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1: 34.3 Basic Properties: 3 j Symbol
§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics
Equations (34.3.19)–(34.3.22) are particular cases of more general results that relate rotation matrices to 3 j symbols, for which see Edmonds (1974, Chapter 4). …
2: 18.38 Mathematical Applications
3: 16.24 Physical Applications
§16.24(iii) 3 j , 6 j , and 9 j Symbols
4: 16.4 Argument Unity
See Raynal (1979) for a statement in terms of 3 j symbols (Chapter 34). …
5: 16.7 Relations to Other Functions
§16.7 Relations to Other Functions
For 3 j , 6 j , 9 j symbols see Chapter 34. …
6: Bibliography M
  • A. J. MacLeod (1993) Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.
  • A. J. MacLeod (1996a) Algorithm 757: MISCFUN, a software package to compute uncommon special functions. ACM Trans. Math. Software 22 (3), pp. 288–301.
  • I. Marquette and C. Quesne (2016) Connection between quantum systems involving the fourth Painlevé transcendent and k -step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..
  • M. Micu (1968) Recursion relations for the 3 - j symbols. Nuclear Physics A 113 (1), pp. 215–220.
  • G. J. Miel (1981) Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.
  • 7: 34.5 Basic Properties: 6 j Symbol
    §34.5 Basic Properties: 6 j Symbol
    §34.5(ii) Symmetry
    Additional symmetries are obtained by applying (34.5.8) to (34.5.9) and (34.5.10). …
    §34.5(iii) Recursion Relations
    §34.5(iv) Orthogonality
    8: 10 Bessel Functions
    … …
    9: Bibliography F
  • H. E. Fettis (1970) On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.
  • J. L. Fields and Y. L. Luke (1963a) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II. J. Math. Anal. Appl. 7 (3), pp. 440–451.
  • 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.
  • J. L. Fields (1965) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. III. J. Math. Anal. Appl. 12 (3), pp. 593–601.
  • 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.
  • 10: Errata
  • 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

  • Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

    The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.

  • 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.

  • 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.

  • Equation (34.3.7)
    34.3.7 ( j 1 j 2 j 3 j 1 j 1 m 3 m 3 ) = ( 1 ) j 1 j 2 m 3 ( ( 2 j 1 ) ! ( j 1 + j 2 + j 3 ) ! ( j 1 + j 2 + m 3 ) ! ( j 3 m 3 ) ! ( j 1 + j 2 + j 3 + 1 ) ! ( j 1 j 2 + j 3 ) ! ( j 1 + j 2 j 3 ) ! ( j 1 + j 2 m 3 ) ! ( j 3 + m 3 ) ! ) 1 2

    In the original equation the prefactor of the above 3j symbol read ( 1 ) j 2 + j 3 + m 3 . It is now replaced by its correct value ( 1 ) j 1 j 2 m 3 .

    Reported 2014-06-12 by James Zibin.