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1: 20.11 Generalizations and Analogs

§20.11 Generalizations and Analogs

… ►It is a discrete analog of theta functions. If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): …This is the discrete analog of the Poisson identity (§1.8(iv)). …

2: 18.39 Applications in the Physical Sciences

… ►The properties of $V(x)$ determine whether the spectrum, this being the set of eigenvalues of $\mathscr{H}$, is discrete, continuous, or mixed, see §1.18. … ►However, in the remainder of this section will will assume that the spectrum is discrete, and that the eigenfunctions of $\mathscr{H}$ form a discrete, normed, and complete basis for a Hilbert space. … ►The spectrum is entirely discrete as in §1.18(v). … ►The spectrum is entirely discrete as in §1.18(v). … ►In the attractive case (18.35.6_4) for the discrete parts of the weight function where with , are also simplified: …

3: Alexander I. Bobenko

… ► Eitner), published by Springer in 2000, and Discrete Differential Geometry: Integrable Structure (with Y. …He is also coeditor of Discrete Integrable Geometry and Physics (with R.  Seiler), published by Oxford University Press in 1999, and Discrete Differential Geometry (with P. …

4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► The analogous orthonormality is … ►

§1.18(v) Point Spectra and Eigenfunction Expansions

… ► … ►The analogs of (1.18.49)–(1.18.52) may be written in a similar fashion each now including contributions from both the discrete and continuous parts of the spectrum, as in (1.18.65). … ► …

5: Vadim B. Kuznetsov

… ►Kuznetsov published papers on special functions and orthogonal polynomials, the quantum scattering method, integrable discrete many-body systems, separation of variables, Bäcklund transformation techniques, and integrability in classical and quantum mechanics. …

6: Bibliography F

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

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Singularity analysis of generating functions. SIAM J. Discrete Math. 3 (2), pp. 216–240.

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External Links:

ISSN 0895-4801, Document, MathReview (E. Rodney Canfield), ZentralBlatt

Referenced by:

§2.10(iv).

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From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

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External Links:

ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§32.7(ii).

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Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142 (2), pp. 313–344.

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External Links:

ISSN 0010-3616, Link, MathReview (Nikolai A. Kostov), ZentralBlatt

Referenced by:

§32.15.

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Continuous and Discrete Painlevé Equations. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 33–47.

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Notes:

Proc. NATO Adv. Res. Workshop, Sainte-Adèle, Canada, 1990

External Links:

MathReview (F. Pempinelli), ZentralBlatt

Referenced by:

§32.15.

Encodings:

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7: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

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Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.

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External Links:

ISSN 1452-8630, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§5.11(ii), §5.11(ii).

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

Encodings:

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Affine Weyl groups, discrete dynamical systems and Painlevé equations. Comm. Math. Phys. 199 (2), pp. 281–295.

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External Links:

ISSN 0010-3616, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

Referenced by:

§32.2(v).

Encodings:

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8: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

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9: 26.22 Software

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GAP (website). A system for computational discrete algebra.

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10: 18.38 Mathematical Applications

… ►

Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … ►Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. …