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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: 26.5 Lattice Paths: Catalan Numbers
β–Ί C ⁑ ( n ) is the Catalan number. …(Sixty-six equivalent definitions of C ⁑ ( n ) are given in Stanley (1999, pp. 219–229).) … β–Ί
26.5.3 C ⁑ ( n + 1 ) = k = 0 n C ⁑ ( k ) ⁒ C ⁑ ( n k ) ,
β–Ί
26.5.4 C ⁑ ( n + 1 ) = 2 ⁒ ( 2 ⁒ n + 1 ) n + 2 ⁒ C ⁑ ( n ) ,
β–Ί
26.5.7 lim n C ⁑ ( n + 1 ) C ⁑ ( n ) = 4 .
3: 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 (1975b) Integrals containing the Fresnel functions S ⁒ ( x ) and C ⁒ ( x ) . Bul. Akad. Ε tiince RSS Moldoven. 1975 (3), pp. 48–60, 93 (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.
  • 4: 34.14 Tables
    §34.14 Tables
    β–ΊTables of exact values of the squares of the 3 ⁒ j and 6 ⁒ j symbols in which all parameters are 8 are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of 3 ⁒ j , 6 ⁒ j , and 9 ⁒ j symbols on pp. … β–ΊSome selected 9 ⁒ j symbols are also given. … 16-17; for 9 ⁒ j symbols on p. … β–Ί 310–332, and for the 9 ⁒ j symbols on pp. …
    5: 19.2 Definitions
    β–ΊThe principal values of K ⁑ ( k ) and E ⁑ ( k ) are even functions. … β–Ί
    §19.2(iv) A Related Function: R C ⁑ ( x , y )
    β–ΊFormulas involving Ξ  ⁑ ( Ο• , Ξ± 2 , k ) that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using R C ⁑ ( x , y ) . … β–ΊWhen x and y are positive, R C ⁑ ( x , y ) is an inverse circular function if x < y and an inverse hyperbolic function (or logarithm) if x > y : …For the special cases of R C ⁑ ( x , x ) and R C ⁑ ( 0 , y ) see (19.6.15). …
    6: 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–Ίβ–Ί
    m n
    0 1 2 3 4 5 6 7 8 9 10
    β–Ί
    β–Ί
    Table 26.6.2: Motzkin numbers M ⁑ ( n ) .
    β–Ί β–Ίβ–Ίβ–Ί
    n M ⁑ ( n ) n M ⁑ ( n ) n M ⁑ ( n ) n M ⁑ ( n ) n M ⁑ ( n )
    0 1 4 9 8 323 12 15511 16 8 53467
    β–Ί
    β–Ί
    Table 26.6.3: Narayana numbers N ⁑ ( n , k ) .
    β–Ί β–Ίβ–Ίβ–Ί
    n k
    0 1 2 3 4 5 6 7 8 9 10
    β–Ί
    7: 16.24 Physical Applications
    β–Ί
    §16.24(iii) 3 ⁒ j , 6 ⁒ j , and 9 ⁒ j Symbols
    β–ΊThey can be expressed as F 2 3 functions with unit argument. …These are balanced F 3 4 functions with unit argument. Lastly, special cases of the 9 ⁒ j symbols are F 4 5 functions with unit argument. …
    8: 1.12 Continued Fractions
    β–Ί C n is called the n th approximant or convergent to C . … β–ΊDefine … β–ΊA contraction of a continued fraction C is a continued fraction C whose convergents { C n } form a subsequence of the convergents { C n } of C . Conversely, C is called an extension of C . … β–ΊThen the convergents C n satisfy …
    9: 16.7 Relations to Other Functions
    β–ΊFor 3 ⁒ j , 6 ⁒ j , 9 ⁒ j symbols see Chapter 34. Further representations of special functions in terms of F q p functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of F q q + 1 functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).
    10: 23 Weierstrass Elliptic and Modular
    Functions