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11: DLMF Project News

error generating summary

12: 26.14 Permutations: Order Notation

… ►The set ${𝔖}_n$ (§26.13) can be viewed as the collection of all ordered lists of elements of $\{1,2,\mathrm{\dots },n\}$: $\{\sigma (1)\sigma (2)\mathrm{\cdots }\sigma (n)\}$. As an example, $35247816$ is an element of ${𝔖}_8.$ The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … ►The permutation $35247816$ has two descents: $52$ and $81$. …For example, $maj(35247816)=2+6=8$. … ►In this subsection $S(n,k)$ is again the Stirling number of the second kind (§26.8), and $B_m$ is the $m$th Bernoulli number (§24.2(i)). …

13: 25.6 Integer Arguments

… ►

$\zeta \left(4\right)={\displaystyle \frac{{\pi }^4}{90}},$

… ►

25.6.2 $$\zeta \left(2n\right)=\frac{{(2\pi )}^{2n}}{2(2n)!}\left|B_{2n}\right|,$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: nonnegative integer

Sources:

Magnus et al. (1966, p. 19); Apostol (1976, (22), p. 266)

A&S Ref:

23.2.16

Referenced by:

(25.11.21)

Permalink:

http://dlmf.nist.gov/25.6.E2

Encodings:

pMML, png, TeX

See also:

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

►

25.6.3 $$\zeta \left(-n\right)=-\frac{B_{n+1}}{n+1},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function and $n$: nonnegative integer

Source:

Apostol (1976, (20), p. 266); with $n\ne 0$

A&S Ref:

23.2.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.6.E3

Encodings:

pMML, png, TeX

See also:

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

… ►

25.6.6 $$\zeta \left(2k+1\right)=\frac{{(-1)}^{k+1}{(2\pi )}^{2k+1}}{2(2k+1)!}{\int }_0^1{B}_{2k+1}\left(t\right)\cot \left(\pi t\right)dt,$$ $k=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral and $k$: nonnegative integer

Source:

Nörlund (1924, (81*), p. 66); with $\nu =k$

A&S Ref:

23.2.17

Permalink:

http://dlmf.nist.gov/25.6.E6

Encodings:

pMML, png, TeX

See also:

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

… ►

25.6.15 $${\zeta }^{\prime }\left(2n\right)=\frac{{(-1)}^{n+1}{(2\pi )}^{2n}}{2(2n)!}\left(2n{\zeta }^{\prime }\left(1-2n\right)-(\psi \left(2n\right)-\ln \left(2\pi \right))B_{2n}\right).$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Source:

Miller and Adamchik (1998, (2), p. 202)

Referenced by:

(25.11.21)

Permalink:

http://dlmf.nist.gov/25.6.E15

Encodings:

pMML, png, TeX

See also:

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

…

14: Bibliography M

… ►

Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd edition, Springer-Verlag, New York-Berlin.

ⓘ

Notes:

Table errata: Math. Comp. v. 23 (1969), p. 471, v. 24 (1970), pp. 240 and 505, v. 25 (1971), no. 113, p. 201, v. 29 (1975), no. 130, p. 672, v. 30 (1976), no. 135, pp. 677–678, v. 32 (1978), no. 141, pp. 319–320, v. 36 (1981), no. 153, pp. 315–317, v. 41 (1983), no. 164, pp. 775–776, and v. 81 (2012), no. 280, p. 2251

External Links:

MathReview (Mary L. Boas), ZentralBlatt

Referenced by:

§10.15, §10.22(vi), §10.32(iv), §10.38, §10.43(vi), §10.60(iv), §10.9(iv), §11.5(iii), Ch.11, §12.12, §12.5(iv), Ch.12, §13.10(vi), §13.23(iv), §14.1, §14.11, §14.12(ii), §14.18(iv), §14.25, §14.3(iii), §14.5(v), §14.7(iv), Ch.14, (25.5.21), §25.5, (25.6.1), (25.6.2), §25.6(i), §8.1, Notations B, Notations I, Notations P, Notations P, Notations Q, Notations Q.

