structure relation

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1: 18.9 Recurrence Relations and Derivatives

… ►and the structure relation …

2: 18.2 General Orthogonal Polynomials

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Degree lowering and raising differentiation formulas and structure relations

… ►Then the OP’s are called semi-classical and (18.2.44) is called a structure relation. …

3: Bibliography K

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The structure relation for Askey-Wilson polynomials. J. Comput. Appl. Math. 207 (2), pp. 214–226.

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External Links:

ISSN 0377-0427, Document, Link, MathReview Entry

Referenced by:

(18.9.18_5).

Encodings:

BibTeX

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4: 37.3 Triangular Region with Weight Function

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37.3.32

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Symbols:

$k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $y$: real variable, $j$: integer ($\in \{-1,0,1\}$), $a_{k,n,j}$: coefficient, $b_{k,n,j}$: coefficient, $c_{k,n,j}$: coefficient, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proof sketch:

The structure of this relation follows from (37.2.19) and (37.3.4). For the explicit values of the coefficients see Dunkl and Xu (2014, §3.3.4), where one has to replace by , and where one has to pass from orthonormal polynomials to polynomials (37.3.3) by using (37.3.4).

Referenced by:

§37.3(vi)

Permalink:

http://dlmf.nist.gov/37.3.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §37.3(vi), §37.3, Part 37.II and Ch.37

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5: Bibliography S

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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External Links:

ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

Encodings:

BibTeX

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6: null

error generating summary

7: 33.22 Particle Scattering and Atomic and Molecular Spectra

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§33.22(i) Schrödinger Equation

►With $\mathrm{e}$ denoting here the elementary charge, the Coulomb potential between two point particles with charges $Z_1\mathrm{e},Z_2\mathrm{e}$ and masses $m_1,m_2$ separated by a distance $s$ is $V(s)=Z_1{Z}_2{\mathrm{e}}^2/(4\pi {\epsilon }_{0}s)=Z_1{Z}_2\alpha \mathrm{\hslash }c/s$, where $Z_j$ are atomic numbers, ${\epsilon }_0$ is the electric constant, $\alpha$ is the fine structure constant, and $\mathrm{\hslash }$ is the reduced Planck’s constant. … ► $R_{\mathrm{\infty }}=m_{\mathrm{e}}c{\alpha }^2/(2\mathrm{\hslash })$. … ►

§33.22(iv) Klein–Gordon and Dirac Equations

… ►The motion of a relativistic electron in a Coulomb field, which arises in the theory of the electronic structure of heavy elements (Johnson (2007)), is described by a Dirac equation. …

8: 37.20 Mathematical Applications

… ►For regular domains, such as square, sphere, ball, simplex, and conic domains, they are used to study convolution structure, maximal functions, and interpolation spaces, as well as localized kernel and localized frames. … ►Hermite polynomials on ${ℝ}^d$ (see §37.17) are closely related to the $d$-dimensional harmonic oscillator, see for instance Aquilanti et al. (1997). … ►The nodes of these cubature rules are closely related to common zeros of OPs and they are often good points for polynomial interpolation. …

9: 15.17 Mathematical Applications

… ►This topic is treated in §§15.10 and 15.11. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … ►The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. …

10: Bibliography P

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Optical properties of layer antiferromagnets with ${\mathrm{K}}_2{\mathrm{NiF}}_4$ structure. J. Phys. C: Solid State Physics 2 (11), pp. 2012–2021.

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External Links:

Document

Referenced by:

§19.35(ii).

Encodings:

BibTeX

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Several new product identities in relation to Rogers-Ramanujan type sums and mock theta functions. Res. Math. Sci. 8 (1), pp. Paper No. 16, 58.

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External Links:

ISSN 2522-0144, Document, Link, MathReview (Jeremy Lovejoy)

Referenced by:

§17.8.

Encodings:

BibTeX

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The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic. Philos. Mag. (7) 37, pp. 311–317.

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

§36.2(ii).

Encodings:

BibTeX

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