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1: 24.1 Special Notation

… ►

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). …

… ►Elliptic integrals appear in lattice models of critical phenomena (Guttmann and Prellberg (1993)); theories of layered materials (Parkinson (1969)); fluid dynamics (Kida (1981)); string theory (Arutyunov and Staudacher (2004)); astrophysics (Dexter and Agol (2009)). …

3: Bibliography D

… ►

H. Davenport (2000) Multiplicative Number Theory. 3rd edition, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York.

ⓘ

Notes:: Revised and with a preface by Hugh L. Montgomery
External Links:: ISBN 0-387-95097-4, MathReview, ZentralBlatt
Referenced by:: §27.12.
Encodings:: BibTeX

… ►

J. B. Dence and T. P. Dence (1999) Elements of the Theory of Numbers. Harcourt/Academic Press, San Diego, CA.

ⓘ

External Links:: ISBN 0-12-209130-2, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: Ch.24.
Encodings:: BibTeX

… ►

J. Dexter and E. Agol (2009) A fast new public code for computing photon orbits in a Kerr spacetime. The Astrophysical Journal 696, pp. 1616–1629.

ⓘ

External Links:: Document
Referenced by:: §19.35(ii).
Encodings:: BibTeX

… ►

K. Dilcher (1996) Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

ⓘ

External Links:: ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt
Referenced by:: §24.14(ii).
Encodings:: BibTeX

►

K. Dilcher (2002) Bernoulli Numbers and Confluent Hypergeometric Functions. In Number Theory for the Millennium, I (Urbana, IL, 2000), pp. 343–363.

ⓘ

External Links:: ISBN 1-56881-126-8, MathReview (Arnold M. Adelberg), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

…

4: About the Project

… ►Since that time there have been a number of developments. … ►Ten Senior Assocate Editors have been named. …

5: Bibliography C

… ►

L. Carlitz (1953) Some congruences for the Bernoulli numbers. Amer. J. Math. 75 (1), pp. 163–172.

ⓘ

External Links:: FullText, MathReview (A. L. Whiteman), ZentralBlatt
Referenced by:: §24.10(i).
Encodings:: BibTeX

►

L. Carlitz (1954a)

q

-Bernoulli and Eulerian numbers. Trans. Amer. Math. Soc. 76 (2), pp. 332–350.

ⓘ

External Links:: FullText, MathReview (A. L. Whiteman), ZentralBlatt
Referenced by:: §17.3(iii), §24.16(iii).
Encodings:: BibTeX

►

L. Carlitz (1954b) A note on Euler numbers and polynomials. Nagoya Math. J. 7, pp. 35–43.

ⓘ

External Links:: MathReview (A. L. Whiteman), ZentralBlatt
Referenced by:: §24.10(iv).
Encodings:: BibTeX

… ►

C. W. Clenshaw and F. W. J. Olver (1984) Beyond floating point. J. Assoc. Comput. Mach. 31 (2), pp. 319–328.

ⓘ

External Links:: Document, MathReview Entry, ZentralBlatt
Referenced by:: §3.1(iv).
Encodings:: BibTeX

… ►

E. T. Copson (1963) On the asymptotic expansion of Airy’s integral. Proc. Glasgow Math. Assoc. 6, pp. 113–115.

ⓘ

External Links:: Document, MathReview (L. Berg), ZentralBlatt
Referenced by:: (9.5.7), §9.5(ii).
Encodings:: BibTeX

…

6: Bibliography B

… ►

B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.

ⓘ

External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §24.16(iii), §24.8(ii).
Encodings:: BibTeX

… ►

D. Bleichenbacher (1996) Efficiency and Security of Cryptosystems Based on Number Theory. Ph.D. Thesis, Swiss Federal Institute of Technology (ETH), Zurich.

ⓘ

External Links:: FullText
Referenced by:: 2nd item.
Encodings:: BibTeX

►

W. E. Bleick and P. C. C. Wang (1974) Asymptotics of Stirling numbers of the second kind. Proc. Amer. Math. Soc. 42 (2), pp. 575–580.

ⓘ

Notes:: Erratum: same journal v. 48 (1975), p. 518
External Links:: FullText, MathReview (H. A. Lauwerier), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

… ►

R. P. Brent (1976) Fast multiple-precision evaluation of elementary functions. J. Assoc. Comput. Mach. 23 (2), pp. 242–251.

ⓘ

External Links:: Document, MathReview (Amnon Barak), ZentralBlatt
Referenced by:: §4.45(i).
Encodings:: BibTeX

… ►

D. Bressoud and S. Wagon (2000) A Course in Computational Number Theory. Key College Publishing, Emeryville, CA.

ⓘ

Notes:: With 1 CD-ROM (Windows, Macintosh and UNIX) containing a collection of Mathematica programs for computations in number theory.
External Links:: ISBN 1-930190-10-7, MathReview (Samuel S. Wagstaff, Jr.), ZentralBlatt
Referenced by:: §27.12, §27.19, §27.21, 2nd item, Profile David M. Bressoud.
Encodings:: BibTeX

…

7: Roderick S. C. Wong

… ►Wong was elected a Fellow of the Royal Society of Canada in 1993, a Foreign Member of the Academy of Science of Turin, Italy, in 2001, a Chevalier dans l’Ordre National de la Légion d’Honneur in 2004, and a Member of the European Academy of Sciences in 2007. …

8: Bibliography K

… ►

M. Kaneko (1997) Poly-Bernoulli numbers. J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.

ⓘ

External Links:: ISSN 1246-7405, PostScript, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

… ►

N. M. Katz (1975) The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers. Math. Ann. 216 (1), pp. 1–4.

ⓘ

External Links:: ISSN 0025-5831, Document, MathReview (M. Basmakov), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

… ►

R. P. Kelisky (1957) On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.

ⓘ

External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §24.15(iv).
Encodings:: BibTeX

… ►

T. Kim and H. S. Kim (1999) Remark on

p

-adic

q

-Bernoulli numbers. Adv. Stud. Contemp. Math. (Pusan) 1, pp. 127–136.

ⓘ

Notes:: Algebraic number theory (Hapcheon/Saga, 1996)
External Links:: ISSN 1229-3067, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

… ►

D. E. Knuth (1986) METAFONT: The Program. Computers and Typesetting, Vol. D, Addison-Wesley, Reading, MA.

ⓘ

External Links:: ISBN 0-201-13438-1, ZentralBlatt
Referenced by:: §3.11(vi).
Encodings:: BibTeX

…

9: 27.18 Methods of Computation: Primes

§27.18 Methods of Computation: Primes

►An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer

x

is given in Crandall and Pomerance (2005, §3.7). …An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … ►These algorithms are used for testing primality of Mersenne numbers,

2^{n}-1

, and Fermat numbers,

2^{2^{n}}+1

. …

10: 26.11 Integer Partitions: Compositions

… ►

c\left(n\right)

denotes the number of compositions of

n

, and

c_{m}\left(n\right)

is the number of compositions into exactly

m

parts.

c\left(\in\!T,n\right)

is the number of compositions of

n

with no 1’s, where again

T=\{2,3,4,\ldots\}

. … ►

26.11.1

c\left(0\right)=c\left(\in\!T,0\right)=1.

ⓘ

Symbols:: $\in$ : element of, $c\left(\NVar{n}\right)$ : number of compositions of $n$ , $c\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : restricted number of compositions of $n$ and $T$ : set
Permalink:: http://dlmf.nist.gov/26.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.11 and Ch.26

… ►The Fibonacci numbers are determined recursively by … ►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).