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  • H. Volkmer and J. J. Wood (2014) A note on the asymptotic expansion of generalized hypergeometric functions. Anal. Appl. (Singap.) 12 (1), pp. 107–115.
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    External Links:
    ISSN 0219-5305, Document, MathReview (Roberto S. Costas-Santos), ZentralBlatt
    Referenced by:
    §16.11(ii), §16.11(ii).
    Encodings:
    BibTeX
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  • H. Volkmer (2004a) 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).
    Encodings:
    BibTeX
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  • H. Volkmer (2008) 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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    External Links:
    ISSN 0377-0427, Document, FullText, MathReview (Javier Segura), ZentralBlatt
    Referenced by:
    item (f), §28.34(ii), §29.20(i).
    Encodings:
    BibTeX
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  • H. Volkmer (2023) Asymptotic expansion of the generalized hypergeometric function F q p ⁒ ( z ) as z → ∞ for p < q . Anal. Appl. (Singap.) 21 (2), pp. 535–545.
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    External Links:
    ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt
    Referenced by:
    §16.11(i), §16.11, Erratum (V1.1.9) for Section 16.11(i).
    Encodings:
    BibTeX
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  • A. P. Vorob’ev (1965) 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).
    Encodings:
    BibTeX
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    2: Bibliography C
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  • T. W. Chaundy (1969) Elementary Differential Equations. Clarendon Press, Oxford.
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    External Links:
    ISBN 0-19-853139-7, MathReview, ZentralBlatt
    Referenced by:
    §16.12.
    Encodings:
    BibTeX
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  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
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    External Links:
    ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
    Referenced by:
    §26.8(vii).
    Encodings:
    BibTeX
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  • E. A. Coddington and N. Levinson (1955) 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).
    Encodings:
    BibTeX
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  • M. Colman, A. Cuyt, and J. Van Deun (2011) 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).
    Encodings:
    BibTeX
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  • M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.
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    External Links:
    Document
    Referenced by:
    5th item.
    Encodings:
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    3: Bibliography W
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  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
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    External Links:
    ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
    Referenced by:
    §29.19(ii).
    Encodings:
    BibTeX
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  • W. Wasow (1965) 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.
    Encodings:
    BibTeX
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  • F. J. W. Whipple (1927) Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.
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    External Links:
    Document, ZentralBlatt
    Referenced by:
    §16.6.
    Encodings:
    BibTeX
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  • R. Wong and H. Y. Zhang (2007) Asymptotic solutions of a fourth order differential equation. Stud. Appl. Math. 118 (2), pp. 133–152.
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    External Links:
    ISSN 0022-2526, Document, MathReview Entry
    Referenced by:
    §2.8(vi).
    Encodings:
    BibTeX
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  • E. M. Wright (1940a) The asymptotic expansion of the generalized hypergeometric function. Proc. London Math. Soc. (2) 46, pp. 389–408.
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    External Links:
    ISSN 0024-6115, Document, MathReview (M. C. Gray), ZentralBlatt
    Referenced by:
    §16.11(ii).
    Encodings:
    BibTeX
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    4: Bibliography K
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  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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    External Links:
    Document, MathReview (Matti Jutila), ZentralBlatt
    Referenced by:
    §25.18(i).
    Encodings:
    BibTeX
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  • H. Ki and Y. Kim (2000) On the zeros of some generalized hypergeometric functions. J. Math. Anal. Appl. 243 (2), pp. 249–260.
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    External Links:
    ISSN 0022-247X, Document, MathReview (M. K. Jain), ZentralBlatt
    Referenced by:
    §16.9.
    Encodings:
    BibTeX
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  • A. V. Kitaev and A. H. Vartanian (2004) 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).
    Encodings:
    BibTeX
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  • U. J. Knottnerus (1960) Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters. J. B. Wolters, Groningen.
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    External Links:
    MathReview (L. J. Slater), ZentralBlatt
    Referenced by:
    §16.11(iii).
    Encodings:
    BibTeX
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  • C. Krattenthaler (1993) 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.
    External Links:
    Code, Document, MathReview (G. Gasper), ZentralBlatt
    Referenced by:
    1st item.
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    5: Bibliography D
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  • A. Debosscher (1998) Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.
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    External Links:
    ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt
    Referenced by:
    §31.17(ii).
    Encodings:
    BibTeX
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  • K. Dekker and J. G. Verwer (1984) Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, Vol. 2, North-Holland Publishing Co., Amsterdam.
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    External Links:
    ISBN 0-444-87634-0, MathReview (A. de Castro Brzezicki), ZentralBlatt
    Referenced by:
    §3.7(v).
    Encodings:
    BibTeX
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  • T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.
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    Notes:
    (This reference has several typographical errors.)
    External Links:
    ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt
    Referenced by:
    §13.21(ii), §13.21(iii), §13.21(iv).
    Encodings:
    BibTeX
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  • T. M. Dunster (1996c) Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.
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    External Links:
    ISSN 1073-2772, Pdf, MathReview (Roderick Wong), ZentralBlatt
    Referenced by:
    §2.11(v).
    Encodings:
    BibTeX
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  • T. M. Dunster (2001a) 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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    External Links:
    ISSN 0022-2526, Document, MathReview, ZentralBlatt
    Referenced by:
    §10.20(ii), §2.8(i).
    Encodings:
    BibTeX
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    6: Bibliography M
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  • A. P. Magnus (1995) Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. J. Comput. Appl. Math. 57 (1-2), pp. 215–237.
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    External Links:
    ISSN 0377-0427, Document, MathReview (Leonid B. GolinskiΔ­), ZentralBlatt
    Referenced by:
    §32.15.
