DLMF: Untitled Document

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§24.15(i) Genocchi Numbers

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24.15.2

G_{n}=2(1-2^{n})B_{n}.

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $n$ : integer and $G_{n}$ : Genocchi numbers
Permalink:: http://dlmf.nist.gov/24.15.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §24.15(i), §24.15 and Ch.24

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§24.15(ii) Tangent Numbers

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24.15.4

T_{2n-1}=(-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}B_{2n},

n=1,2,\dots

,

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $n$ : integer and $T_{n}$ : tangent numbers
Referenced by:: §24.19(i)
Permalink:: http://dlmf.nist.gov/24.15.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.15(ii), §24.15 and Ch.24

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§24.15(iii) Stirling Numbers

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	Open Source						With Book			Commercial
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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
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M. Hauss (1997) An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to

\zeta(2m+1)

. Commun. Appl. Anal. 1 (1), pp. 15–32.

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External Links:: ISSN 1083-2564, MathReview (Pavel Guerzhoy), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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K. Horata (1989) An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.

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External Links:: ZentralBlatt
Referenced by:: §24.6.
Encodings:: BibTeX

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K. Horata (1991) On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.

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External Links:: ZentralBlatt
Referenced by:: §24.15(iii).
Encodings:: BibTeX

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F. T. Howard (1996a) Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.

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External Links:: ISSN 0012-365X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(i).
Encodings:: BibTeX

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I. Huang and S. Huang (1999) Bernoulli numbers and polynomials via residues. J. Number Theory 76 (2), pp. 178–193.

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

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J. W. Tanner and S. S. Wagstaff (1987) New congruences for the Bernoulli numbers. Math. Comp. 48 (177), pp. 341–350.

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External Links:: ISSN 0025-5718, FullText, MathReview (Wells Johnson), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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N. M. Temme (1995b) Bernoulli polynomials old and new: Generalizations and asymptotics. CWI Quarterly 8 (1), pp. 47–66.

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External Links:: ISSN 0922-5366, FullText, MathReview (Michael Hauss), ZentralBlatt
Referenced by:: §24.11, §24.16(i).
Encodings:: BibTeX

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P. G. Todorov (1991) Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.

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External Links:: ISSN 0025-5858, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.4(v), §24.6.
Encodings:: BibTeX

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P. G. Todorov (1984) On the theory of the Bernoulli polynomials and numbers. J. Math. Anal. Appl. 104 (2), pp. 309–350.

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External Links:: ISSN 0022-247X, Document, MathReview (Louis Shapiro), ZentralBlatt
Referenced by:: §24.15(iii).
Encodings:: BibTeX

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… ►In (4.19.3)–(4.19.9),

B_{n}

are the Bernoulli numbers and

E_{n}

are the Euler numbers (§§24.2(i)–24.2(ii)). ►

4.19.3

\tan z=z+\frac{z^{3}}{3}+\frac{2}{15}z^{5}+\frac{17}{315}z^{7}+\cdots+\frac{(-% 1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,

|z|<\frac{1}{2}\pi

,

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\tan\NVar{z}$ : tangent function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.67
Referenced by:: §4.19
Permalink:: http://dlmf.nist.gov/4.19.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

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4.19.4

\csc z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}z^{3}+\frac{31}{15120}z^{5}+\cdots% +\frac{(-1)^{n-1}2(2^{2n-1}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,

0<|z|<\pi

,

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $!$ : factorial (as in $n!$ ), $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.68
Referenced by:: §4.33
Permalink:: http://dlmf.nist.gov/4.19.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

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4.19.6

\cot z=\frac{1}{z}-\frac{z}{3}-\frac{z^{3}}{45}-\frac{2}{945}z^{5}-\cdots-% \frac{(-1)^{n-1}2^{2n}B_{2n}}{(2n)!}z^{2n-1}-\cdots,

0<|z|<\pi

,

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $!$ : factorial (as in $n!$ ), $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.70
Permalink:: http://dlmf.nist.gov/4.19.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

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4.19.7

\ln\left(\frac{\sin z}{z}\right)=\sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}B_{2% n}}{n(2n)!}z^{2n},

|z|<\pi

,

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.71
Permalink:: http://dlmf.nist.gov/4.19.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

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D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

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D. J. Leeming (1989) The real zeros of the Bernoulli polynomials. J. Approx. Theory 58 (2), pp. 124–150.

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External Links:: ISSN 0021-9045, Document, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §24.12(i), §24.9.
Encodings:: BibTeX

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J. L. López and N. M. Temme (1999b) Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.

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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

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J. L. López and N. M. Temme (1999c) Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions. Stud. Appl. Math. 103 (3), pp. 241–258.

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External Links:: ISSN 0022-2526, Document, Link, MathReview, ZentralBlatt
Referenced by:: §24.11, §24.11.
Encodings:: BibTeX

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J. L. López and N. M. Temme (2010b) Large degree asymptotics of generalized Bernoulli and Euler polynomials. J. Math. Anal. Appl. 363 (1), pp. 197–208.

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External Links:: ISSN 0022-247X, Document, Link, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §24.16(i), §24.16(i).
Encodings:: BibTeX

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4.33.3

\tanh z=z-\frac{z^{3}}{3}+\frac{2}{15}z^{5}-\frac{17}{315}z^{7}+\cdots+\frac{2% ^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,

|z|<\frac{1}{2}\pi

.

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\tanh\NVar{z}$ : hyperbolic tangent function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.5.64
Permalink:: http://dlmf.nist.gov/4.33.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.33 and Ch.4

►For

B_{2n}

see §24.2(i). …

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§24.20 Tables

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A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

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A. Adelberg (1996) Congruences of

p

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

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External Links:: ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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H. Alzer (2000) Sharp bounds for the Bernoulli numbers. Arch. Math. (Basel) 74 (3), pp. 207–211.

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External Links:: ISSN 0003-889X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.9.
Encodings:: BibTeX

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T. M. Apostol (2006) Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

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Referenced by:: §24.4(iii).
Encodings:: BibTeX

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T. M. Apostol (2008) A primer on Bernoulli numbers and polynomials. Math. Mag. 81 (3), pp. 178–190.

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External Links:: ISSN 0025-570X, MathReview Entry, ZentralBlatt
Referenced by:: §24.5(ii), Ch.24.
Encodings:: BibTeX

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as Bernoulli or Euler polynomials

31—40 of 148 matching pages

31: 24.15 Related Sequences of Numbers

§24.15(i) Genocchi Numbers

§24.15(ii) Tangent Numbers

§24.15(iii) Stirling Numbers

32: Software Index

33: Bibliography H

34: Bibliography T

35: 4.19 Maclaurin Series and Laurent Series

36: Bibliography L

37: 4.33 Maclaurin Series and Laurent Series

38: B. L. J. Braaksma

39: 24.20 Tables

§24.20 Tables

40: Bibliography