Encodings:

BibTeX

… ►

On the Representation of Meijer’s $G$-Function in the Vicinity of Singular Unity. In Complex Analysis and Applications ’81 (Varna, 1981), pp. 383–398.

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External Links:

MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§16.21.

Encodings:

BibTeX

… ►

Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

ⓘ

External Links:

FullText

Referenced by:

§8.24(i).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§18.27(v).

Encodings:

BibTeX

… ►

Lattice Statistics. In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.), University of California Engineering and Physical Sciences Extension Series, pp. 96–143.

ⓘ

Notes:

Reprinted by Robert E. Krieger Publishing Co. in 1981.

External Links:

ZentralBlatt

Referenced by:

§26.20.

Encodings:

BibTeX

…

15: Bibliography S

… ►

Hypergeometric Functions and Their Applications. Texts in Applied Mathematics, Vol. 8, Springer-Verlag, New York.

ⓘ

External Links:

ISBN 0-387-97558-6, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

Erratum (V1.2.0) Section 18.39.

Encodings:

BibTeX

… ►

Decoherent dynamics of a two-level system coupled to a sea of spins. Phys. Rev. Lett. 81 (26), pp. 5710–5713.

ⓘ

External Links:

Document

Referenced by:

§11.12.

Encodings:

BibTeX

… ►

Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comp. 81 (280), pp. 2159–2173.

ⓘ

External Links:

ISSN 0025-5718, Document, Link, MathReview (Vittoria Demichelis), ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

BibTeX

… ►

On the calculation of complex zeros of the modified Bessel function of the second kind. Dokl. Akad. Nauk SSSR 280 (2), pp. 296–299.

ⓘ

Notes:

English translation in Soviet Math. Dokl. 31(1), pp. 78–81

External Links:

ISSN 0002-3264, MathReview (I. F. Kushnirchuk), ZentralBlatt

Referenced by:

§10.74(vi).

Encodings:

BibTeX

… ►

Liouville-Green-Olver approximations for complex difference equations. J. Approx. Theory 96 (2), pp. 301–322.

ⓘ

External Links:

ISSN 0021-9045, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

§2.9(iii).

Encodings:

BibTeX

…

16: Bibliography D

… ►

The principal frequencies of vibrating systems with elliptic boundaries. Quart. J. Mech. Appl. Math. 8 (3), pp. 361–372.

ⓘ

External Links:

Document, MathReview (M. A. Hyman), ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

… ►

Integralen voor de $\zeta$-functie van Riemann. Mathematica (Zutphen) B5, pp. 170–180 (Dutch).

ⓘ

Notes:

In Dutch.

External Links:

ZentralBlatt

Referenced by:

(25.5.15), (25.5.16), (25.5.17), (25.5.18), (25.5.19), §25.5(ii).

Encodings:

BibTeX

… ►

Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1837, pp. 45–81 (German).

ⓘ

Notes:

Reprinted in G. Lejeune Dirichlet’s Werke, Band I, pp. 313–342, Reimer, Berlin, 1889 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.

External Links:

ISBN 0-8284-0225-6, Link, MathReview

Referenced by:

§25.15(i).

Encodings:

BibTeX

… ►

Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge.

ⓘ

External Links:

ISBN 0-521-80043-9, MathReview (Marcel de Jeu), ZentralBlatt

Referenced by:

§18.37(ii), §18.37(i).

Encodings:

BibTeX

… ►

Lamé instantons. J. High Energy Phys. 2000 (1), pp. Paper 19, 8.

ⓘ

Notes:

(electronic only, article id. JHEP01(2000)019, 8 pp.).

External Links:

ISSN 1029-8479, Document, MathReview, ZentralBlatt

Referenced by:

§22.11.

