Bernoulli and Euler polynomials

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21: Software Index

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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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22: 17.3 $q$-Elementary and $q$-Special Functions

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§17.3(iii) Bernoulli Polynomials; Euler and Stirling Numbers

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23: 18.2 General Orthogonal Polynomials

… ►The Bernoulli polynomials $B_n\left(x\right)$ and Euler polynomials $E_n\left(x\right)$ are examples of Sheffer polynomials which are not OP’s, see the generating functions (24.2.3) and (24.2.8). For other examples of Sheffer polynomials, not in DLMF, see Roman (1984). …

24: Bibliography H

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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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25: Bibliography D

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Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials. J. Approx. Theory 49 (4), pp. 321–330.

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External Links:

ISSN 0021-9045, Document, MathReview (Guillermo López Lagomasino), ZentralBlatt

Referenced by:

§24.11.

Encodings:

BibTeX

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Zeros of Bernoulli, generalized Bernoulli and Euler polynomials. Mem. Amer. Math. Soc. 73 (386), pp. iv+94.

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External Links:

ISSN 0065-9266, MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i), §24.12(iii), §24.16(ii).

Encodings:

BibTeX

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26: Bibliography L

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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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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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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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27: Bibliography T

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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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28: 5.11 Asymptotic Expansions

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5.11.8 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)\sim \left(z+h-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=2}^{\mathrm{\infty }}\frac{{(-1)}^k{B}_k\left(h\right)}{k(k-1)z^{k-1}},$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§12.10(ii), (5.11.8), §5.11(i), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8), Erratum (V1.0.11) for Equation (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E8

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.11):

Originally $h$ in (5.11.8) was unnecessarily restricted to lie in the interval $[0,1]$. In fact, $h$ may lie anywhere in the complex plane.

Suggested 2015-02-28 by Nico Temme

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+h\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

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29: 25.6 Integer Arguments

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§25.6(i) Function Values

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30: 3.11 Approximation Techniques

… ►For splines based on Bernoulli and Euler polynomials, see §24.17(ii). …