Bernoulli lemniscate

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1: 24.1 Special Notation

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Bernoulli Numbers and Polynomials

►The origin of the notation

B_{n}

B_{n}\left(x\right)

, is not clear. …

2: 19.30 Lengths of Plane Curves

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§19.30(iii) Bernoulli’s Lemniscate

►For

0\leq\theta\leq\tfrac{1}{4}\pi

, the arclength

s

of Bernoulli’s lemniscate …The perimeter length

P

of the lemniscate is given by …

3: 24.18 Physical Applications

§24.18 Physical Applications

►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). …

4: Bibliography T

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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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J. Todd (1975) The lemniscate constants. Comm. ACM 18 (1), pp. 14–19.

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Notes:: Collection of articles honoring Alston S. Householder. For corrigendum see same journal v. 18 (1975), no. 8, p. 462
External Links:: ISSN 0001-0782, Document, MathReview (L. Fox), ZentralBlatt
Referenced by:: §19.20(i), §19.20(iv), §19.36(iii).
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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5: 24.3 Graphs

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See accompanying text — Figure 24.3.1: Bernoulli polynomials $B_{n}\left(x\right)$ , $n=2,3,\dots,6$ . Magnify

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6: 24.4 Basic Properties

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§24.4(ii) Symmetry

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§24.4(v) Multiplication Formulas

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Raabe’s Theorem

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§24.4(vii) Derivatives

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§24.4(ix) Relations to Other Functions

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7: 24.16 Generalizations

§24.16 Generalizations

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Bernoulli Numbers of the Second Kind

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Degenerate Bernoulli Numbers

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§24.16(ii) Character Analogs

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§24.16(iii) Other Generalizations

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8: 24.19 Methods of Computation

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§24.19(i) Bernoulli and Euler Numbers and Polynomials

… ►For algorithms for computing

B_{n}

E_{n}

B_{n}\left(x\right)

, and

E_{n}\left(x\right)

see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180). ►

§24.19(ii) Values of $B_{n}$ Modulo $p$

… ►We list here three methods, arranged in increasing order of efficiency. ►

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Tanner and Wagstaff (1987) derives a congruence $\pmod{p}$ for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).

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9: 24.14 Sums

§24.14 Sums

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§24.14(i) Quadratic Recurrence Relations

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24.14.2

\sum_{k=0}^{n}{n\choose k}B_{k}B_{n-k}=(1-n)B_{n}-nB_{n-1}.

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(i), §24.14 and Ch.24

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§24.14(ii) Higher-Order Recurrence Relations

… ►For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).

10: 24.13 Integrals

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§24.13(i) Bernoulli Polynomials

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24.13.4

\int_{0}^{1/2}B_{n}\left(t\right)\,\mathrm{d}t=\frac{1-2^{n+1}}{2^{n}}\frac{B_% {n+1}}{n+1},

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex
Permalink:: http://dlmf.nist.gov/24.13.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(i), §24.13 and Ch.24

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24.13.6

\int_{0}^{1}B_{n}\left(t\right)B_{m}\left(t\right)\,\mathrm{d}t=\frac{(-1)^{n-% 1}m!n!}{(m+n)!}B_{m+n}.

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $m$ : integer, $n$ : integer and $t$ : real or complex
A&S Ref:: 23.1.12
Referenced by:: §24.13(i)
Permalink:: http://dlmf.nist.gov/24.13.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(i), §24.13 and Ch.24

►For integrals of the form

\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)\,\mathrm{d}t

and

\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)B_{k}\left(t\right)\,\mathrm% {d}t

see Agoh and Dilcher (2011). … ►

§24.13(iii) Compendia

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