐸𝑠 ν m ( z , k 2 ) . …
6: 34.1 Special Notation
2 j 1 , 2 j 2 , 2 j 3 , 2 l 1 , 2 l 2 , 2 l 3

nonnegative integers.

( j 1 j 2 j 3 m 1 m 2 m 3 ) ,
{ j 1 j 2 j 3 l 1 l 2 l 3 } ,
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 ) ;
7: 34.14 Tables
 33–36. Tables of 3 j and 6 j symbols in which all parameters are 17 / 2 are given in Appel (1968) to 6D. …
8: 30.9 Asymptotic Approximations and Expansions
§30.9(i) Prolate Spheroidal Wave Functions
As γ 2 + , with q = 2 ( n m ) + 1 , … The asymptotic behavior of λ n m ( γ 2 ) and a n , k m ( γ 2 ) as n in descending powers of 2 n + 1 is derived in Meixner (1944). …The asymptotic behavior of 𝖯𝗌 n m ( x , γ 2 ) and 𝖰𝗌 n m ( x , γ 2 ) as x ± 1 is given in Erdélyi et al. (1955, p. 151). The behavior of λ n m ( γ 2 ) for complex γ 2 and large | λ n m ( γ 2 ) | is investigated in Hunter and Guerrieri (1982). …
9: 28.11 Expansions in Series of Mathieu Functions
Let f ( z ) be a 2 π -periodic function that is analytic in an open doubly-infinite strip S that contains the real axis, and q be a normal value (§28.7). …See Meixner and Schäfke (1954, §2.28), and for expansions in the case of the exceptional values of q see Meixner et al. (1980, p. 33). …
28.11.3 1 = 2 n = 0 A 0 2 n ( q ) ce 2 n ( z , q ) ,
28.11.4 cos 2 m z = n = 0 A 2 m 2 n ( q ) ce 2 n ( z , q ) , m 0 ,
28.11.7 sin ( 2 m + 2 ) z = n = 0 B 2 m + 2 2 n + 2 ( q ) se 2 n + 2 ( z , q ) .
10: 26.2 Basic Definitions
Thus 231 is the permutation σ ( 1 ) = 2 , σ ( 2 ) = 3 , σ ( 3 ) = 1 . … Here σ ( 1 ) = 2 , σ ( 2 ) = 5 , and σ ( 5 ) = 1 . … A lattice path is a directed path on the plane integer lattice { 0 , 1 , 2 , } × { 0 , 1 , 2 , } . … As an example, { 1 , 3 , 4 } , { 2 , 6 } , { 5 } is a partition of { 1 , 2 , 3 , 4 , 5 , 6 } . … As an example, { 1 , 1 , 1 , 2 , 4 , 4 } is a partition of 13. …