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1: 34.6 Definition: 9 ⁒ j Symbol
§34.6 Definition: 9 ⁒ j Symbol
β–ΊThe 9 ⁒ j symbol may be defined either in terms of 3 ⁒ j symbols or equivalently in terms of 6 ⁒ j symbols: β–Ί
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 ) ,
β–Ί
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 } .
β–ΊThe 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: 19.37 Tables
β–ΊTabulated for Ο• = 0 ⁒ ( 5 ∘ ) ⁒ 90 ∘ , k 2 = 0 ⁒ ( .01 ) ⁒ 1 to 10D by Fettis and Caslin (1964). β–ΊTabulated for Ο• = 0 ⁒ ( 1 ∘ ) ⁒ 90 ∘ , k 2 = 0 ⁒ ( .01 ) ⁒ 1 to 7S by BeliΝ‘akov et al. (1962). … β–ΊTabulated for Ο• = 0 ⁒ ( 5 ∘ ) ⁒ 90 ∘ , k = 0 ⁒ ( .01 ) ⁒ 1 to 10D by Fettis and Caslin (1964). … β–ΊTabulated (with different notation) for Ο• = 0 ⁒ ( 15 ∘ ) ⁒ 90 ∘ , Ξ± 2 = 0 ⁒ ( .1 ) ⁒ 1 , arcsin ⁑ k = 0 ⁒ ( 15 ∘ ) ⁒ 90 ∘ to 5D by Abramowitz and Stegun (1964, Chapter 17), and for Ο• = 0 ⁒ ( 15 ∘ ) ⁒ 90 ∘ , Ξ± 2 = 0 ⁒ ( .1 ) ⁒ 1 , arcsin ⁑ k = 0 ⁒ ( 15 ∘ ) ⁒ 90 ∘ to 7D by Zhang and Jin (1996, pp. 676–677). β–ΊTabulated for Ο• = 5 ∘ ⁒ ( 5 ∘ ) ⁒ 80 ∘ ⁒ ( 2.5 ∘ ) ⁒ 90 ∘ , Ξ± 2 = 1 ⁒ ( .1 ) 0.1 , 0.1 ⁒ ( .1 ) ⁒ 1 , k 2 = 0 ⁒ ( .05 ) ⁒ 0.9 ⁒ ( .02 ) ⁒ 1 to 10D by Fettis and Caslin (1964) (and warns of inaccuracies in Selfridge and Maxfield (1958) and Paxton and Rollin (1959)). …
3: 8.23 Statistical Applications
β–ΊThe function B x ⁑ ( a , b ) and its normalization I x ⁑ ( a , b ) play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of Q ⁑ ( a , x ) ; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …
4: 27.2 Functions
β–Ί
27.2.9 d ⁑ ( n ) = d | n 1
β–ΊIt is the special case k = 2 of the function d k ⁑ ( n ) that counts the number of ways of expressing n as the product of k factors, with the order of factors taken into account. …Note that Οƒ 0 ⁑ ( n ) = d ⁑ ( n ) . … β–ΊTable 27.2.2 tabulates the Euler totient function Ο• ⁑ ( n ) , the divisor function d ⁑ ( n ) ( = Οƒ 0 ⁑ ( n ) ), and the sum of the divisors Οƒ ⁑ ( n ) ( = Οƒ 1 ⁑ ( n ) ), for n = 1 ⁒ ( 1 ) ⁒ 52 . β–Ί
Table 27.2.1: Primes.
β–Ί β–Ίβ–Ί
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
β–Ί
5: 26.6 Other Lattice Path Numbers
β–Ί
Delannoy Number D ⁑ ( m , n )
β–Ί D ⁑ ( m , n ) is the number of paths from ( 0 , 0 ) to ( m , n ) that are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . … β–Ί
Table 26.6.1: Delannoy numbers D ⁑ ( m , n ) .
β–Ί β–Ίβ–Ί
m n
β–Ί
β–Ί
26.6.4 r ⁑ ( n ) = D ⁑ ( n , n ) D ⁑ ( n + 1 , n 1 ) , n 1 .
β–Ί
26.6.10 D ⁑ ( m , n ) = D ⁑ ( m , n 1 ) + D ⁑ ( m 1 , n ) + D ⁑ ( m 1 , n 1 ) , m , n 1 ,
6: Bibliography
β–Ί
  • H. Alzer (1997a) A harmonic mean inequality for the gamma function. J. Comput. Appl. Math. 87 (2), pp. 195–198.
  • β–Ί
  • D. E. Amos (1990) Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.
  • β–Ί
  • G. E. Andrews and A. Berkovich (1998) A trinomial analogue of Bailey’s lemma and N = 2 superconformal invariance. Comm. Math. Phys. 192 (2), pp. 245–260.
  • β–Ί
  • T. M. Apostol (2006) Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.
  • β–Ί
  • M. J. Atia, A. Martínez-Finkelshtein, P. Martínez-González, and F. Thabet (2014) Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.
  • 7: 22.7 Landen Transformations
    8: Bibliography H
    β–Ί
  • P. I. HadΕΎi (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Ε tiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
  • β–Ί
  • P. I. HadΕΎi (1976a) Expansions for the probability function in series of ČebyΕ‘ev polynomials and Bessel functions. Bul. Akad. Ε tiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).
  • β–Ί
  • P. I. HadΕΎi (1976b) Integrals that contain a probability function of complicated arguments. Bul. Akad. Ε tiince RSS Moldoven. 1976 (1), pp. 8084, 96 (Russian).
  • β–Ί
  • P. I. HadΕΎi (1978) Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 8084, 95 (Russian).
  • β–Ί
  • D. R. Hartree (1936) Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.
  • 9: 3.4 Differentiation
    β–ΊThe B k n are the differentiated Lagrangian interpolation coefficients: …where A k n is as in (3.3.10). … β–Ί
    B 2 7 = 1 240 ⁒ ( 72 + 36 ⁒ t 267 ⁒ t 2 80 ⁒ t 3 + 90 ⁒ t 4 + 12 ⁒ t 5 7 ⁒ t 6 ) ,
    β–Ίwhere C is a simple closed contour described in the positive rotational sense such that C and its interior lie in the domain of analyticity of f , and x 0 is interior to C . Taking C to be a circle of radius r centered at x 0 , we obtain …
    10: 10.31 Power Series