# as eigenfunctions of a q-difference operator

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##### 1: 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 …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 … ►

28.30.4
$${f}_{m}=\frac{1}{2\pi}{\int}_{0}^{2\pi}f(x){w}_{m}(x)dx.$$

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##### 2: 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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##### 3: 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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##### 4: 18.39 Physical Applications

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►For a harmonic oscillator, the potential energy is given by
…The corresponding eigenfunctions are
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►The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials.
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►For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323).
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►For applications of Legendre polynomials in fluid dynamics to study the flow around the outside of a puff of hot gas rising through the air, see Paterson (1983).
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##### 5: 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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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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A proof of the asymptotic series for log $\mathrm{\Gamma}(z)$ and log $\mathrm{\Gamma}(z+a)$
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Ann. of Math. (2) 32 (1), pp. 10–16.
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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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The effective resistance and inductance of a concentric main, and methods of computing the $\mathrm{ber}$ and $\mathrm{bei}$ and allied functions.
Philos. Mag. (6) 17, pp. 524–552.
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##### 6: 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 $w(z+\pi )=-w(z)$. …##### 7: 28.12 Definitions and Basic Properties

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►As a function of $\nu $ with fixed $q$ ($\ne 0$), ${\lambda}_{\nu}\left(q\right)$ is discontinuous at $\nu =\pm 1,\pm 2,\mathrm{\dots}$.
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###### §28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu}(z,q)$

►Two eigenfunctions correspond to each eigenvalue $a={\lambda}_{\nu}\left(q\right)$. …The other eigenfunction is ${\mathrm{me}}_{\nu}(-z,q)$, a Floquet solution with respect to $-\nu $ with $a={\lambda}_{\nu}\left(q\right)$. If $q$ is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of $z$ and $q$ by the normalization …##### 8: 29.3 Definitions and Basic Properties

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►They are denoted by ${a}_{\nu}^{2m}\left({k}^{2}\right)$, ${a}_{\nu}^{2m+1}\left({k}^{2}\right)$, ${b}_{\nu}^{2m+1}\left({k}^{2}\right)$, ${b}_{\nu}^{2m+2}\left({k}^{2}\right)$, where $m=0,1,2,\mathrm{\dots}$; see Table 29.3.1.
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►If $\nu $ is a nonnegative integer, then
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►If $\nu $ is a nonnegative integer and $2p>\nu $, then (29.3.10) has only the solutions (29.3.9) with $2m>\nu $.
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►The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by ${\mathit{Ec}}_{\nu}^{2m}(z,{k}^{2})$, ${\mathit{Ec}}_{\nu}^{2m+1}(z,{k}^{2})$, ${\mathit{Es}}_{\nu}^{2m+1}(z,{k}^{2})$, ${\mathit{Es}}_{\nu}^{2m+2}(z,{k}^{2})$.
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##### 9: 3.7 Ordinary Differential Equations

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►Let $(a,b)$ be a finite or infinite interval and $q(x)$ be a real-valued continuous (or piecewise continuous) function on the closure of $(a,b)$.
The

*Sturm–Liouville eigenvalue problem*is the construction of a nontrivial solution of the system …with limits taken in (3.7.16) when $a$ or $b$, or both, are infinite. The values ${\lambda}_{k}$ are the*eigenvalues*and the corresponding solutions ${w}_{k}$ of the differential equation are the*eigenfunctions*. The eigenvalues ${\lambda}_{k}$ are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy …##### 10: 18.28 Askey–Wilson Class

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►The Askey–Wilson polynomials form a system of OP’s $\{{p}_{n}(x)\}$, $n=0,1,2,\mathrm{\dots}$, that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set.
…$y$) such that ${P}_{n}(z)={p}_{n}(\frac{1}{2}(z+{z}^{-1}))$ in the Askey–Wilson case, and ${P}_{n}(y)={p}_{n}({q}^{-y}+c{q}^{y+1})$ in the $q$-Racah case, and both are eigenfunctions of a second-order $q$-difference operator similar to (18.27.1).
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►Assume $a,b,c,d$ are all real, or two of them are real and two form a conjugate pair, or none of them are real but they form two conjugate pairs.
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►Also, if $$, then (18.28.10) holds with $a,b$ interchanged.
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