    Encodings:
    BibTeX
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  • W. Magnus and S. Winkler (1966) 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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  • R. S. Maier (2005) On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.
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    External Links:
    ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt
    Referenced by:
    §31.7(i).
    Encodings:
    BibTeX
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  • A. R. Miller (1997) A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.
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    External Links:
    ISSN 0377-0427, Document, MathReview, ZentralBlatt
    Referenced by:
    §16.10.
    Encodings:
    BibTeX
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  • A. R. Miller (2003) On a Kummer-type transformation for the generalized hypergeometric function F 2 2 . J. Comput. Appl. Math. 157 (2), pp. 507–509.
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    External Links:
    ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
    Referenced by:
    §16.6.
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    BibTeX
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    7: Bibliography B
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  • W. N. Bailey (1928) Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.
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    External Links:
    Document, ZentralBlatt
    Referenced by:
    §16.12.
    Encodings:
    BibTeX
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  • W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.
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    External Links:
    Document, ZentralBlatt
    Referenced by:
    §16.6.
    Encodings:
    BibTeX
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  • W. N. Bailey (1964) Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.
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    External Links:
    MathReview
    Referenced by:
    §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §17.1, §17.4(i), Ch.17.
    Encodings:
    BibTeX
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  • S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
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    External Links:
    MathReview (S. C. van Veen), ZentralBlatt
    Referenced by:
    §35.5(i).
    Encodings:
    BibTeX
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  • J. C. Butcher (1987) The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods. John Wiley & Sons Ltd., Chichester.
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    External Links:
    ISBN 0-471-91046-5, MathReview (Peter Alfeld), ZentralBlatt
    Referenced by:
    §3.7(v).
    Encodings:
    BibTeX
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    8: Bibliography N
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  • J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.
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    External Links:
    ISSN 0022-2488, Document, MathReview, ZentralBlatt
    Referenced by:
    §13.28(i).
    Encodings:
    BibTeX
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  • J. J. Nestor (1984) 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).
    Encodings:
    BibTeX
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  • V. A. Noonburg (1995) A separating surface for the Painlevé differential equation x ′′ = x 2 − t . J. Math. Anal. Appl. 193 (3), pp. 817–831.
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    External Links:
    ISSN 0022-247X, Document, MathReview (W. S. Loud), ZentralBlatt
    Referenced by:
    §32.17.
    Encodings:
    BibTeX
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  • L. N. Nosova and S. A. Tumarkin (1965) Tables of Generalized Airy Functions for the Asymptotic Solution of the Differential Equations Ο΅ ⁒ ( p ⁒ y ′ ) ′ + ( q + Ο΅ ⁒ r ) ⁒ y = f . Pergamon Press, Oxford.
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    Notes:
    D. E. Brown, translator.
    External Links:
    MathReview, ZentralBlatt
    Referenced by:
    4th item.
    Encodings:
    BibTeX
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  • V. Yu. Novokshënov (1985) The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).
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    Notes:
    In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.
    External Links:
    ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt
    Referenced by:
    §32.11(iv).
    Encodings:
    BibTeX
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    9: Bibliography S
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  • L. J. Slater (1966) Generalized Hypergeometric Functions. Cambridge University Press, Cambridge.
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    External Links:
    ISBN 9780521090612, MathReview (S. C. van Veen), ZentralBlatt
    Referenced by:
    Ch.15, §16.2(iii), §16.4(ii), §16.4(iii), §16.4(v), Ch.16, §17.1, §17.4(i), Ch.17.
    Encodings:
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  • F. C. Smith (1939a) On the logarithmic solutions of the generalized hypergeometric equation when p = q + 1 . Bull. Amer. Math. Soc. 45 (8), pp. 629–636.
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    External Links:
    Document, MathReview (H. Bateman), ZentralBlatt
    Referenced by:
    §16.8(ii).
    Encodings:
    BibTeX
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  • F. C. Smith (1939b) 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).
    Encodings:
    BibTeX
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  • K. Srinivasa Rao (1981) Computation of angular momentum coefficients using sets of generalized hypergeometric functions. Comput. Phys. Comm. 22 (2-3), pp. 297–302.
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    External Links:
    Document
    Referenced by:
    §34.13.
    Encodings:
    BibTeX
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  • F. Stenger (1966b) Error bounds for asymptotic solutions of differential equations. II. The general case. J. Res. Nat. Bur. Standards Sect. B 70B, pp. 187–210.
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    External Links:
    MathReview (F. W. J. Olver), ZentralBlatt
    Referenced by:
    §2.7(ii).
    Encodings:
    BibTeX
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    10: Bibliography
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  • A. S. Abdullaev (1985) Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation. Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).
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    Notes:
    English translation: Soviet Math. Dokl. 31(1985), no. 1, pp. 45–47
    External Links:
    ISSN 0002-3264, MathReview (M. S. Agranovich), ZentralBlatt
    Referenced by:
    §32.11(iv).
    Encodings:
    BibTeX
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  • S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.
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    External Links:
    ISSN 0024-1318, MathReview (James M. Horner)
    Referenced by:
    §13.9(i).
    Encodings:
    BibTeX
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  • G. D. Anderson, S.-L. Qiu, M. K. Vamanamurthy, and M. Vuorinen (2000) Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.
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    External Links:
    ISSN 0030-8730, FullText, MathReview (Bruce C. Berndt), ZentralBlatt
    Referenced by:
    §19.35(i).
    Encodings:
    BibTeX
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  • U. M. Ascher and L. R. Petzold (1998) Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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    ISBN 0-89871-412-5, MathReview (Michael Hanke), ZentralBlatt
    Referenced by:
    §3.7(i).
    Encodings:
    BibTeX
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  • R. Askey and J. Wilson (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
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