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1: 34.6 Definition: $\mathit{9j}$ Symbol

§34.6 Definition: $\mathit{9j}$ Symbol

►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►

34.6.2

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.13
Permalink:: http://dlmf.nist.gov/34.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►The

\mathit{9j}

symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

3: Bibliography H

… ►

P. I. Hadži (1975b) Integrals containing the Fresnel functions

S(x)

and

C(x)

. Bul. Akad. Štiince RSS Moldoven. 1975 (3), pp. 48–60, 93 (Russian).

ⓘ

External Links:: MathReview (J. Kaczmarski)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

►

P. I. Hadži (1976a) Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

ⓘ

External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

J. R. Herndon (1961b) Algorithm 56: Complete elliptic integral of the second kind. Comm. ACM 4 (4), pp. 180–181.

ⓘ

Notes:: Includes Algol program, maximum accuracy 8D. For certification see same journal 9 (1966), p. 12
External Links:: ISSN 0001-0782, Document
Referenced by:: §19.39(ii).
Encodings:: BibTeX

… ►

M. H. Hull and G. Breit (1959) Coulomb Wave Functions. In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.), pp. 408–465.

ⓘ

External Links:: MathReview (G. Källén)
Referenced by:: §33.2(iii), §33.23(vii), §33.23(vii), §33.24, §33.5(ii), §33.7, Ch.33.
Encodings:: BibTeX

►

T. E. Hull and A. Abrham (1986) Variable precision exponential function. ACM Trans. Math. Software 12 (2), pp. 79–91.

ⓘ

External Links:: ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt
Referenced by:: §4.48(iii).
Encodings:: BibTeX

…

A. Decarreau, M.-Cl. Dumont-Lepage, P. Maroni, A. Robert, and A. Ronveaux (1978a) Formes canoniques des équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.

ⓘ

External Links:: MathReview (A. E. Danese), ZentralBlatt
Referenced by:: §31.12.
Encodings:: BibTeX

►

A. Decarreau, P. Maroni, and A. Robert (1978b) Sur les équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (3), pp. 151–189.

ⓘ

External Links:: ISSN 0037-959X, MathReview, ZentralBlatt
Referenced by:: §31.12.
Encodings:: BibTeX

… ►

G. Doetsch (1955) Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

ⓘ

External Links:: MathReview (I. I. Hirschman), ZentralBlatt
Referenced by:: §2.5(i).
Encodings:: BibTeX

… ►

B. A. Dubrovin (1981) Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

ⓘ

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

… ►

L. Durand (1978) Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.

ⓘ

External Links:: Document, MathReview (R. C. Varma), ZentralBlatt
Referenced by:: §18.17(iii).
Encodings:: BibTeX

…

5: 23.21 Physical Applications

… ►The Weierstrass function

\wp

plays a similar role for cubic potentials in canonical form

g_{3}+g_{2}x-4x^{3}

. … ►For applications to soliton solutions of the Korteweg–de Vries (KdV) equation see McKean and Moll (1999, p. 91), Deconinck and Segur (2000), and Walker (1996, §8.1). … ►where

x, y, z

are the corresponding Cartesian coordinates and

e_{1}

e_{2}

e_{3}

are constants. … ►

23.21.3

f(\rho)=2\left((\rho-e_{1})(\rho-e_{2})(\rho-e_{3})\right)^{1/2}.

ⓘ

Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $\rho$ : roots and $f(\rho)$ : function
Permalink:: http://dlmf.nist.gov/23.21.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

►Another form is obtained by identifying

e_{1}

e_{2}

e_{3}

as lattice roots (§23.3(i)), and setting …

6: 18.13 Continued Fractions

… ►

18.13.3

\cfrac{a_{1}}{x+\cfrac{-\frac{1}{2}}{\frac{3}{2}x+\cfrac{-\frac{2}{3}}{\frac{5% }{3}x+\cfrac{-\frac{3}{4}}{\frac{7}{4}x+}}}}\cdots,

ⓘ

Symbols:: $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.13.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §18.13, §18.13 and Ch.18

… ►

18.13.4

\cfrac{a_{1}}{1-x+\cfrac{-\frac{1}{2}}{\frac{1}{2}(3-x)+\cfrac{-\frac{2}{3}}{% \frac{1}{3}(5-x)+\cfrac{-\frac{3}{4}}{\frac{1}{4}(7-x)+}}}}\cdots,

ⓘ

Symbols:: $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.13.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.13, §18.13 and Ch.18

… ►See also Cuyt et al. (2008, pp. 91–99).

7: 24.2 Definitions and Generating Functions

… ►

B_{2n+1}=0

… ►

24.2.4

B_{n}=B_{n}\left(0\right),

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials and $n$ : integer
A&S Ref:: 23.1.2
Referenced by:: (25.11.7)
Permalink:: http://dlmf.nist.gov/24.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.2(i), §24.2 and Ch.24

►

24.2.5

B_{n}\left(x\right)=\sum_{k=0}^{n}{n\choose k}B_{k}x^{n-k}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §24.2(i), §24.2 and Ch.24

… ►

\widetilde{B}_{n}\left(x\right)=B_{n}\left(x\right)

… ►

\widetilde{B}_{n}\left(x+1\right)=\widetilde{B}_{n}\left(x\right),

…

8: Bibliography R

… ►

S. Ramanujan (1921) Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.

ⓘ

External Links:: ISSN 0025-5874, ISSN 0025-5874, Document, MathReview Entry, ZentralBlatt
Referenced by:: §27.14(v).
Encodings:: BibTeX

