relation to Bernoulli numbers

SearchAdvancedHelp

(0.003 seconds)

1—10 of 30 matching pages

1: 24.15 Related Sequences of Numbers

… ►

§24.15(i) Genocchi Numbers

… ►

§24.15(ii) Tangent Numbers

… ►

§24.15(iii) Stirling Numbers

…

2: 24.4 Basic Properties

… ►

§24.4(ix) Relations to Other Functions

►For the relation of Bernoulli numbers to the Riemann zeta function see §25.6, and to the Eulerian numbers see (26.14.11).

3: 26.8 Set Partitions: Stirling Numbers

… ►

§26.8(vi) Relations to Bernoulli Numbers

…

4: 24.17 Mathematical Applications

… ►

§24.17(iii) Number Theory

…

5: 24.19 Methods of Computation

… ►

§24.19(i) Bernoulli and Euler Numbers and Polynomials

►Equations (24.5.3) and (24.5.4) enable $B_n$ and $E_n$ to be computed by recurrence. … ►

§24.19(ii) Values of $B_n$ Modulo $p$

►For number-theoretic applications it is important to compute $B_{2n}(modp)$ for $2n\le p-3$; in particular to find the irregular pairs $(2n,p)$ for which $B_{2n}\equiv 0(modp)$. We list here three methods, arranged in increasing order of efficiency. …

6: 25.6 Integer Arguments

… ►

§25.6(i) Function Values

…

7: 24.16 Generalizations

§24.16 Generalizations

… ►

Degenerate Bernoulli Numbers

… ►

§24.16(ii) Character Analogs

… ►Generalized Bernoulli numbers and polynomials belonging to $\chi$ are defined by … ►In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); $p$-adic integer order Bernoulli numbers (Adelberg (1996)); $p$-adic $q$-Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).

8: 25.1 Special Notation

… ► ►►►►►►

$k,m,n$	nonnegative integers.
$p$	prime number.
…
$B_n,B_n\left(x\right)$	Bernoulli number and polynomial (§24.2(i)).
${\tilde{B}}_n\left(x\right)$	periodic Bernoulli function $B_n\left(x-\lfloor x\rfloor \right)$.
…
primes	on function symbols: derivatives with respect to argument.

… ►The main related functions are the Hurwitz zeta function $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.

9: Bibliography T

… ►

New congruences for the Bernoulli numbers. Math. Comp. 48 (177), pp. 341–350.

ⓘ

External Links:

ISSN 0025-5718, FullText, MathReview (Wells Johnson), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Bernoulli polynomials old and new: Generalizations and asymptotics. CWI Quarterly 8 (1), pp. 47–66.

ⓘ

External Links:

ISSN 0922-5366, FullText, MathReview (Michael Hauss), ZentralBlatt

Referenced by:

§24.11, §24.16(i).

Encodings:

BibTeX

… ►

Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.

ⓘ

External Links:

ISSN 0025-5858, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.4(v), §24.6.

Encodings:

BibTeX

… ►

On the theory of the Bernoulli polynomials and numbers. J. Math. Anal. Appl. 104 (2), pp. 309–350.

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview (Louis Shapiro), ZentralBlatt

Referenced by:

§24.15(iii).

Encodings:

BibTeX

►

Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.

ⓘ

External Links:

Document

Referenced by:

§31.17(ii).

Encodings:

BibTeX

…

10: Bibliography S

… ►

A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt

Referenced by:

§28.32(i).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§24.10(ii).

Encodings:

BibTeX

►

About von Staudt congruences for Bernoulli numbers. Comment. Math. Univ. St. Paul. 48 (2), pp. 137–144.

ⓘ

External Links:

ISSN 0010-258X, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.10(ii).

Encodings:

BibTeX

►

On the generalized Bernoulli numbers that belong to unequal characters. Rev. Mat. Iberoamericana 16 (3), pp. 459–475.

ⓘ

External Links:

ISSN 0213-2230, MathReview (Aichi Kudo), ZentralBlatt

Referenced by:

§24.4(iii).

Encodings:

BibTeX

►

Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

Encodings:

BibTeX

…