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1: 24.1 Special Notation

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Bernoulli Numbers and Polynomials

►The origin of the notation $B_n$, $B_n\left(x\right)$, is not clear. … ►

Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations $E_n$, $E_n\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

2: 6.17 Physical Applications

… ►Geller and Ng (1969) cites work with applications from diffusion theory, transport problems, the study of the radiative equilibrium of stellar atmospheres, and the evaluation of exchange integrals occurring in quantum mechanics. …

3: 26.22 Software

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World Combinatorics Exchange (website).

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4: Bibliography W

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Prime Divisors of the Bernoulli and Euler Numbers. In Number Theory for the Millennium, III (Urbana, IL, 2000), pp. 357–374.

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External Links:

FullText, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.20.

Encodings:

BibTeX

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Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.

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External Links:

ISSN 0006-3835, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.5(iv).

Encodings:

BibTeX

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Generating functions of class-numbers. Compositio Math. 1, pp. 39–68.

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External Links:

ISSN 0010-437X, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§20.7(v), §20.8.

Encodings:

BibTeX

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Notes:

A free resource from Wolfram Research built with Mathematica technology.

External Links:

Home Page

Referenced by:

8th item, § ‣ Software Cross Index.

Encodings:

BibTeX

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Notes:

A service of the Electronic Journal of Combinatorics with links to databases, software, and other resources.

External Links:

World Combinatorics Exchange

Referenced by:

8th item.

Encodings:

BibTeX

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5: 19.16 Definitions

… ►It should be noted that the integrals (19.16.1)–(19.16.2_5) have been normalized so that $R_F(1,1,1)=R_J(1,1,1,1)=R_G(1,1,1)=1$. … ►which is homogeneous and of degree $-a$ in the $z$’s, and unchanged when the same permutation is applied to both sets of subscripts $1,\mathrm{\dots },n$. … ► $R_{-a}(𝐛;𝐳)$ is an elliptic integral iff the $z$’s are distinct and exactly four of the parameters $a,a^{\prime },b_1,\mathrm{\dots },b_n$ are half-odd-integers, the rest are integers, and none of $a$, $a^{\prime }$, $a+a^{\prime }$ is zero or a negative integer. The only cases that are integrals of the first kind are the four in which each of $a$ and $a^{\prime }$ is either $\frac{1}{2}$ or 1 and each $b_j$ is $\frac{1}{2}$. … ►The last $R$-function has $a=a^{\prime }=\frac{1}{2}$. …

6: 19.20 Special Cases

… ► Schneider that this is a transcendental number. … ►

$p=wy+(1-w)z,$

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… ► Schneider that this is a transcendental number. … ►Define $c={\sum }_{j=1}^n{b}_j$. …

7: Preface

… ►The new DLMF (Digital Library of Mathematical Functions) will appear in a hardcover edition and as a free electronic publication on the World Wide Web. The authors will review the relevant published literature and produce approximately twice the number of formulas that were contained in the original Handbook. …

8: 31.14 General Fuchsian Equation

… ►The general second-order Fuchsian equation with $N+1$ regular singularities at $z=a_j$, $j=1,2,\mathrm{\dots },N$, and at $\mathrm{\infty }$, is given by …The exponents at the finite singularities $a_j$ are $\{0,1-{\gamma }_j\}$ and those at $\mathrm{\infty }$ are $\{\alpha ,\beta \}$, where …The three sets of parameters comprise the singularity parameters $a_j$, the exponent parameters $\alpha ,\beta ,{\gamma }_j$, and the $N-2$ free accessory parameters $q_j$. With $a_1=0$ and $a_2=1$ the total number of free parameters is $3N-3$. … ►

31.14.3 $$w(z)=\left(\prod _{j=1}^N{(z-a_j)}^{-{\gamma }_j/2}\right)W(z),$$

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $W(z)$: solution

Permalink:

http://dlmf.nist.gov/31.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

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9: Notices

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Within each of the DLMF chapters themselves we will provide a list of research software for the functions discussed in that chapter. The purpose of these listings is to provide references to the research literature on the engineering of software for special functions. To qualify for listing, the development of the software must have been the subject of a research paper published in the peer-reviewed literature. If such software is available online for free download we will provide a link to the software.

In general, we will not index other software within DLMF chapters unless the software is unique in some way, such as being the only known software for computing a particular function.

… ►Please contact us if you have software that you believe should be listed.

10: Bibliography N

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The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-66289-8, MathReview (D. R. Heath-Brown), ZentralBlatt

Referenced by:

§27.12.

Encodings:

BibTeX

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Complex standard functions and their implementation in the CoStLy library. ACM Trans. Math. Softw. 33 (1), pp. Article 2.

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Notes:

For the software see CoStLy (free C-XSC library)

External Links:

ISSN 0098-3500, Document

Referenced by:

3rd item, CoStLy (free C-XSC library).

Encodings:

BibTeX

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Notes:

Double precision. In Japanese, with some English.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

Encodings:

BibTeX

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Notes:

Single and double precision. See Kahaner et al. (1989).

External Links:

GAMS (package NMS)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Notes:

References and links to software for factorization and primality testing.

External Links:

Home Page

Referenced by:

6th item.

Encodings:

BibTeX

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