Van Vleck theorem for continued fractions

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21: 6.18 Methods of Computation

… ►Lastly, the continued fraction (6.9.1) can be used if $|z|$ is bounded away from the origin. … ►For a comprehensive survey of computational methods for the functions treated in this chapter, see van der Laan and Temme (1984, Ch. IV).

22: 6.17 Physical Applications

… ►For applications in astrophysics, see also van de Hulst (1980). …

23: Bibliography L

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Démonstration d’un Théoréme d’Arithmétique. Nouveau Mém. Acad. Roy. Sci. Berlin, pp. 123–133 (French).

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Notes:

Reprinted in Oeuvres, Vol. 3, pp. 189–201, Gauthier-Villars, Paris, 1867

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FullText

Referenced by:

§27.13(iii).

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Generating Bessel functions in Mie scattering calculations using continued fractions. Applied Optics 15 (3), pp. 668–671.

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Referenced by:

§3.10(iii), §3.10(iii), Erratum (V1.0.24) for Paragraph Steed’s Algorithm (in §3.10(iii)).

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Orthogonal Polynomials for Exponential Weights. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 4, Springer-Verlag, New York.

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External Links:

ISBN 0-387-98941-2, MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§18.32.

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Polylogarithms and Associated Functions. North-Holland Publishing Co., New York.

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Notes:

With a foreword by A. J. Van der Poorten

External Links:

ISBN 0-444-00550-1, MathReview, ZentralBlatt

Referenced by:

(25.12.10), (25.12.11), §25.12(ii), §25.12(i), §25.12(i), §25.12(ii), §25.12.

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Continued Fractions with Applications. Studies in Computational Mathematics, North-Holland Publishing Co., Amsterdam.

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External Links:

ISBN 0-444-89265-6, MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§1.12(ii), §1.12(iv), §1.12(v), §15.7, §18.13, §3.10(ii), §3.10(iii), §3.11(iv), §4.25, §4.39, §4.9(i), §4.9(ii), §5.10, §6.9, §7.18(v), §7.9.

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24: 10.55 Continued Fractions

§10.55 Continued Fractions

►For continued fractions for ${𝗃}_{n+1}\left(z\right)/{𝗃}_n\left(z\right)$ and ${𝗂}_{n+1}^{(1)}\left(z\right)/{𝗂}_n^{(1)}\left(z\right)$ see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).

25: Bibliography K

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Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§19.12.

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Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)

Referenced by:

§18.32.

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A general addition theorem for spheroidal wave functions. SIAM J. Math. Anal. 4 (1), pp. 149–160.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§30.10.

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A Fortran computer program for calculating the prolate and oblate angle functions of the first kind and their first and second derivatives. NRL Report No. 7161 Naval Res. Lab.  Washingtion, D.C..

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Referenced by:

§30.18(iii).

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A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

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External Links:

Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(i).

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26: 25.6 Integer Arguments

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25.6.4 $$\zeta \left(-2n\right)=0,$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function and $n$: nonnegative integer

Source:

Apostol (1976, Theorem 12.16, p. 266)

A&S Ref:

23.2.14

Permalink:

http://dlmf.nist.gov/25.6.E4

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Annotations for §25.6(i), §25.6 and Ch.25

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25.6.8 $$\zeta \left(2\right)=3\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $k$: nonnegative integer

Keywords:

infinite series

Source:

van der Poorten (1980, (2), p. 271)

Permalink:

http://dlmf.nist.gov/25.6.E8

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Annotations for §25.6(i), §25.6 and Ch.25

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25.6.9 $$\zeta \left(3\right)=\frac{5}{2}\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^{k-1}}{k^3\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $k$: nonnegative integer

Keywords:

infinite series

Source:

van der Poorten (1980, (3), p. 271)

Permalink:

http://dlmf.nist.gov/25.6.E9

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Annotations for §25.6(i), §25.6 and Ch.25

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25.6.10 $$\zeta \left(4\right)=\frac{36}{17}\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^4\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $k$: nonnegative integer

Keywords:

infinite series

Source:

van der Poorten (1980, (5), p. 274)

Permalink:

http://dlmf.nist.gov/25.6.E10

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Annotations for §25.6(i), §25.6 and Ch.25

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27: Bibliography S

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Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:

FullText, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§3.5(viii).

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Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung III. Numer. Math. 8 (1), pp. 68–71.

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Notes:

Includes Algol program

External Links:

ISSN 0029-599X, Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§28.36(ii).

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Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung I. Numer. Math. 3 (1), pp. 30–38.

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External Links:

ISSN 0029-599X, Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

item (b).

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Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83 (2), pp. 333–382.

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External Links:

ISSN 0020-9910, Document, MathReview (Bert van Geemen), ZentralBlatt

Referenced by:

§21.9.

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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.

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28: 30.18 Software

… ►See also King et al. (1970), King and Van Buren (1970), Van Buren et al. (1970), and Van Buren et al. (1972).

29: Bibliography T

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COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 36 (4), pp. 363–372.

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Notes:

Double-precision Fortran, minimum accuracy: 14D. See also Thompson (2004)

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Document, Code, ZentralBlatt

Referenced by:

1st item, §33.23(vi), 3rd item.

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Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§33.23(vi), I. J. Thompson and A. R. Barnett (1985).

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Asymptotische Eigenschaften der unvollständigen Gammafunktion. Math. Z. 53, pp. 136–148 (German).

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External Links:

ISSN 0025-5874, Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§8.11(iii), §8.11(iv), §8.13(i), §8.13(i), §8.13(iii), §8.15.

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Sugli zeri delle funzioni di cui si conosce una rappresentazione asintotica. Ann. Mat. Pura Appl. (4) 26, pp. 283–300 (Italian).

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External Links:

Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§13.9(i).

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30: Tom H. Koornwinder

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