… ►

W. H. Reid (1974a) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

ⓘ

External Links:: MathReview (T. W. Kao), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

… ►

W. P. Reinhardt (2021a) Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.

ⓘ

External Links:: ISSN 1521-9615, Document, Link
Referenced by:: §18.39(iv), §18.40(ii).
Encodings:: BibTeX

… ►

R. Reynolds and A. Stauffer (2021) Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).

ⓘ

External Links:: Link, ISSN 2227-7390, Document
Referenced by:: §8.15.
Encodings:: BibTeX

… ►

H. Rosengren (1999) Another proof of the triple sum formula for Wigner

9j

-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt
Referenced by:: §34.6.
Encodings:: BibTeX

…

9: Bibliography N

… ►

G. Nemes (2013c) Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

ⓘ

External Links:: ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt
Referenced by:: §5.11(i), §5.11(i).
Encodings:: BibTeX

… ►

G. Nemes (2015b) On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

ⓘ

External Links:: ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt
Referenced by:: §11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).
Encodings:: BibTeX

… ►

E. Neuman (1969b) On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

… ►

V. Yu. Novokshënov (1985) The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

ⓘ

Notes:: In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.
External Links:: ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

… ►

H. M. Nussenzveig (1965) High-frequency scattering by an impenetrable sphere. Ann. Physics 34 (1), pp. 23–95.

ⓘ

External Links:: Document, MathReview (E. Ar)
Referenced by:: §14.31(iii).
Encodings:: BibTeX

…

10: Bibliography

… ►

W. A. Al-Salam and L. Carlitz (1959) Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

ⓘ

External Links:: FullText, MathReview (D. P. Banerjee), ZentralBlatt
Referenced by:: §24.14(ii).
Encodings:: BibTeX

… ►

G. E. Andrews (1966a) On basic hypergeometric series, mock theta functions, and partitions. II. Quart. J. Math. Oxford Ser. (2) 17, pp. 132–143.

ⓘ

External Links:: Document, MathReview (B. R. Bhonsle), ZentralBlatt
Referenced by:: §17.6(ii).
Encodings:: BibTeX

… ►

G. E. Andrews (2000) Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.

ⓘ

Notes:: In memory of Gian-Carlo Rota
External Links:: ISSN 0097-3165, Document, MathReview (Alessandro Di Bucchianico), ZentralBlatt
Referenced by:: §17.12.
Encodings:: BibTeX

… ►

T. M. Apostol (1952) Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

ⓘ

External Links:: MathReview (N. G. de Bruijn), ZentralBlatt
Referenced by:: (25.11.41), (25.11.42), §25.11(xii).
Encodings:: BibTeX

… ►

F. M. Arscott (1959) A new treatment of the ellipsoidal wave equation. Proc. London Math. Soc. (3) 9, pp. 21–50.

ⓘ

External Links:: ISSN 0024-6115, Document, MathReview (J. Meixner), ZentralBlatt
Referenced by:: §29.11.
Encodings:: BibTeX

…

.%E7%BD%91%E7%BB%9C%E7%9B%B4%E6%92%AD%E4%B8%96%E7%95%8C%E6%9D%papp%E3%80%8E%E7%BD%91%E5%9D%80%3Amxsty.cc%E3%80%8F.2022%E4%B8%8B%E5%B1%8A%E4%B8%96%E7%95%8C%E6%9D%AF%E5%9C%A8%E5%93%AA%E4%B8%AA%E5%9B%BD%E5%AE%B6-m6q3s2-2022%E5%B9%411%E6%9C%882%E6%97%256%E6%97%623%E5%88%861%E7%A7%92wmcsgwckm.com

1—10 of 754 matching pages

1: 34.6 Definition: $\mathit{9j}$ Symbol

§34.6 Definition: $\mathit{9j}$ Symbol

2: 19.2 Definitions

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

3: Bibliography H

4: Bibliography D

5: 23.21 Physical Applications

6: 18.13 Continued Fractions

7: 24.2 Definitions and Generating Functions

8: Bibliography R

9: Bibliography N

10: Bibliography

.%E7%BD%91%E7%BB%9C%E7%9B%B4%E6%92%AD%E4%B8%96%E7%95%8C%E6%9D%papp%E3%80%8E%E7%BD%91%E5%9D%80%3Amxsty.cc%E3%80%8F.2022%E4%B8%8B%E5%B1%8A%E4%B8%96%E7%95%8C%E6%9D%AF%E5%9C%A8%E5%93%AA%E4%B8%AA%E5%9B%BD%E5%AE%B6-m6q3s2-2022%E5%B9%411%E6%9C%882%E6%97%256%E6%97%623%E5%88%861%E7%A7%92wmcsgwckm.com

1—10 of 754 matching pages

1: 34.6 Definition: 9 ⁢ j Symbol

§34.6 Definition: 9 ⁢ j Symbol

2: 19.2 Definitions

§19.2(iv) A Related Function: R C ⁡ ( x , y )

3: Bibliography H

4: Bibliography D

5: 23.21 Physical Applications

6: 18.13 Continued Fractions

7: 24.2 Definitions and Generating Functions

8: Bibliography R

9: Bibliography N

10: Bibliography

1: 34.6 Definition: $\mathit{9j}$ Symbol

§34.6 Definition: $\mathit{9j}$ Symbol

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$