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21: Bibliography C

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Determination of $\nu$-zeros of Hankel functions. Comput. Phys. Comm. 32 (3), pp. 333–339.

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Notes:

Double-precision Fortran, maximum accuracy 7S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

1st item.

Encodings:

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Some inequalities for hypergeometric functions. Proc. Amer. Math. Soc. 17 (1), pp. 32–39.

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External Links:

FullText, MathReview (Gabor Szegő), ZentralBlatt

Referenced by:

§19.24(i), §19.24(i), §19.24(ii), §19.9(i), §19.9(ii).

Encodings:

BibTeX

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Asymptotics of Racah coefficients and polynomials. J. Phys. A 32 (3), pp. 537–553.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§34.8, §34.8.

Encodings:

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A dispersion analysis for difference schemes: Tables of generalized Airy functions. Math. Comp. 32 (144), pp. 1163–1170.

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External Links:

ISSN 0025-5718, FullText, MathReview (Philip Brenner), ZentralBlatt

Referenced by:

§9.13(ii).

Encodings:

BibTeX

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Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.

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External Links:

ISSN 0002-9920, MathReview (Willard Parker)

Referenced by:

§19.8(i).

Encodings:

BibTeX

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22: 24.15 Related Sequences of Numbers

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24.15.11 $$\sum _{k=0}^{\lfloor n/2\rfloor }\left(\genfrac{}{}{0pt}{}{n}{2k}\right){\left(\frac{5}{9}\right)}^k{B}_{2k}{u}_{n-2k}=\frac{n}{6}{v}_{n-1}+\frac{n}{3^n}{v}_{2n-2},$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $k$: integer, $n$: integer, $u_n$: Fibonacci numbers and $v_n$: Lucas numbers

Permalink:

http://dlmf.nist.gov/24.15.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iv), §24.15 and Ch.24

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23: 27.13 Functions

… ►For example, $g\left(3\right)=9$, $g\left(4\right)=19$, $g\left(5\right)=37$, $g\left(6\right)=73$, $g\left(7\right)=143$, and $g\left(8\right)=279$. …

24: 9.8 Modulus and Phase

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9.8.22 $$\theta \left(x\right)\sim \frac{\pi }{4}+\frac{2}{3}{(-x)}^{3/2}\left(1+\frac{5}{32}\frac{1}{x^3}+\frac{1105}{6144}\frac{1}{x^6}+\frac{82825}{65536}\frac{1}{x^9}+\frac{\mathrm{12820\hspace{0.33em}31525}}{\mathrm{587\hspace{0.33em}20256}}\frac{1}{x^{12}}+\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\theta \left(z\right)$: Airy phase function and $x$: real variable

Proof sketch:

Combine (9.8.9)–(9.8.12) with §10.18(iii). See also Nemes (2021, Theorem 2.1 and p. 4333).

Referenced by:

§9.8(iv), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv)

Permalink:

http://dlmf.nist.gov/9.8.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(iv), §9.8 and Ch.9

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9.8.23 $$\varphi \left(x\right)\sim -\frac{\pi }{4}+\frac{2}{3}{(-x)}^{3/2}\left(1-\frac{7}{32}\frac{1}{x^3}-\frac{1463}{6144}\frac{1}{x^6}-\frac{\mathrm{4\hspace{0.33em}95271}}{\mathrm{3\hspace{0.33em}27680}}\frac{1}{x^9}-\frac{\mathrm{2065\hspace{0.33em}30429}}{\mathrm{83\hspace{0.33em}88608}}\frac{1}{x^{12}}-\mathrm{\cdots }\right).$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\varphi \left(z\right)$: Airy phase function and $x$: real variable

Proof sketch:

Combine (9.8.9)–(9.8.12) with §10.18(iii). See also Nemes (2021, Theorem 2.1 and p. 4333).

Referenced by:

§9.8(iv), §9.8(iv), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv)

Permalink:

http://dlmf.nist.gov/9.8.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(iv), §9.8 and Ch.9

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25: Bibliography T

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An algorithm with ALGOL 60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives. J. Comput. Phys. 32 (2), pp. 270–279.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.74(vi), §10.77(x).

