higher-order 3nj symbols

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11: Bibliography

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On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

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Sharp bounds for the Bernoulli numbers. Arch. Math. (Basel) 74 (3), pp. 207–211.

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External Links:

ISSN 0003-889X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.9.

Encodings:

BibTeX

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Computation of exponential integrals. ACM Trans. Math. Software 6 (3), pp. 365–377.

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External Links:

ISSN 0098-3500, Document, MathReview (M. M. Chawla), ZentralBlatt

Referenced by:

§8.25(v).

Encodings:

BibTeX

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$q$-identities of Auluck, Carlitz, and Rogers. Duke Math. J. 33 (3), pp. 575–581.

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External Links:

Document, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§17.9(iv).

Encodings:

BibTeX

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Numerical Tables for Angular Correlation Computations in $\alpha$-, $\beta$- and $\gamma$-Spectroscopy: $3j$-, $6j$-, $9j$-Symbols, F- and $\mathrm{\Gamma }$-Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.

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External Links:

ISBN 978-3-540-04218-1

Referenced by:

§34.14.

Encodings:

BibTeX

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12: 2.11 Remainder Terms; Stokes Phenomenon

… ►As noted in §2.11(i), poor accuracy is yielded by this expansion as $\mathrm{ph}z$ approaches $\frac{3}{2}\pi$ or $-\frac{3}{2}\pi$. …Since the ray $\mathrm{ph}z=\frac{3}{2}\pi$ is well away from the new boundaries, the compound expansion (2.11.7) yields much more accurate results when $\mathrm{ph}z\to \frac{3}{2}\pi$. In effect, (2.11.7) “corrects” (2.11.6) by introducing a term that is relatively exponentially small in the neighborhood of $\mathrm{ph}z=\pi$, is increasingly significant as $\mathrm{ph}z$ passes from $\pi$ to $\frac{3}{2}\pi$, and becomes the dominant contribution after $\mathrm{ph}z$ passes $\frac{3}{2}\pi$. … ►For higher-order Stokes phenomena see Olde Daalhuis (2004b) and Howls et al. (2004). … ►For higher-order differential equations, see Olde Daalhuis (1998a, b). …

13: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►For $n=3$: …Higher-order determinants are natural generalizations. … ►

1.3.15

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Symbols:

$det$: determinant, $k$: integer, $n$: nonnegative integer and ${\omega }_j$: roots of unity

Permalink:

http://dlmf.nist.gov/1.3.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

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14: 8.19 Generalized Exponential Integral

… ►For $n=1,2,3,\mathrm{\dots }$, … ►The right-hand sides are replaced by their limiting forms when $p=1,2,3,\mathrm{\dots }$. … ►For $j=1,2,3,\mathrm{\dots }$, … ►For $n=1,2,3,\mathrm{\dots }$ and $x>0$, … ►For higher-order generalized exponential integrals see Meijer and Baken (1987) and Milgram (1985).

15: 1.13 Differential Equations

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1.13.20 $$\{z,\zeta \}=-2{\dot{z}}^{1/2}\frac{d^2}{{d\zeta }^2}({\dot{z}}^{-1/2})=\frac{\stackrel{\dot{}\dot{}\dot{}}{z}}{\dot{z}}-\frac{3}{2}{\left(\frac{\ddot{z}}{\dot{z}}\right)}^2.$$

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Defines:

$\{z,\zeta \}$: Schwarzian derivative

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: variable, $\zeta (z)$: thrice-differentiable function, $\dot{z}$: derivative of $z$ with respect to $\zeta$ , $\ddot{z}$: 2nd derivative of $z$ with respect to $\zeta$ and $\stackrel{\dot{}\dot{}\dot{}}{z}$: 3rd derivative of $z$ with respect to $\zeta$

Permalink:

http://dlmf.nist.gov/1.13.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

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1.13.23 $$\frac{d^{3}w}{{dz}^3}+3f\frac{d^{2}w}{{dz}^2}+(2f^2+f^{\prime }+4g)\frac{dw}{dz}+(4fg+2g^{\prime })w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: variable, $f(z)$: analytic coefficient, $g(z)$: analytic coefficient and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/1.13.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §1.13(v), §1.13 and Ch.1

… ►For an extensive collection of solutions of differential equations of the first, second, and higher orders see Kamke (1977). …

16: 2.7 Differential Equations

… ►when $s=1,2,3,\mathrm{\dots }$. … ►For extensions to higher-order differential equations see Stenger (1966a, b), Olver (1997a, 1999), and Olde Daalhuis and Olver (1998). … ►Then there are solutions $w_3(x)$, $w_4(x)$, such that …Similarly for $w_2(x)$ and $w_3(x)$ as $x\to a_2-$. … ►as $x\to +\mathrm{\infty }$, $w_2(x)$ being recessive and $w_3(x)$ dominant. …

17: 15.8 Transformations of Variable

… ►The hypergeometric functions that correspond to Groups 3 and 4 have a nonlinear function of $z$ as variable. … ►

Group 1 $⟶$ Group 3

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Group 2 $⟶$ Group 3

… ►which is a quadratic transformation between two cases in Group 3. … ►For further examples and higher-order transformations see Goursat (1881), Watson (1910), Vidūnas (2005), and Tu and Yang (2013); see also Erdélyi et al. (1953a, pp. 67 and 113–114). …

18: 34.12 Physical Applications

§34.12 Physical Applications

►The angular momentum coupling coefficients ($\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols) are essential in the fields of nuclear, atomic, and molecular physics. …$\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).

19: 34.10 Zeros

§34.10 Zeros

►In a $\mathit{3}j$ symbol, if the three angular momenta $j_1,j_2,j_3$ do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the $\mathit{3}j$ symbol is zero. Similarly the $\mathit{6}j$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3}j$ symbols in the summation. …However, the $\mathit{3}j$ and $\mathit{6}j$ symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. Such zeros are called nontrivial zeros. …

20: 34.14 Tables

§34.14 Tables

►Tables of exact values of the squares of the $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols on pp. … ►Tables of $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 17/2$ are given in Appel (1968) to 6D. …Other tabulations for $\mathit{3}j$ symbols are listed on pp.  11-12; for $\mathit{6}j$ symbols on pp. …