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11: 34.5 Basic Properties: $\mathit{6}j$ Symbol

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§34.5(ii) Symmetry

… ►Equations (34.5.9) and (34.5.10) are called Regge symmetries. Additional symmetries are obtained by applying (34.5.8) to (34.5.9) and (34.5.10). …

12: 34.3 Basic Properties: $\mathit{3}j$ Symbol

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§34.3(ii) Symmetry

… ►Equations (34.3.11) and (34.3.12) are called Regge symmetries. Additional symmetries are obtained by applying (34.3.8)–(34.3.10) to (34.3.11)) and (34.3.12). …

13: Bibliography N

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Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).

Encodings:

BibTeX

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Symmetries in the fourth Painlevé equation and Okamoto polynomials. Nagoya Math. J. 153, pp. 53–86.

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External Links:

ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(iv).

Encodings:

BibTeX

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Painlevé Equations through Symmetry. Translations of Mathematical Monographs, Vol. 223, American Mathematical Society, Providence, RI.

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Notes:

Translated from the 2000 Japanese original by the author

External Links:

ISBN 0-8218-3221-2, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.2(v).

Encodings:

BibTeX

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14: 24.4 Basic Properties

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§24.4(ii) Symmetry

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15: 10.73 Physical Applications

… ►Bessel functions of the first kind, $J_n\left(x\right)$, arise naturally in applications having cylindrical symmetry in which the physics is described either by Laplace’s equation ${\nabla }^{2}V=0$, or by the Helmholtz equation $({\nabla }^2+k^2)\psi =0$. … ►Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation. …

16: 36.7 Zeros

… ►Near $z=z_n$, and for small $x$ and $y$, the modulus $|{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)|$ has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose $z$ and $x$ repeat distances are given by …In the symmetry planes (e. …

17: Bibliography Z

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“Hidden symmetry” of Askey-Wilson polynomials. Theoret. and Math. Phys. 89 (2), pp. 1146–1157.

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External Links:

MathReview (Vadim B. Kuznetsov), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

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18: Bibliography C

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Symmetry in c, d, n of Jacobian elliptic functions. J. Math. Anal. Appl. 299 (1), pp. 242–253.

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External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§19.25(v), §19.25(v), §22.13(i), §22.6(ii), §22.6(iii), §22.8(i), §22.8(ii), Profile Bille C. Carlson.

Encodings:

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Permutation symmetry for theta functions. J. Math. Anal. Appl. 378 (1), pp. 42–48.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Mohammad Ahmad), ZentralBlatt

Referenced by:

§20.11(v), §20.7(i), §20.7(i), §20.7(ii), §20.7(ii), §20.7(ii), §20.7(iii), §20.7(iii), §20.7(vii), §20.7(vii), Profile Bille C. Carlson.

Encodings:

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Elliptic Integrals: Symmetry and Symbolic Integration. In Tricomi’s Ideas and Contemporary Applied Mathematics (Rome/Turin, 1997), Atti dei Convegni Lincei, Vol. 147, pp. 161–181.

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External Links:

MathReview (S. L. Kalla), ZentralBlatt

Referenced by:

§19.26(iii), §19.29(i).

Encodings:

BibTeX

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Nonclassical Symmetry Reductions and Exact Solutions for Physically Significant Nonlinear Evolution Equations. In Nonlinear and Chaotic Phenomena in Plasmas, Solids and Fluids (Edmonton, AB, 1990), W. Rozmus and J. A. Tuszynski (Eds.), pp. 72–79.

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External Links:

MathReview

Referenced by:

§29.19(ii).

Encodings:

BibTeX

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Multi-Integral Representations for Associated Legendre and Ferrers Functions. Symmetry 12 (10).

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External Links:

Link, ISSN 2073-8994, Document, ZentralBlatt

Referenced by:

§14.6(ii), §14.6(ii), Erratum (V1.1.1) for Subsection 14.6(ii).

Encodings:

BibTeX

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19: 36.2 Catastrophes and Canonical Integrals

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§36.2(iii) Symmetries

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20: 20.9 Relations to Other Functions

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