Encodings:

BibTeX

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The universal Askey-Wilson algebra and DAHA of type $(C_1^{\vee },C_1)$ . SIGMA 9, pp. Paper 047, 40 pp..

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External Links:

ISSN 1815-0659, Link, MathReview (Renato Álvarez-Nodarse), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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Algorithm 926: incomplete gamma functions with negative arguments. ACM Trans. Math. Software 39 (2), pp. Art. 14, 9.

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External Links:

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

Referenced by:

7th item, §8.28(ii).

Encodings:

BibTeX

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Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.

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Notes:

With 1 Macintosh floppy disk (3.5 inch; DD). Programs for the calculation of $3j,6j,9j$ symbols are included.

External Links:

ISBN 0-471-55264-X, MathReview (J. F. Cornwell), ZentralBlatt

Referenced by:

§34.3(vii).

Encodings:

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Theory of the Fresnel integral. USSR Comput. Math. and Math. Phys. 9 (4), pp. 271–279.

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Notes:

Translation of the paper in Zh. Vychisl. Mat. Mat. Fiz., Vol. 9, No. 4, 938-944, 1969

External Links:

Document, ZentralBlatt

Referenced by:

§7.13(iv).

Encodings:

BibTeX

26: Bibliography

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Ly$\alpha$ line formation in starbursting galaxies. I. Moderately thick, dustless, and static H i media. Astrophysical J. 554, pp. 604–614.

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External Links:

FullText

Referenced by:

§7.21.

Encodings:

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Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

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External Links:

MathReview (N. G. de Bruijn), ZentralBlatt

Referenced by:

(25.11.41), (25.11.42), §25.11(xii).

Encodings:

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Numerical Tables for Angular Correlation Computations in $\alpha$-, $\beta$- and $\gamma$-Spectroscopy: $3j$-, $6j$-, $9j$-Symbols, F- and $\mathrm{\Gamma }$-Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.

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External Links:

ISBN 978-3-540-04218-1

Referenced by:

§34.14.

Encodings:

BibTeX

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A new treatment of the ellipsoidal wave equation. Proc. London Math. Soc. (3) 9, pp. 21–50.

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External Links:

ISSN 0024-6115, Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§29.11.

Encodings:

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Matching higher conserved charges for strings and spins. J. High Energy Phys. 2004 (3).

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Notes:

(electronic only, article id. JHEP03(2004)004, 32 pp.).

External Links:

ISSN 1126-6708, Document, MathReview

Referenced by:

§19.35(ii).

Encodings:

BibTeX

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27: 34 3j, 6j, 9j Symbols

Chapter 34 $\mathit{3}j,\mathit{6}j,\mathit{9}j$ Symbols

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28: 34.14 Tables

§34.14 Tables

►Tables of exact values of the squares of the $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols on pp. … ►Some selected $\mathit{9}j$ symbols are also given. … 16-17; for $\mathit{9}j$ symbols on p. … ► 310–332, and for the $\mathit{9}j$ symbols on pp. …

29: Bibliography R

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Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.

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External Links:

ISSN 0025-5874, ISSN 0025-5874, Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(v).

Encodings:

BibTeX

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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).

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External Links:

Link, ISSN 2227-7390, Document

Referenced by:

§8.15.

Encodings:

BibTeX

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Another proof of the triple sum formula for Wigner $9j$-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

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External Links:

ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt

Referenced by:

§34.6.

Encodings:

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A proof of the asymptotic series for log $\mathrm{\Gamma }(z)$ and log $\mathrm{\Gamma }(z+a)$ . Ann. of Math. (2) 32 (1), pp. 10–16.

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External Links:

ISSN 0003-486X, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§5.11(i).

Encodings:

BibTeX

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30: 18.5 Explicit Representations

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$T_6\left(x\right)=32x^6-48x^4+18x^2-1.$

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$U_5\left(x\right)=32x^5-32x^3+6x,$

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$H_5\left(x\right)=32x^5-160x^3+120x,$

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