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1: Bibliography M

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Hill’s Equation. Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.

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Notes:

Reprinted by Dover Publications, Inc., New York, 1979.

External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§1.3(iii), §28.29(i), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii), §28.30(i), Ch.28, §29.3(i), §29.9.

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On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

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ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

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Quadratic relations for confluent hypergeometric functions. Tohoku Math. J. (2) 52 (4), pp. 489–513.

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ISSN 0040-8735, Document, MathReview (Takeshi Sasaki), ZentralBlatt

Referenced by:

§13.12.

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An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.

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ISBN 0-471-58884-9, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii).

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

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2: Bibliography V

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Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

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ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8, §29.8.

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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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ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

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Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. J. Comput. Appl. Math. 213 (2), pp. 488–500.

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ISSN 0377-0427, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

item (f), §28.34(ii), §29.20(i).

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Asymptotic expansion of the generalized hypergeometric function ${}_p{}^{}F_q^{}(z)$ as $z\to \mathrm{\infty }$ for . Anal. Appl. (Singap.) 21 (2), pp. 535–545.

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ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§16.11(i), §16.11, Erratum (V1.1.9) for Section 16.11(i).

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On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).

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Notes:

In Russian. English translation: Differential Equations 1(1), pp. 58–59.

External Links:

MathReview, ZentralBlatt

Referenced by:

§32.8(ii).

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3: Bibliography W

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

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Asymptotic Expansions for Ordinary Differential Equations. Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney.

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Notes:

Reprinted by Krieger, New York, 1976 and Dover, New York, 1987.

External Links:

MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

Ch.2, Ch.9.

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Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules. Ann. Inst. Statist. Math. 45 (3), pp. 467–475.

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ISSN 0020-3157, Document, MathReview (N. K. Thakare), ZentralBlatt

Referenced by:

§35.9.

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

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Asymptotic solutions of a fourth order differential equation. Stud. Appl. Math. 118 (2), pp. 133–152.

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ISSN 0022-2526, Document, MathReview Entry

Referenced by:

§2.8(vi).

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4: 20.11 Generalizations and Analogs

… ►However, in this case $q$ is no longer regarded as an independent complex variable within the unit circle, because $k$ is related to the variable $\tau =\tau (k)$ of the theta functions via (20.9.2). … ►The first of equations (20.9.2) can also be written …For applications to rapidly convergent expansions for $\pi$ see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). … ►The importance of these combined theta functions is that sets of twelve equations for the theta functions often can be replaced by corresponding sets of three equations of the combined theta functions, plus permutation symmetry. Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

5: Bibliography K

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

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On differential equations for Sobolev-type Laguerre polynomials. Trans. Amer. Math. Soc. 350 (1), pp. 347–393.

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ISSN 0002-9947, FullText, MathReview (Hans W. Volkmer), ZentralBlatt

Referenced by:

§18.36(ii).

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An algorithm for solving second order linear homogeneous differential equations. J. Symbolic Comput. 2 (1), pp. 3–43.

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ISSN 0747-7171, MathReview, ZentralBlatt

Referenced by:

§31.14(ii).

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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

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Code, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

1st item.

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The Korteweg-de Vries Equation and Related Evolution Equations. In Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), A. C. Newell (Ed.), Lectures in Appl. Math., Vol. 15, pp. 61–83.

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External Links:

MathReview (G. L. Lamb, Jr.), ZentralBlatt

Referenced by:

§22.19(iii).

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6: Bibliography D

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Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London Ser. A 440, pp. 37–54.

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ISSN 0962-8444, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§10.41(iii), §2.8(i).

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Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.

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External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§2.8(vi), §8.12, §8.12.

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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.

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ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§10.20(ii), §2.8(i).

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Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions. Stud. Appl. Math. 113 (3), pp. 245–270.

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ISSN 0022-2526, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§14.15(iii), §2.8(i).

