Bernoulli%20numbers

SearchAdvancedHelp

(0.001 seconds)

1—10 of 21 matching pages

1: Bibliography L

… ►

An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

ⓘ

External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

BibTeX

…

2: 25.6 Integer Arguments

… ►

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

ⓘ

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

…

3: 24.2 Definitions and Generating Functions

§24.2 Definitions and Generating Functions

►

§24.2(i) Bernoulli Numbers and Polynomials

… ►

§24.2(ii) Euler Numbers and Polynomials

… ►

§24.2(iii) Periodic Bernoulli and Euler Functions

… ►

Table 24.2.1: Bernoulli and Euler numbers.

► ►►

$n$	$B_n$	$E_n$
…

►

…

4: Bibliography I

… ►

The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

ⓘ

External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

Encodings:

BibTeX

… ►

A Classical Introduction to Modern Number Theory. 2nd edition, Springer-Verlag, New York.

ⓘ

External Links:

ISBN 0-387-97329-X, MathReview (Glenn Stevens), ZentralBlatt

Referenced by:

§24.10(i), §24.10(ii), §24.10(iii), §24.17(iii).

Encodings:

BibTeX

…

5: Software Index

… ► ►►►►►►

	Open Source													With Book						Commercial
…
20 Theta Functions
…
24 Bernoulli and Euler Polynomials
24.21(ii) $B_n$, $B_n\left(x\right)$, $E_n$, $E_n\left(x\right)$	✓	✓						✓		✓		a	✓	✓			✓		✓		✓	✓	✓		Derive, MuPAD
…
27 Functions of Number Theory
…

… ►

An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

…

6: 24.20 Tables

§24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. …

7: Bibliography

… ►

On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

ⓘ

External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

►

Congruences of $p$-adic integer order Bernoulli numbers. J. Number Theory 59 (2), pp. 374–388.

ⓘ

External Links:

ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

… ►

Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

ⓘ

External Links:

FullText, MathReview (D. P. Banerjee), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

… ►

Sharp bounds for the Bernoulli numbers. Arch. Math. (Basel) 74 (3), pp. 207–211.

ⓘ

External Links:

ISSN 0003-889X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.9.

Encodings:

BibTeX

… ►

A primer on Bernoulli numbers and polynomials. Math. Mag. 81 (3), pp. 178–190.

ⓘ

External Links:

ISSN 0025-570X, MathReview Entry, ZentralBlatt

Referenced by:

§24.5(ii), Ch.24.

Encodings:

BibTeX

…

8: Bibliography C

… ►

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

►

$q$-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

… ►

Expansions of $q$-Bernoulli numbers. Duke Math. J. 25 (2), pp. 355–364.

ⓘ

External Links:

Document, MathReview (A. L. Whiteman), ZentralBlatt

Referenced by:

§17.3(iii).

Encodings:

BibTeX

… ►

Accélération de calcul de nombres de Bernoulli. J. Number Theory 28 (3), pp. 347–362 (French).

ⓘ

External Links:

ISSN 0022-314X, Document, MathReview (Louis Shapiro), ZentralBlatt

Referenced by:

§24.19(i).

Encodings:

BibTeX

►

Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

ⓘ

External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

…

9: Bibliography K

… ►

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

… ►

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

… ►

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

… ►

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

… ►

Computation of tangent, Euler, and Bernoulli numbers. Math. Comp. 21 (100), pp. 663–688.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§24.15(ii), §24.19(i).

Encodings:

BibTeX

…

10: 26.14 Permutations: Order Notation

… ►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 Eulerian number, denoted , is the number of permutations in ${𝔖}_n$ with exactly $k$ descents. … ►

§26.14(iii) Identities

►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)). …