# as eigenfunctions of a q-difference operator

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##### 1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

###### §1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► … ►###### §1.18(v) Point Spectra and Eigenfunction Expansions

… ►###### §1.18(vi) Continuous Spectra and Eigenfunction Expansions: Simple Cases

… ►Note that eigenfunctions for distinct (necessarily real) eigenvalues of a self-adjoint operator are mutually orthogonal. …##### 2: 28.30 Expansions in Series of Eigenfunctions

###### §28.30 Expansions in Series of Eigenfunctions

►###### §28.30(i) Real Variable

►Let ${\widehat{\lambda}}_{m}$, $m=0,1,2,\mathrm{\dots}$, be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let ${w}_{m}(x)$, $m=0,1,2,\mathrm{\dots}$, be the*eigenfunctions*, that is, an orthonormal set of $2\pi $-periodic solutions; thus … ►

28.30.2
$$\frac{1}{2\pi}{\int}_{0}^{2\pi}{w}_{m}(x){w}_{n}(x)dx={\delta}_{m,n}.$$

►Then every continuous $2\pi $-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi ]$ can be expanded in a uniformly and absolutely convergent series
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##### 3: 31.17 Physical Applications

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►We use vector notation $[\mathbf{s},\mathbf{t},\mathbf{u}]$ (respective scalar $(s,t,u)$) for any one of the three spin operators (respective spin values).
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►for the common eigenfunction
$\mathrm{\Psi}(\mathbf{x})=\mathrm{\Psi}({x}_{s},{x}_{t},{x}_{u})$, where $a$ is the coupling parameter of interacting spins.
…The operators
${\mathbf{J}}^{2}$ and ${\mathit{H}}_{s}$ admit separation of variables in ${z}_{1},{z}_{2}$, leading to the following factorization of the eigenfunction
$\mathrm{\Psi}(\mathbf{x})$:
…where $w(z)$ satisfies Heun’s equation (31.2.1) with $a$ as in (31.17.1) and the other parameters given by
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►For application of biconfluent Heun functions in a model of an equatorially trapped Rossby wave in a shear flow in the ocean or atmosphere see Boyd and Natarov (1998).
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##### 4: Howard S. Cohl

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► 1968 in Paterson, New Jersey) is a Mathematician in the Applied and Computational Mathematics Division at the National Institute of Standards and Technology in Gaithersburg, Maryland.
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►He obtained a B.
… in astronomy and astrophysics from Indiana University, Bloomington, Indiana, a M.
… in physics from Louisiana State University, Baton Rouge, Louisiana, and a Ph.
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►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series.
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##### 5: 18.39 Applications in the Physical Sciences

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►The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of self-adjoint second order differential operators is overviewed in §1.18.
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►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being ${L}^{2}$ and forming a complete set.
Also presented are the analytic solutions for the ${L}^{2}$, bound state, eigenfunctions and eigenvalues of the Morse oscillator which also has analytically known non-normalizable continuum eigenfunctions, thus providing an example of a mixed spectrum.
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►The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV).
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►This is illustrated in Figure 18.39.1 where the first and fourth excited state eigenfunctions of the Schrödinger operator with the rationally extended harmonic potential, of (18.39.19), are shown, and compared with the first and fourth excited states of the harmonic oscillator eigenfunctions of (18.39.14) of paragraph a), above.
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##### 6: 18.38 Mathematical Applications

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►Define a further operator
${K}_{2}$ by
…A further operator, the so-called

*Casimir operator*… ►Eigenvalue equations involving Dunkl type operators have as eigenfunctions nonsymmetric analogues of multivariable special functions associated with root systems. … ► … ►The Dunkl type operator is a $q$-difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial ${R}_{n}(z;a,b,c,d|q)$ and the ‘anti-symmetric’ Laurent polynomial ${z}^{-1}(1-az)(1-bz){R}_{n-1}(z;qa,qb,c,d|q)$, where ${R}_{n}(z)$ is given in (18.28.1_5). …##### 7: Bibliography T

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Eigenfunction Expansions Associated with Second-Order Differential Equations.
Clarendon Press, Oxford.
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Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations.
Clarendon Press, Oxford.
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Eigenfunction expansions associated with second-order differential equations. Part I.
Second edition, Clarendon Press, Oxford.
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Dunkl shift operators and Bannai-Ito polynomials.
Adv. Math. 229 (4), pp. 2123–2158.
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Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades.
J. Appl. Math. Mech. 23, pp. 1549–1565.
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##### 8: Bibliography R

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Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric.
IEEE Trans. Antennas and Propagation 39 (2), pp. 218–223.
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Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators.
Academic Press, New York.
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On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators.
Z. Angew. Math. Mech. 78 (1), pp. 66–72.
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On the foundations of combinatorial theory. VIII. Finite operator calculus.
J. Math. Anal. Appl. 42, pp. 684–760.
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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media.
In Differential Operators and Related Topics, Vol. I (Odessa,
1997),
Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
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##### 9: 1.13 Differential Equations

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###### §1.13(ii) Equations with a Parameter

… ►###### §1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

… ►###### Eigenvalues and Eigenfunctions

… ►The functions $u(x)$ which correspond to these being*eigenfunctions*. … ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, $\lambda $; (ii) the corresponding (real) eigenfunctions, $u(x)$ and $w(t)$, have the same number of zeros, also called*nodes*, for $t\in (0,c)$ as for $x\in (a,b)$; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …##### 10: 28.2 Definitions and Basic Properties

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►The Fourier series of a Floquet solution
…leads to a Floquet solution.
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###### §28.2(vi) Eigenfunctions

►Table 28.2.2 gives the notation for the eigenfunctions corresponding to the eigenvalues in Table 28.2.1.*Period*$\pi $ means that the eigenfunction has the property $w(z+\pi )=w(z)$, whereas*antiperiod*$\pi $ means that $w(z+\pi )=-w(z)$. …