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21: Bibliography

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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:

Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.

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Code, Document

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1st item.

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$q$-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. CBMS Regional Conference Series in Mathematics, Vol. 66, Amer. Math. Soc., Providence, RI.

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ISBN 0-8218-0716-1, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.14, §17.14, §17.16, Ch.17, Profile George E. Andrews.

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Modular Functions and Dirichlet Series in Number Theory. 2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.

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ISBN 0-387-97127-0, MathReview, ZentralBlatt

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§17.2(i), §23.10(iv), §23.15(i), §23.17(ii), §23.18, §23.18, §23.19, §23.20(i), §23.20(v), Ch.23, §27.14(iii), §27.14(iv), §27.14(vi), §27.7, Ch.27.

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Orthogonal Polynomials and Special Functions. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 21, Society for Industrial and Applied Mathematics, Philadelphia, PA.

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ISBN 0-89871-018-9, MathReview (G. Gasper), ZentralBlatt

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§18.10(ii), §18.18(vii), Profile Richard A. Askey.

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Continuous $q$-Hermite Polynomials when $q>1$ . In $q$-series and Partitions (Minneapolis, MN, 1988), IMA Vol. Math. Appl., Vol. 18, pp. 151–158.

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External Links:

MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§18.28(vii).

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22: 6.16 Mathematical Applications

… ►Consider the Fourier series …uniformly for $x\in [-\pi ,\pi ]$. … ►Compare Figure 6.16.1. … ►It occurs with Fourier-series expansions of all piecewise continuous functions. … …

23: 11.6 Asymptotic Expansions

§11.6 Asymptotic Expansions

… ►For re-expansions of the remainder terms in (11.6.1) and (11.6.2), see Dingle (1973, p. 445). … ►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). … ►Here …

24: Bibliography L

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New method to obtain small parameter power series expansions of Mathieu radial and angular functions. Math. Comp. 78 (265), pp. 255–274.

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ISSN 0025-5718, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

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§28.24, §28.24.

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New series expansions for the confluent hypergeometric function $M(a,b,z)$ . Appl. Math. Comput. 235, pp. 26–31.

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ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

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§13.11, §13.11.

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New series expansions of the Gauss hypergeometric function. Adv. Comput. Math. 39 (2), pp. 349–365.

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ISSN 1019-7168, Document, FullText, MathReview (Jochen Denzler), ZentralBlatt

Referenced by:

§15.19(i), §15.19(i).

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Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions. Numer. Math. 116 (2), pp. 269–289.

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ISSN 0029-599X, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§13.11, §13.11.

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Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput. 13 (68), pp. 261–271.

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FullText, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§13.24(ii).

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25: Bibliography P

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On the use of Hadamard expansions in hyperasymptotic evaluation. I. Real variables. Proc. Roy. Soc. London Ser. A 457 (2016), pp. 2835–2853.

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ISSN 1364-5021, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.40(iv), §2.11(v).

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On the use of Hadamard expansions in hyperasymptotic evaluation. II. Complex variables. Proc. Roy. Soc. London Ser. A 457, pp. 2855–2869.

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ISSN 1364-5021, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.40(iv), §2.11(v).

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Double-precision Fortran, maximum accuracy 20S.

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ISSN 0010-4655, Document, Code

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6th item.

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Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

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Integrals and Series: Special Functions, Vol. 2. Gordon & Breach Science Publishers, New York.

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A second edition was published in 1988. Table errata: Math. Comp. v. 65 (1996), no. 215, pp. 1382–1383

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ISBN 2-88124-097-6, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §10.22(vi), §10.23(iv), §10.43(v), §10.43(vi), §10.44(iv), §10.60(iv), §10.71(iii), §12.12, §12.13(ii), §18.17(ix), §18.18(ix), §25.7, §25.8, §5.13, §6.14(iii), §6.15, §7.14(iii), §7.15, §8.14, §8.15.

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26: 2.11 Remainder Terms; Stokes Phenomenon

… ►Secondly, the asymptotic series represents an infinite class of functions, and the remainder depends on which member we have in mind. … ►For illustration, we give re-expansions of the remainder terms in the expansions (2.7.8) arising in differential-equation theory. … ►In this way we arrive at hyperasymptotic expansions. … ►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. … ►The process just used is equivalent to re-expanding the remainder term of the original asymptotic series (2.11.24) in powers of $1/(x+5)$ and truncating the new series optimally. …

27: 25.11 Hurwitz Zeta Function

… ►The function $\zeta (s,a)$ was introduced in Hurwitz (1882) and defined by the series expansion … ►

§25.11(iv) Series Representations

… ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►

§25.11(x) Further Series Representations

… ►As $a\to \mathrm{\infty }$ in the sector , with $s(\ne 1)$ and $\delta$ fixed, we have the asymptotic expansion …

28: Bibliography G

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Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.

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Notes:

A revised and updated edition, with three new chapters, was published in 2004 by Cambridge University Press (Encyclopedia of Mathematics and its Applications, vol. 96).

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ISBN 0-521-35049-2, MathReview (Shaun Cooper), ZentralBlatt

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(17.9.3), Erratum (V1.0.23) for Equation (17.9.3).

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Includes Fortran program claiming 12S accuracy

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

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3rd item, §10.77(ix).

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Matrices, moments and quadrature with applications. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ.

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ISBN 978-0-691-14341-5, MathReview (G. A. Evans)

Referenced by:

§18.2(ix), §18.40(ii).

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Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA.

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CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26

External Links:

MathReview (O. Widlund), ZentralBlatt

Referenced by:

§18.38(i).

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Multilateral summation theorems for ordinary and basic hypergeometric series in $\mathrm{U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

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ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

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