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11: 12.16 Mathematical Applications

… ►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). …

12: Bibliography H

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Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 14, Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-60452-9, MathReview, ZentralBlatt

Referenced by:

§3.7(v).

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13: 21.7 Riemann Surfaces

… ►Equation (21.7.1) determines a plane algebraic curve in ${ℂ}^2$, which is made compact by adding its points at infinity. …

14: 31.8 Solutions via Quadratures

… ►For $𝐦=(m_0,0,0,0)$, these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. …

15: 18.38 Mathematical Applications

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Zhedanov Algebra

… ►See Zhedanov (1991), Granovskiĭ et al. (1992, §3), Koornwinder (2007a, §2) and Terwilliger (2011). Similar algebras can be associated with all families of OP’s in the $q$-Askey scheme and the Askey scheme. … ►Algebraic structures were built of which special representations involve Dunkl type operators. In the $q$-case this algebraic structure is called the double affine Hecke algebra (DAHA), introduced by Cherednik. …

16: Bibliography K

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An algebraic-geometrical construction of the Zakharov-Shabat equations and their periodic solutions. Sov. Math. Doklady 17, pp. 394–397.

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External Links:

ZentralBlatt

Referenced by:

Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2.

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17: Bibliography M

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On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46 (1), pp. 121–171.

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External Links:

ISSN 0532-8721, FullText, MathReview (Andrew Pickering), ZentralBlatt

Referenced by:

§32.9(iii).

Encodings:

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18: 32.9 Other Elementary Solutions

… ►Then ${\text{P}}_{\text{III}}$ has algebraic solutions iff … ► ${\text{P}}_{\text{V}}$, with $\delta =0$, has algebraic solutions if either … ►An elementary algebraic solution of ${\text{P}}_{\text{VI}}$ is … ►For further examples of algebraic solutions see Andreev and Kitaev (2002), Boalch (2005, 2006), Gromak et al. (2002, §48), Hitchin (2003), Masuda (2003), and Mazzocco (2001b).

19: Bibliography B

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The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. Dover Publications, New York.

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Notes:

Two volumes. A reproduction of the seventh edition [vol. 1, Longmans, Green, London 1912; vol. 2, Longmans, Green, London 1928].

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.11(i), §1.11(ii), §1.11(iv).

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20: Bibliography E

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On algebraic Lamé functions. Philos. Mag. (7) 32, pp. 348–350.

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External Links:

MathReview (G. Szegő), ZentralBlatt

Referenced by:

§29.17(ii).

Encodings:

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Integral equations for Heun functions. Quart. J. Math., Oxford Ser. 13, pp. 107–112.

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External Links:

Document, MathReview (E. Rothe), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

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The Fuchsian equation of second order with four singularities. Duke Math. J. 9 (1), pp. 48–58.

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External Links:

Document, MathReview (R. E. Langer), ZentralBlatt

Referenced by:

§31.11(i).

Encodings:

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Certain expansions of solutions of the Heun equation. Quart. J. Math., Oxford Ser. 15, pp. 62–69.

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External Links:

Document, MathReview (R. E. Langer), ZentralBlatt

Referenced by:

§31.11(i), §31.11(i), §31.9(i).

Encodings:

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A catalogue of Sturm-Liouville differential equations. In Sturm-Liouville theory, pp. 271–331.

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External Links:

Document, Link, MathReview Entry

Referenced by:

§1.18(ix), §1.18(x).

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