# structure relation

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## 1—10 of 12 matching pages

##### 1: 18.9 Recurrence Relations and Derivatives

##### 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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##### 4: 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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##### 5: 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. …##### 6: 15.17 Mathematical Applications

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

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►Both contributions concerned the electronic structure of molecules and solids.
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►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.
This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions.
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##### 8: Preface

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►A summary of the responsibilities of these groups may help in understanding the structure and results of this project.
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►Boisvert and Clark were responsible for advising and assisting in matters related to the use of information technology and applications of special functions in the physical sciences (and elsewhere); they also participated in the resolution of major administrative problems when they arose.
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##### 9: 18.38 Mathematical Applications

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►It has elegant structures, including $N$-soliton solutions, Lax pairs, and Bäcklund transformations.
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►Algebraic structures were built of which special representations involve Dunkl type operators.
In the $q$-case this algebraic structure is called the

*double affine Hecke algebra*(DAHA), introduced by Cherednik. …This gives also new structures and results in the one-variable case, but the obtained nonsymmetric special functions can now usually be written as a linear combination of two known special functions. …##### 10: Richard A. Askey

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►Over his career his primary research areas were in Special Functions and Orthogonal Polynomials, but also included other topics from Classical Analysis and related areas.
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►Askey was a member of the original editorial committee for the DLMF project, serving as an Associate Editor advising on all aspects of the project from the mid-1990’s to the mid-2010’s when the organizational structure of the DLMF project was reconstituted; see About the Project.