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Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.

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ISSN 0219-5305, Document, Link, MathReview (Thomas Mbah Acho), ZentralBlatt

Referenced by:

5th item, §14.32.

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7: Bibliography

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Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1), pp. 95–148.

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External Links:

MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§23.21(ii).

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Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt

Referenced by:

(25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.

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Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§29.8.

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Periodic Differential Equations. An Introduction to Mathieu, Lamé, and Allied Functions. International Series of Monographs in Pure and Applied Mathematics, Vol. 66, Pergamon Press, The Macmillan Co., New York.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§28.1, §28.1, §28.10(i), §28.11, §28.12(ii), §28.17, §28.2(ii), §28.2(iii), §28.2(iv), §28.28(i), §28.28(ii), §28.29(i), §28.32(i), §28.32(ii), §28.4(i), §28.5(i), Ch.28, §29.1, §29.11, §29.12(i), §29.12(ii), §29.12(iii), §29.14, §29.15(ii), §29.15(ii), §29.17(i), §29.18(iii), Ch.29, §30.1, §30.10, §30.11(i), §30.11(vi), §30.2(ii), §30.4(iii), §30.6, §30.8(i), §30.8(ii), Ch.30, §31.4, §31.5, §31.9(ii), Notations F, Notations G, Notations M, Notations N.

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Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-412-5, MathReview (Michael Hanke), ZentralBlatt

Referenced by:

§3.7(i).

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8: Bibliography N

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Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§28.1, §28.26(i), 6th item, Notations S, Notations S.

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Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.

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ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§13.28(i).

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BibTeX

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Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

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Referenced by:

§2.8(vi).

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A separating surface for the Painlevé differential equation $x^{\prime \prime }=x^2-t$ . J. Math. Anal. Appl. 193 (3), pp. 817–831.

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ISSN 0022-247X, Document, MathReview (W. S. Loud), ZentralBlatt

Referenced by:

§32.17.

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Tables of Generalized Airy Functions for the Asymptotic Solution of the Differential Equations $ϵ{(p{y}^{\prime })}^{\prime }+(q+ϵr)y=f$ . Pergamon Press, Oxford.

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Notes:

D. E. Brown, translator.

External Links:

MathReview, ZentralBlatt

Referenced by:

4th item.

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9: Bibliography C

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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.

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ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

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Elementary Differential Equations. Clarendon Press, Oxford.

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External Links:

ISBN 0-19-853139-7, MathReview, ZentralBlatt

Referenced by:

§16.12.

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Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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External Links:

MathReview (M. Zlámal)

Referenced by:

§1.18(ix), §1.18(x).

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:

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

Referenced by:

§13.29(v), §15.19(v).

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Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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Document

Referenced by:

5th item.

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10: Bibliography S

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Algorithm 721: MTIEU1 and MTIEU2: Two subroutines to compute eigenvalues and solutions to Mathieu’s differential equation for noninteger and integer order. ACM Trans. Math. Software 19 (3), pp. 391–406.

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Notes:

Double-precision Fortran, maximum accuracy 18D

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ISSN 0098-3500, Document, GAMS (module 721; package TOMS), ZentralBlatt

Referenced by:

3rd item, 5th item.

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

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Relations among the fundamental solutions of the generalized hypergeometric equation when $p=q+1$. II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.

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External Links:

Document, MathReview (C. S. Meijer), ZentralBlatt

Referenced by:

§16.8(ii).

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Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§13.23(iv), Ch.14, §31.3(iii).

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The relation between asymptotic properties of the second Painlevé equation in different directions towards infinity. Differ. Uravn. 23 (5), pp. 834–842 (Russian).

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Notes:

In Russian. English translation: Differential Equations 23(1987), no. 5, pp. 569–576.

External Links:

ISSN 0374-0641, MathReview (T. Chanturiya), ZentralBlatt

Referenced by:

§32.11(ii).

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