Bernoulli and Euler numbers and polynomials

SearchAdvancedHelp

(0.004 seconds)

21—30 of 42 matching pages

21: Bibliography W

… ►

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

ⓘ

External Links:

FullText, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.20.

Encodings:

BibTeX

… ►

The zeros of Euler’s psi function and its derivatives. J. Math. Anal. Appl. 332 (1), pp. 607–616.

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview, ZentralBlatt (Mehdi Hassani)

Referenced by:

§5.4(iii).

Encodings:

BibTeX

… ►

Global asymptotics of the Meixner polynomials. Asymptotic Analysis 75 (3-4), pp. 211–231.

ⓘ

External Links:

ISSN 0921-7134, Document, MathReview (Mario Pérez Riera), ZentralBlatt

Referenced by:

§18.24, §18.24.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§20.7(v), §20.8.

Encodings:

BibTeX

… ►

Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

ⓘ

Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

Encodings:

BibTeX

…

22: 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

►

The real zeros of the Bernoulli polynomials. J. Approx. Theory 58 (2), pp. 124–150.

ⓘ

External Links:

ISSN 0021-9045, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§24.12(i), §24.9.

Encodings:

BibTeX

… ►

Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.34(iii), §24.11, §24.16(i), §24.16(i).

Encodings:

BibTeX

►

Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions. Stud. Appl. Math. 103 (3), pp. 241–258.

ⓘ

External Links:

ISSN 0022-2526, Document, Link, MathReview, ZentralBlatt

Referenced by:

§24.11, §24.11.

Encodings:

BibTeX

… ►

Large degree asymptotics of generalized Bernoulli and Euler polynomials. J. Math. Anal. Appl. 363 (1), pp. 197–208.

ⓘ

External Links:

ISSN 0022-247X, Document, Link, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§24.16(i), §24.16(i).

Encodings:

BibTeX

…

23: 2.10 Sums and Sequences

… ►

§2.10(i) Euler–Maclaurin Formula

►As in §24.2, let $B_n$ and $B_n\left(x\right)$ denote the $n$th Bernoulli number and polynomial, respectively, and ${\tilde{B}}_n\left(x\right)$ the $n$th Bernoulli periodic function $B_n\left(x-\lfloor x\rfloor \right)$. … ►This is the Euler–Maclaurin formula. … ►From §24.12(i), (24.2.2), and (24.4.27), ${\tilde{B}}_{2m}\left(x\right)-B_{2m}$ is of constant sign ${(-1)}^m$. … ►

Example

…

24: 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

►

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

►

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

…

25: 25.11 Hurwitz Zeta Function

… ►

25.11.6 $$\zeta (s,a)=\frac{1}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $s\ne 1$, $\mathrm{\Re }s>-1$, $a>0$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Source:

Apostol (1985a, (25), p. 231)

Referenced by:

Erratum (V1.0.12) for Equations (25.11.6), (25.11.19), and (25.11.20)

Permalink:

http://dlmf.nist.gov/25.11.E6

Encodings:

pMML, png, TeX

Errata (effective with 1.0.12):

Originally the integrand was incorrect because its numerator contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$.

Reported 2016-05-08 by Clemens Heuberger

See also:

Annotations for §25.11(iii), §25.11 and Ch.25

►

25.11.7 $$\zeta (s,a)=\frac{1}{a^s}+\frac{1}{{(1+a)}^s}\left(\frac{1}{2}+\frac{1+a}{s-1}\right)+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}\frac{1}{{(1+a)}^{s+2k-1}}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{{(x+a)}^{s+2n+1}}dx,$$ $s\ne 1$, $a>0$, $n=1,2,3,\mathrm{\dots }$, $\mathrm{\Re }s>-2n$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Proof sketch:

Derivable from (25.11.5) by setting $N=1$ and repeatedly integrating by parts using (1.2.6), (24.2.2), (24.2.4), (24.2.11), (24.2.12), (24.4.34).

Permalink:

http://dlmf.nist.gov/25.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(iii), §25.11 and Ch.25

…

26: 25.1 Special Notation

… ► ►►►►►►►

$k,m,n$	nonnegative integers.
$p$	prime number.
…
$\gamma$	Euler’s constant (§5.2(ii)).
$\psi \left(x\right)$	digamma function ${\mathrm{\Gamma }}^{\prime }\left(x\right)/\mathrm{\Gamma }\left(x\right)$ except in §25.16. See §5.2(i).
$B_n,B_n\left(x\right)$	Bernoulli number and polynomial (§24.2(i)).
${\tilde{B}}_n\left(x\right)$	periodic Bernoulli function $B_n\left(x-\lfloor x\rfloor \right)$.
…

…

27: Bibliography T

… ►

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

…

28: 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

…

29: 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

…

30: Karl Dilcher

… ►Dilcher’s research interests include classical analysis, special functions, and elementary, combinatorial, and computational number theory. Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. … ►

24 Bernoulli and Euler Polynomials

…