Encodings:

BibTeX

…

17: Bibliography K

… ►

A proof of the $q$-Macdonald-Morris conjecture for $B{C}_n$ . Mem. Amer. Math. Soc. 108 (516), pp. vi+80.

ⓘ

External Links:

ISSN 0065-9266, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§17.14.

Encodings:

BibTeX

… ►

Special functions and the Bieberbach conjecture. Amer. Math. Monthly 95 (8), pp. 689–696.

ⓘ

External Links:

ISSN 0002-9890, FullText, MathReview (S. L. Shukla), ZentralBlatt

Referenced by:

§16.23(iii).

Encodings:

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… ►

Elliptic asymptotics of the first and second Painlevé transcendents. Uspekhi Mat. Nauk 49 (1(295)), pp. 77–140 (Russian).

ⓘ

Notes:

In Russian. English translation: Russian Math. Surveys 49(1994), no. 1, pp. 81–150.

External Links:

ISSN 0042-1316, Document, MathReview (V. A. Golubeva), ZentralBlatt

Referenced by:

§32.11(i), §32.12(ii).

Encodings:

BibTeX

… ►

Dark optical solitons: Physics and applications. Physics Reports 298 (2-3), pp. 81–197.

ⓘ

External Links:

Document

Referenced by:

§22.19(iii).

Encodings:

BibTeX

… ►

The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

ⓘ

External Links:

Document, MathReview (M. Mikolas), ZentralBlatt

Referenced by:

(18.14.8), §18.14(i), §18.17(ii), §18.18(ii).

Encodings:

BibTeX

…

18: 10.41 Asymptotic Expansions for Large Order

… ►The curve $E_{1}B{E}_2$ in the $z$-plane is the upper boundary of the domain $𝐊$ depicted in Figure 10.20.3 and rotated through an angle $-\frac{1}{2}\pi$. Thus $B$ is the point $z=c$, where $c$ is given by (10.20.18). … ►The series (10.41.3)–(10.41.6) can also be regarded as generalized asymptotic expansions for large $|z|$. … ►Similar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20. … ►It needs to be noted that the results (10.41.14) and (10.41.15) do not apply when $z\to 0+$ or equivalently $\zeta \to +\mathrm{\infty }$. …

19: 1.11 Zeros of Polynomials

… ►Set $z=w-\frac{1}{3}a$ to reduce $f(z)=z^3+a{z}^2+bz+c$ to $g(w)=w^3+pw+q$, with $p=(3b-a^2)/3$, $q=(2a^3-9ab+27c)/27$. … ► $f(z)=z^3-6z^2+6z-2$, $g(w)=w^3-6w-6$, $A=3\sqrt[3]{4}$, $B=3\sqrt[3]{2}$. … ►

$p=(-3a^2+8b)/8,$

►

$q=(a^3-4ab+8c)/8,$

… ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. …

20: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The cofactor $A_{jk}$ of $a_{jk}$ is … ►The determinant of an upper or lower triangular, or diagonal, square matrix $𝐀$ is the product of the diagonal elements $det(𝐀)={\prod }_{i=1}^n{a}_{ii}$. … ►Let $a_{j,k}$ be defined for all integer values of $j$ and $k$, and ${𝐷}_n[a_{j,k}]$ denote the $(2n+1)\times (2n+1)$ determinant … ►Square matices can be seen as linear operators because $𝐀(\alpha 𝐚+\beta 𝐛)=\alpha 𝐀𝐚+\beta 𝐀𝐛$ for all $\alpha ,\beta \in ℂ$ and $𝐚,𝐛\in {𝐄}_n$, the space of all $n$-dimensional vectors. … ►The adjoint of a matrix $𝐀$ is the matrix ${𝐀}^{\ast }$ such that $⟨𝐀𝐚,𝐛⟩=⟨𝐚,{𝐀}^{\ast }𝐛⟩$ for all $𝐚,𝐛\in {𝐄}_n$. …