general%20properties

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1: Bibliography R

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

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2: Bibliography D

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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction $\zeta (s)$ de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

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Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§13.29(iv), §13.29(iv).

Encodings:

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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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On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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External Links:

ZentralBlatt

Referenced by:

§15.4(ii).

Encodings:

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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.)

External Links:

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

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

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3: Bibliography N

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

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External Links:

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

Referenced by:

§19.37(iv).

Encodings:

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The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

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External Links:

ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

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The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.

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External Links:

ISSN 1848-5979, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

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The resurgence properties of the incomplete gamma function II. Stud. Appl. Math. 135 (1), pp. 86–116.

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External Links:

ISSN 0022-2526, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.11(iii).

Encodings:

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The resurgence properties of the incomplete gamma function, I. Anal. Appl. (Singap.) 14 (5), pp. 631–677.

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External Links:

ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.11(iii), §8.11(v), §8.11(v), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).

Encodings:

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4: 5.11 Asymptotic Expansions

… ►The scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ is defined in (5.11.3) and its main property is ${\mathrm{\Gamma }}^{\ast }\left(z\right)\sim 1$ as $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$. Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. … ►In terms of generalized Bernoulli polynomials $B_n^{(\mathrm{\ell })}\left(x\right)$ (§24.16(i)), we have for $k=0,1,\mathrm{\dots }$, ►

5.11.17 $$G_k(a,b)=\left(\genfrac{}{}{0.0pt}{}{a-b}{k}\right)B_k^{(a-b+1)}\left(a\right),$$

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Defines:

$G_k(a,b)$: coefficients (locally)

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $B_n^{(\mathrm{\ell })}\left(x\right)$: generalized Bernoulli polynomials, $k$: nonnegative integer, $a$: real or complex variable and $b$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.11.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

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5.11.18 $$H_k(a,b)=\left(\genfrac{}{}{0.0pt}{}{a-b}{2k}\right)B_{2k}^{(a-b+1)}\left(\frac{a-b+1}{2}\right).$$

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Defines:

$H_k(a,b)$: coefficients (locally)

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $B_n^{(\mathrm{\ell })}\left(x\right)$: generalized Bernoulli polynomials, $k$: nonnegative integer, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.11(iii)

Permalink:

http://dlmf.nist.gov/5.11.E18

Encodings:

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See also:

Annotations for §5.11(iii), §5.11 and Ch.5

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5: Bibliography G

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Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.

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Notes:

Translated from the Russian by Eugene Saletan. A paperbound reprinting by the same publisher appeared in 1977

External Links:

MathReview, ZentralBlatt

Referenced by:

§2.6(i).

Encodings:

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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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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

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

Referenced by:

3rd item, §10.77(ix).

Encodings:

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Generalized Fermi-Dirac functions and derivatives: Properties and evaluation. Comput. Phys. Comm. 136 (3), pp. 294–309.

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Notes:

Double-precision Fortran, maximum accuracy 20D

External Links:

Document, Code, ZentralBlatt

Referenced by:

7th item.

Encodings:

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Do integrable mappings have the Painlevé property?. Phys. Rev. Lett. 67 (14), pp. 1825–1828.

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External Links:

ISSN 0031-9007, Document, MathReview, ZentralBlatt

Referenced by:

§32.16.

Encodings:

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A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.

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External Links:

ISSN 0019-3577, Document, MathReview (Sumitra Purkayastha), ZentralBlatt

Referenced by:

§35.9.

Encodings:

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6: Bibliography S

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A property of the zeros of cross-product Bessel functions of different orders. Z. Angew. Math. Mech. 56 (2), pp. 120–121.

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External Links:

Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

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Some properties of polynomial sets of type zero. Duke Math. J. 5, pp. 590–622.

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External Links:

ISSN 0012-7094, Link, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.2(xii).

Encodings:

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Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.

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External Links:

ISSN 0285-4821, MathReview (Peter M. Neumann), ZentralBlatt

Referenced by:

§24.10(ii).

Encodings:

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Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 (1), pp. 76–115.

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External Links:

ISSN 0001-8708, Document, MathReview (Dennis White), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

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7: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

Encodings:

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Integrable Hamiltonian systems and the Painlevé property. Phys. Rev. A (3) 25 (3), pp. 1257–1264.

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External Links:

ISSN 0556-2791, Document, MathReview

Referenced by:

§32.16.

Encodings:

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Discrete Cosine and Sine Transforms. General Properties, Fast Algorithms and Integer Approximations. Elsevier/Academic Press, Amsterdam.

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External Links:

ISBN 978-0-12-373624-6; 0-12-373624-2, MathReview (Gerd Teschke)

Referenced by:

§18.3.

Encodings:

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8: 18.5 Explicit Representations

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18.5.5 $$p_n(x)=\frac{1}{{\kappa }_{n}w(x)}\frac{d^n}{{dx}^n}\left(w(x){(F(x))}^n\right).$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $p_n(x)$: polynomial of degree $n$ , $w(x)$: weight function, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.1.6 (incorrectly stated as property of general OP’s)

Referenced by:

§18.17(v), §18.17(vi), §18.17(vii), item 3., §18.39(ii), §18.39(ii), Table 18.5.1, §18.9(iii)

Permalink:

http://dlmf.nist.gov/18.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(ii), §18.5 and Ch.18

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§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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Hermite

… ►However, in these circumstances the orthogonality property (18.2.1) disappears. … …