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1: 31 Heun Functions
Chapter 31 Heun Functions
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: Vadim B. Kuznetsov
5: George E. Andrews
Andrews served as President of the AMS from February 1, 2009 to January 31, 2011, and became a Fellow of the AMS in 2012. …
6: Brian D. Sleeman
7: Staff
  • Vadim B. Kuznetsov, University of Leeds, Chap. 31

  • Brian D. Sleeman, University of Leeds, Chap. 31

  • Brian D. Sleeman, University of Leeds, for Chap. 31

  • 8: 1.3 Determinants
    1.3.2 det [ a j k ] = | a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 | = a 11 | a 22 a 23 a 32 a 33 | - a 12 | a 21 a 23 a 31 a 33 | + a 13 | a 21 a 22 a 31 a 32 | = a 11 a 22 a 33 - a 11 a 23 a 32 - a 12 a 21 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 - a 13 a 22 a 31 .
    9: 26.2 Basic Definitions
    Table 26.2.1: Partitions p ( n ) .
    n p ( n ) n p ( n ) n p ( n )
    14 135 31 6842 48 1 47273
    10: 27.2 Functions
    Table 27.2.1: Primes.
    n p n p n + 10 p n + 20 p n + 30 p n + 40 p n + 50 p n + 60 p n + 70 p n + 80 p n + 90
    1 2 31 73 127 179 233 283 353 419 467
    Table 27.2.2: Functions related to division.
    n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n )
    3 2 2 4 16 8 5 31 29 28 2 30 42 12 8 96
    5 4 2 6 18 6 6 39 31 30 2 32 44 20 6 84
    12 4 6 28 25 20 3 31 38 18 4 60 51 32 4 72