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1: Bibliography D

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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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ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt

Referenced by:

§31.17(ii).

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Formes canoniques des équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.

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External Links:

MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§31.12.

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Bernoulli Numbers and Confluent Hypergeometric Functions. In Number Theory for the Millennium, I (Urbana, IL, 2000), pp. 343–363.

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ISBN 1-56881-126-8, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.16(iii).

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

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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.)

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ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

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

Referenced by:

§28.26(ii).

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

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

Referenced by:

§19.37(iv).

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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).

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Hypergeometric functions. Acta Math. 94, pp. 289–349.

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Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§16.10, §16.8(ii), §16.8(ii).

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3: Bibliography C

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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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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

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

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

Referenced by:

§13.29(v), §15.19(v).

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On zeros of the hypergeometric function. Serdica 7 (3), pp. 243–249.

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ISSN 0204-4140, MathReview (R. G. Buschman), ZentralBlatt

Referenced by:

§15.13.

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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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External Links:

Document

Referenced by:

5th item.

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4: Bibliography V

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Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.

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ISSN 0377-9017, Document, MathReview (Laurent Habsieger), ZentralBlatt

Referenced by:

§19.35(i).

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Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.

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ISSN 0377-0427, Document, Link, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§15.8(v).

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A note on the asymptotic expansion of generalized hypergeometric functions. Anal. Appl. (Singap.) 12 (1), pp. 107–115.

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ISSN 0219-5305, Document, MathReview (Roberto S. Costas-Santos), ZentralBlatt

Referenced by:

§16.11(ii), §16.11(ii).

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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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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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5: Bibliography K

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

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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).

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

Referenced by:

§32.11(iv).

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

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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.

External Links:

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

Referenced by:

1st item.

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6: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

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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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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

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7: Bibliography O

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Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{\mathrm{VI}}$ . Ann. Mat. Pura Appl. (4) 146, pp. 337–381.

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ISSN 0003-4622, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(vi), §32.6(v), §32.7(vii), §32.7(viii).

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Studies on the Painlevé equations. II. Fifth Painlevé equation $P_{\mathrm{V}}$ . Japan. J. Math. (N.S.) 13 (1), pp. 47–76.

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ISSN 0289-2316, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(v), §32.6(v), §32.7(viii).

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Studies on the Painlevé equations. IV. Third Painlevé equation $P_{\mathrm{III}}$ . Funkcial. Ekvac. 30 (2-3), pp. 305–332.

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ISSN 0532-8721, FullText, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(iii), §32.2(iii), §32.6(iv), §32.7(viii).

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

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Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function. SIAM J. Math. Anal. 24 (3), pp. 756–767.

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External Links:

ISSN 0036-1410, Document, MathReview (Hussain S. Nur), ZentralBlatt

Referenced by:

§10.17(v), §13.7(iii), (9.7.18), (9.7.19), (9.7.20), (9.7.21), (9.7.22), (9.7.23), §9.7(v).

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8: Bibliography F

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

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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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Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

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Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

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External Links:

FullText, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.10.

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

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9: Software Index

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	Open Source													With Book						Commercial
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13 Confluent Hypergeometric Functions
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15 Hypergeometric Function
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16 Generalized Hypergeometric Functions & Meijer G-Function
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20 Theta Functions
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28 Mathieu Functions and Hill’s Equation
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10: Errata

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Section: 15.9(v) Complete Elliptic Integrals. Equations: (11.11.9_5), (11.11.13_5), Intermediate equality in (15.4.27) which relates to $F(a,a;a+1;\frac{1}{2})$, (15.4.34), (19.5.4_1), (19.5.4_2) and (19.5.4_3).

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Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${}_2{}^{}F_1^{}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

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There were clarifications made in the conditions on the parameter $a$ in $U(a,b,z)$ of those equations.

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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13.18.7 $$W_{-\frac{1}{4},\pm \frac{1}{4}}\left(z^2\right)={\mathrm{e}}^{\frac{1}{2}{z}^2}\sqrt{\pi z}\mathrm{erfc}\left(z\right)$$

Originally the left-hand side was given correctly as $W_{-\frac{1}{4},-\frac{1}{4}}\left(z^2\right)$; the equation is true also for $W_{-\frac{1}{4},+\frac{1}{4}}\left(z^2\right)$.

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