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41: 17.17 Physical Applications

… ►See Kassel (1995). ►A substantial literature on $q$-deformed quantum-mechanical Schrödinger equations has developed recently. …

42: 30.10 Series and Integrals

… ►Integrals and integral equations for ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ are given in Arscott (1964b, §8.6), Erdélyi et al. (1955, §16.13), Flammer (1957, Chapter 5), and Meixner (1951). …

43: 31.18 Methods of Computation

§31.18 Methods of Computation

… ►The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.

44: Bibliography U

… ►

Solutions of the third Painlevé equation. I. Nagoya Math. J. 151, pp. 1–24.

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External Links:

ISSN 0027-7630, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

Referenced by:

§32.10(iii).

Encodings:

BibTeX

►

On the transformation group of the second Painlevé equation. Nagoya Math. J. 157, pp. 15–46.

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External Links:

ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.7(viii).

Encodings:

BibTeX

… ►

Theory of the Whittaker-Hill equation. Proc. Roy. Soc. Edinburgh Sect. A 69, pp. 28–44.

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External Links:

MathReview (R. K. Rathy), ZentralBlatt

Referenced by:

§28.31(i), §28.31(ii), §28.32(ii).

Encodings:

BibTeX

►

Integral equations for paraboloidal wave functions. I. Quart. J. Math. Oxford Ser. (2) 15, pp. 309–315.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.31(iii), §28.31(iii).

Encodings:

BibTeX

►

Integral equations for the paraboloidal wave functions. II. Quart. J. Math. Oxford Ser. (2) 16, pp. 257–262.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.31(iii), §28.31(iii).

Encodings:

BibTeX

…

45: 10.36 Other Differential Equations

§10.36 Other Differential Equations

… ►Differential equations for products can be obtained from (10.13.9)–(10.13.11) by replacing $z$ by $\mathrm{i}z$.

46: 28.7 Analytic Continuation of Eigenvalues

§28.7 Analytic Continuation of Eigenvalues

… ►The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). … ► … ►

28.7.1 $$\sum _{n=0}^{\mathrm{\infty }}\left(a_{2n}\left(q\right)-{(2n)}^2\right)=0,$$

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Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.7 and Ch.28

… ►

28.7.4 $$\sum _{n=0}^{\mathrm{\infty }}\left(b_{2n+2}\left(q\right)-{(2n+2)}^2\right)=0.$$

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Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.7 and Ch.28

47: 31.17 Physical Applications

§31.17 Physical Applications

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§31.17(i) Addition of Three Quantum Spins

… ►where $w(z)$ satisfies Heun’s equation (31.2.1) with $a$ as in (31.17.1) and the other parameters given by … ►For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. …

48: 34.12 Physical Applications

§34.12 Physical Applications

… ► $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).

49: 36.15 Methods of Computation

… ►

§36.15(v) Differential Equations

►For numerical solution of partial differential equations satisfied by the canonical integrals see Connor et al. (1983).

50: 15.17 Mathematical Applications

… ►

§15.17(i) Differential Equations

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … … ►The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. …These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …