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1: 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 ) ,
2: 34.4 Definition: 6 ⁒ j Symbol
§34.4 Definition: 6 ⁒ j Symbol
β–ΊThe 6 ⁒ j symbol can be expressed as the finite sumβ–ΊFor 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).
3: 34.5 Basic Properties: 6 ⁒ j Symbol
β–ΊExamples are provided by: …
4: Bibliography R
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  • J. Raynal (1979) On the definition and properties of generalized 6 - j  symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
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  • H. Rosengren (1999) Another proof of the triple sum formula for Wigner 9 ⁒ j -symbols. J. Math. Phys. 40 (12), pp. 6689–6691.
  • β–Ί
  • G. Rota, D. Kahaner, and A. Odlyzko (1973) On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.
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  • M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr. (1959) The 3 - j and 6 - j Symbols. The Technology Press, MIT, Cambridge, MA.
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  • K. Rottbrand (2000) Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.
  • 5: Mathematical Introduction
    β–Ί β–Ίβ–Ίβ–Ίβ–Ίβ–Ί
    β„‚ complex plane (excluding infinity).
    empty sums zero.
    n !! double factorial: 2 4 6 ⁒ β‹― ⁒ n if n = 2 , 4 , 6 , ; 1 3 5 ⁒ β‹― ⁒ n if n = 1 , 3 , 5 , ; 1 if n = 0 , 1 .
    < is finite, or converges.
    β–Ί β–Ίβ–Ίβ–Ί
    ( a , b ] or [ a , b ) half-closed intervals.
    [ a j , k ] or [ a j ⁒ k ] matrix with ( j , k ) th element a j , k or a j ⁒ k .
    β–Ί J. …
    6: 10.22 Integrals
    β–Ί
    §10.22(ii) Integrals over Finite Intervals
    β–ΊWhen Ξ± = m = 1 , 2 , 3 , the left-hand side of (10.22.36) is the m th repeated integral of J Ξ½ ⁑ ( x ) (§§1.4(v) and 1.15(vi)). … β–Ίwhere j Ξ½ , β„“ and j Ξ½ , m are zeros of J Ξ½ ⁑ ( x ) 10.21(i)), and Ξ΄ β„“ , m is Kronecker’s symbol. … β–Ίwhere Ξ± β„“ and Ξ± m are positive zeros of a ⁒ J Ξ½ ⁑ ( x ) + b ⁒ x ⁒ J Ξ½ ⁑ ( x ) . … …
    7: Errata
    β–Ί
  • Subsection 31.11(iv)

    Just below (31.11.17), P j has been replaced with P j 6 .

  • β–Ί
  • Chapters 10 Bessel Functions, 18 Orthogonal Polynomials, 34 3j, 6j, 9j Symbols

    The Legendre polynomial P n was mistakenly identified as the associated Legendre function P n in §§10.54, 10.59, 10.60, 18.18, 18.41, 34.3 (and was thus also affected by the bug reported below). These symbols now link correctly to their definitions. Reported by Roy Hughes on 2022-05-23

  • β–Ί
  • Section 34.1

    The relation between Clebsch-Gordan and 3 ⁒ j symbols was clarified, and the sign of m 3 was changed for readability. The reference Condon and Shortley (1935) for the Clebsch-Gordan coefficients was replaced by Edmonds (1974) and Rotenberg et al. (1959) and the references for 3 ⁒ j , 6 ⁒ j , 9 ⁒ j symbols were made more precise in §34.1.

  • β–Ί
  • Section 34.1

    The reference for Clebsch-Gordan coefficients, Condon and Shortley (1935), was replaced by Edmonds (1974) and Rotenberg et al. (1959). The references for 3 ⁒ j , 6 ⁒ j , 9 ⁒ j symbols were made more precise.

  • β–Ί
  • Equation (34.7.4)
    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 )

    Originally the third 3 ⁒ j symbol in the summation was written incorrectly as ( j 31 j 32 j 33 m 13 m 23 m 33 ) .

    Reported 2015-01-19 by Yan-Rui Liu.

  • 8: Bibliography F
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  • J. L. Fields and Y. L. Luke (1963b) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. J. Math. Anal. Appl. 6 (3), pp. 394–403.
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  • J. L. Fields (1983) Uniform asymptotic expansions of a class of Meijer G -functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.
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  • J. P. M. Flude (1998) The Edmonds asymptotic formulas for the 3 ⁒ j and 6 ⁒ j symbols. J. Math. Phys. 39 (7), pp. 3906–3915.
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  • P. J. Forrester and N. S. Witte (2002) Application of the Ο„ -function theory of Painlevé equations to random matrices: P V , P III , the LUE, JUE, and CUE. Comm. Pure Appl. Math. 55 (6), pp. 679–727.
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  • T. Fort (1948) Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.
  • 9: Bibliography W
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  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
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  • G. N. Watson (1935b) The surface of an ellipsoid. Quart. J. Math., Oxford Ser. 6, pp. 280–287.
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  • J. K. G. Watson (1999) Asymptotic approximations for certain 6 - j and 9 - j symbols. J. Phys. A 32 (39), pp. 6901–6902.
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  • D. V. Widder (1941) The Laplace Transform. Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, NJ.
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  • C. Y. Wu (1982) A series of inequalities for Mills’s ratio. Acta Math. Sinica 25 (6), pp. 660–670.
  • 10: 25.11 Hurwitz Zeta Function
    β–Ί
    25.11.10 ΢ ⁑ ( s , a ) = n = 0 ( s ) n n ! ⁒ ΢ ⁑ ( n + s ) ⁒ ( 1 a ) n , s 1 , | a 1 | < 1 .
    β–Ί
    25.11.36Removed because it is just (25.15.1) combined with (25.15.3).
    β–Ί
    §25.11(xi) Sums
    β–ΊFor further sums see Prudnikov et al. (1990, pp. 396–397) and Hansen (1975, pp. 358–360). … β–Ί
    25.11.43 ΢ ⁑ ( s , a ) a 1 s s 1 1 2 ⁒ a s k = 1 B 2 ⁒ k ( 2 ⁒ k ) ! ⁒ ( s ) 2 ⁒ k 1 ⁒ a 1 s 2 ⁒ k .