Regge symmetries

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1: 34.5 Basic Properties: $\mathit{6j}$ 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). …

2: 34.3 Basic Properties: $\mathit{3j}$ 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). …

3: 19.15 Advantages of Symmetry

§19.15 Advantages of Symmetry

… ►Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). … … ►Symmetry makes possible the reduction theorems of §19.29(i), permitting remarkable compression of tables of integrals while generalizing the interval of integration. … ►

4: Bille C. Carlson

… ►The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few. …This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. … ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. …In Permutation symmetry for theta functions (2011) he found an analogous hidden symmetry between theta functions. …

5: Peter A. Clarkson

… ►Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations. … Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. …

6: 18.6 Symmetry, Special Values, and Limits to Monomials

§18.6 Symmetry, Special Values, and Limits to Monomials

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§18.6(i) Symmetry and Special Values

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Table 18.6.1: Classical OP’s: symmetry and special values.

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$p_{n}(x)$	$p_{n}(-x)$	$p_{n}(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime}(0)$
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7: 7.4 Symmetry

§7.4 Symmetry

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\mathrm{g}\left(-z\right)=\sqrt{2}\sin\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^% {2}\right)-\mathrm{g}\left(z\right).

8: 14.31 Other Applications

… ►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). …

9: 21.3 Symmetry and Quasi-Periodicity

§21.3 Symmetry and Quasi-Periodicity

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§21.3(i) Riemann Theta Functions

… ►For Riemann theta functions with half-period characteristics, …

10: 4.3 Graphics

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See accompanying text — Figure 4.3.1: $\ln x$ and $e^{x}$ . Parallel tangent lines at $(1,0)$ and $(0,1)$ make evident the mirror symmetry across the line $y=x$ , demonstrating the inverse relationship between the two functions. Magnify

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