as eigenfunctions of a q-difference operator

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11: 28.12 Definitions and Basic Properties

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

§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 …

12: Bibliography K

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Lowering and Raising Operators for Some Special Orthogonal Polynomials. In Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, pp. 227–238.

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External Links:

FullText, MathReview Entry, ZentralBlatt

Referenced by:

§18.9(iii).

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

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Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

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External Links:

ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

Referenced by:

§18.14(i).

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Algorithm 910: a portable C++ multiple-precision system for special-function calculations. ACM Trans. Math. Software 37 (4), pp. Art. 45, 27.

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Notes:

This system provides a uniform interface in C++, named e_float, to perform arithmetic operations and to calculate mathematical functions that are implemented in several different MP packages. It is a free, open-source, multiple precision package that allows the computation of many special functions. It supports calculations with 30….300 decimal digits and is interoperable with Microsoft’s CLR, Python and Mathematica.

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ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

Erratum (V1.0.9) for References, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

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Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions. J. Math. Anal. Appl. 324 (1), pp. 285–303.

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External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§18.36(i).

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13: 29.3 Definitions and Basic Properties

… ►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. … ►If $\nu$ is a nonnegative integer, then … ►If $\nu$ is a nonnegative integer and $2p>\nu$, then (29.3.10) has only the solutions (29.3.9) with $2m>\nu$. … ►The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$, ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$. …

14: 3.7 Ordinary Differential Equations

… ►where $𝐀(\tau ,z)$ is the matrix … ►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 …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 …

15: 18.28 Askey–Wilson Class

… ►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). … ►where the operator $L$ is defined by … ►In Tsujimoto et al. (2012) an extension of the Bannai–Ito polynomials occurs as eigenfunctions of a Dunkl type operator. …

16: DLMF Project News

error generating summary

17: 30.9 Asymptotic Approximations and Expansions

… ►For the eigenfunctions see Meixner and Schäfke (1954, §3.251) and Müller (1963). … ►For the eigenfunctions see Meixner and Schäfke (1954, §3.252) and Müller (1962). … ►The asymptotic behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ and $a_{n,k}^m({\gamma }^2)$ as $n\to \mathrm{\infty }$ in descending powers of $2n+1$ is derived in Meixner (1944). …

18: Bibliography J

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Fonctions de Mathieu et fonctions propres de l’oscillateur relativiste. Ann. Fac. Sci. Toulouse Math. (6) 7 (3), pp. 465–495 (French).

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Notes:

Translated title: Mathieu functions and eigenfunctions of the relativistic oscillator.

External Links:

ISSN 0240-2963, Pdf, MathReview (Jean-Michel Ghez), ZentralBlatt

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7th item.

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A note on sampling expansion for a transform with parabolic cylinder kernel. Inform. Sci. 26 (2), pp. 155–158.

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ISSN 0020-0255, Document, MathReview, ZentralBlatt

Referenced by:

§12.16.

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REMES2 — a Fortran program to calculate rational minimax approximations to a given function. Technical Report Technical Report AECL-4210, Atomic Energy of Canada Limited. Chalk River Nuclear Laboratories, Chalk River, Ontario.

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Referenced by:

§3.11(iii).

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Introduction to Asymptotics: A Treatment Using Nonstandard Analysis. World Scientific Publishing Co. Inc., River Edge, NJ.

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External Links:

ISBN 981-02-2915-1, MathReview (W. A. J. Luxemburg), ZentralBlatt

Referenced by:

Ch.2.

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Operator Techniques in Atomic Spectroscopy. Princeton University Press, Princeton, NJ.

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External Links:

ISBN 0-691-05901-2

Referenced by:

§34.12, Profile Brian R. Judd.

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19: Bibliography V

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Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation

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External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, A. L. Van Buren and J. E. Boisvert (2002), A. L. Van Buren and J. E. Boisvert (2004), A. L. Van Buren and J. E. Boisvert (2007).

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Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1466-2, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§18.38(iii).

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Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions. Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1493-X, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§18.38(iii), Ch.35.

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Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1492-1, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§18.38(iii).

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

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20: 1.17 Integral and Series Representations of the Dirac Delta

… ►This is a symbolic function with the properties: …Such a sequence is called a delta sequence and we write, symbolically, … ►An example of a delta sequence is provided by … ►For a generalization of (1.17.14) see Maximon (1991). … ►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non-$L^2$ improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. …