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21: 34.7 Basic Properties: 9 ⁢ j Symbol
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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 } .
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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 ′ .
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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 } .
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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 ) .
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Symbols:
{ j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } : 9 ⁢ j symbol, ( j 1 j 2 j 3 m 1 m 2 m 3 ) : 3 ⁢ j symbol, j , j r : non-negative integers or non-negative integers plus one half. and r : nonnegative integer
Referenced by:
Erratum (V1.0.10) for Equation (34.7.4)
Permalink:
http://dlmf.nist.gov/34.7.E4
Encodings:
pMML, png, TeX
Errata (effective with 1.0.10):
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

See also:
Annotations for §34.7(vi), §34.7 and Ch.34
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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 } .
22: Simon Ruijsenaars
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    • 23: Bibliography L
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  • A. Leitner and J. Meixner (1960) Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.
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    External Links:
    ISSN 0003-9268, Document, MathReview (M. J. O. Strutt), ZentralBlatt
    Referenced by:
    §30.12.
    Encodings:
    BibTeX
    • … ►
  • Y. Lin and R. Wong (2013) Global asymptotics of the Hahn polynomials. Anal. Appl. (Singap.) 11 (3), pp. 1350018, 47.
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    External Links:
    ISSN 0219-5305, Document, FullText, MathReview (Yamilet Quintana), ZentralBlatt
    Referenced by:
    §18.24, §18.24.
    Encodings:
    BibTeX
    • … ►
  • H. Lotsch and M. Gray (1964) Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.
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    Notes:
    Includes Algol program, maximum accuracy 12D.
    External Links:
    ISSN 0001-0782, Document
    Referenced by:
    §7.25(iv).
    Encodings:
    BibTeX
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  • N. A. Lukaševič (1967b) On the theory of Painlevé’s third equation. Differ. Uravn. 3 (11), pp. 1913–1923 (Russian).
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    Notes:
    In Russian. English translation: Differential Equations 3(11), pp. 994–999
    External Links:
    MathReview (Z. Nehari), ZentralBlatt
    Referenced by:
    §32.10(iii), §32.8(iii), §32.9(i), §32.9(ii).
    Encodings:
    BibTeX
    • … ►
  • Y. L. Luke (1977a) Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.
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    External Links:
    MathReview (F. Gotze), ZentralBlatt
    Referenced by:
    §13.31(iii).
    Encodings:
    BibTeX
    • …
    24: Bibliography D
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  • M. D’Ocagne (1904) Sur une classe de nombres rationnels réductibles aux nombres de Bernoulli. Bull. Sci. Math. (2) 28, pp. 29–32 (French).
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    External Links:
    ZentralBlatt
    Referenced by:
    §24.1.
    Encodings:
    BibTeX
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  • P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou (1999b) Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.
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    External Links:
    ISSN 0010-3640, Document, MathReview (D. S. Lubinsky), ZentralBlatt
    Referenced by:
    §18.32.
    Encodings:
    BibTeX
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  • E. Dorrer (1968) Algorithm 322. F-distribution. Comm. ACM 11 (2), pp. 116–117.
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    Notes:
    Includes Algol program, maximum accuracy 10S. For certification see same journal 12(1969), p. 39
    External Links:
    ISSN 0001-0782, Document
    Referenced by:
    §8.28(iv).
    Encodings:
    BibTeX
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  • B. A. Dubrovin (1981) Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).
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    Notes:
    With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)
    External Links:
    ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt
    Referenced by:
    §21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.
    Encodings:
    BibTeX
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  • J. Dutka (1981) The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.
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    External Links:
    ISSN 0003-9519, MathReview (Willard Parker), ZentralBlatt
    Referenced by:
    §8.17(i).
    Encodings:
    BibTeX
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    25: 28.29 Definitions and Basic Properties
    … ►Its order of growth for | λ | → ∞ is exactly 1 2 ; see Magnus and Winkler (1966, Chapter II, pp. 19–28). … ►For this purpose the discriminant can be expressed as an infinite determinant involving the Fourier coefficients of Q ⁡ ( x ) ; see Magnus and Winkler (1966, §2.3, pp. 28–36). …
    26: 10 Bessel Functions
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    27: 34.9 Graphical Method
    … ►For specific examples of the graphical method of representing sums involving the 3 ⁢ j , 6 ⁢ j , and 9 ⁢ j symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).
    28: Richard B. Paris
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    29: Bibliography M
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  • M. Mazzocco (2001a) Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.
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    External Links:
    ISSN 0305-4470, Document, MathReview (Shun Shimomura), ZentralBlatt
    Referenced by:
    §32.8(vi).
    Encodings:
    BibTeX
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  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
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    External Links:
    ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
    Referenced by:
    §18.27(v).
    Encodings:
    BibTeX
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  • D. Müller, B. G. Kelly, and J. J. O’Brien (1994) Spheroidal eigenfunctions of the tidal equation. Phys. Rev. Lett. 73 (11), pp. 1557–1560.
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    External Links:
    ISSN 0031-9007, Document, MathReview, ZentralBlatt
    Referenced by:
    §30.13(v).
    Encodings:
    BibTeX
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  • Y. Murata (1985) Rational solutions of the second and the fourth Painlevé equations. Funkcial. Ekvac. 28 (1), pp. 1–32.
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    External Links:
    ISSN 0532-8721, FullText, MathReview (Hélène Airault), ZentralBlatt
    Referenced by:
    §32.8(iv).
    Encodings:
    BibTeX
    • … ►
  • L. A. Muraveĭ (1976) Zeros of the function A ⁢ i ′ ⁢ ( z ) − σ ⁢ A ⁢ i ⁢ ( z ) . Differential Equations 11, pp. 797–811.
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    External Links:
    ZentralBlatt
    Referenced by:
    §9.9(i).
    Encodings:
    BibTeX
    • …
    30: Gerhard Wolf
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