Bernoulli%20numbers

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11: Bibliography R

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Topics in Analytic Number Theory. Springer-Verlag, New York.

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Notes:

Edited by E. Grosswald, J. Lehner and M. Newman, Die Grundlehren der mathematischen Wissenschaften, Band 169

External Links:

MathReview (H. Halberstam), ZentralBlatt

Referenced by:

§1.8(iv), §20.7(viii), §20.7(viii), Ch.24.

Encodings:

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Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,

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Referenced by:

§24.7(i).

Encodings:

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

BibTeX

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Elementary Number Theory and its Applications. 5th edition, Addison-Wesley, Reading, MA.

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External Links:

ISBN 0-201-87073-8, MathReview (Norman J. Richert)

Referenced by:

§27.17.

Encodings:

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12: 5.11 Asymptotic Expansions

… ►For the Bernoulli numbers $B_{2k}$, see §24.2(i). … ►Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. …For explicit formulas for $g_k$ in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of $g_k$ as $k\to \mathrm{\infty }$ see Boyd (1994) and Nemes (2015a). … ►where $h(\in ℂ)$ is fixed, and $B_k\left(h\right)$ is the Bernoulli polynomial defined in §24.2(i). … ►In terms of generalized Bernoulli polynomials $B_n^{(\mathrm{\ell })}\left(x\right)$ (§24.16(i)), we have for $k=0,1,\mathrm{\dots }$, …

13: Bibliography G

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

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A theorem on the numerators of the Bernoulli numbers. Amer. Math. Monthly 97 (2), pp. 136–138.

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External Links:

ISSN 0002-9890, FullText, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.10(iv), §24.16(iii).

Encodings:

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Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.

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External Links:

ISSN 0026-9255, MathReview (Peter Stevenhagen), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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External Links:

FullText, MathReview (B. M. Stewart), ZentralBlatt

Referenced by:

§24.15(iii), §24.6.

Encodings:

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

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14: 25.11 Hurwitz Zeta Function

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

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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

… ►For ${\tilde{B}}_n\left(x\right)$ see §24.2(iii). … ►

25.11.19 $${\zeta }^{\prime }(s,a)=-\frac{\ln a}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{{(s-1)}^2}+\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2)\ln \left(x+a\right)}{{(x+a)}^{s+2}}dx-\frac{(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$.

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Symbols:

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

Keywords:

derivative

Source:

Apostol (1985a, fourth equation, p. 231); with $k=1$, ${\phi }_2(x)=\frac{1}{2}({\tilde{B}}_2\left(x\right)-B_2)$

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.E19

Encodings:

pMML, png, TeX

Errata (effective with 1.0.12):

Originally both integrands were incorrect because their numerators contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$.

Reported 2016-06-27 by Gergő Nemes

See also:

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

… ►where $H_n$ are the harmonic numbers: … ►

25.11.34 $$n{\int }_0^a{\zeta }^{\prime }(1-n,x)dx={\zeta }^{\prime }(-n,a)-{\zeta }^{\prime }\left(-n\right)+\frac{B_{n+1}-B_{n+1}\left(a\right)}{n(n+1)},$$ $n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>0$.

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $a$: real or complex parameter

Keywords:

definite integral, integral representation

Source:

Adamchik (1998, (15), p. 196)

Permalink:

http://dlmf.nist.gov/25.11.E34

Encodings:

pMML, png, TeX

See also:

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

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15: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

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Traité Élémentaire des Nombres de Bernoulli. Gauthier-Villars, Paris.

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External Links:

ZentralBlatt

Referenced by:

§24.1.

Encodings:

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Mémoire sur les polynomes de Bernoulli. Acta Math. 43, pp. 121–196 (French).

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External Links:

Document, ZentralBlatt

Referenced by:

§24.13(i), §24.13(ii), §24.14(i), Ch.24.

Encodings:

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Notes:

References and links to software for factorization and primality testing.

External Links:

Home Page

Referenced by:

6th item.

Encodings:

BibTeX

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16: Bibliography D

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Bernoulli Numbers. Bibliography (1713–1990). Queen’s Papers in Pure and Applied Mathematics, Vol. 87, Queen’s University, Kingston, ON.

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Notes:

A frequently updated version, with links to reviews, exists online

External Links:

Online version, MathReview, ZentralBlatt

Referenced by:

Ch.24.

Encodings:

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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:

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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

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External Links:

ISSN 0022-314X, Document, MathReview (T. Metsänkylä)

Referenced by:

§24.12(iii).

Encodings:

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Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

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External Links:

ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

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Bernoulli Numbers and Confluent Hypergeometric Functions. In Number Theory for the Millennium, I (Urbana, IL, 2000), pp. 343–363.

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External Links:

ISBN 1-56881-126-8, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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17: Bibliography F

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

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Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

Encodings:

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Faster computation of Bernoulli numbers. J. Algorithms 13 (3), pp. 431–445.

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External Links:

ISSN 0196-6774, MathReview (P. K. Subramanian), ZentralBlatt

Referenced by:

§24.19(i).

Encodings:

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

Encodings:

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18: Bibliography W

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

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External Links:

FullText, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.20.

Encodings:

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

Encodings:

BibTeX

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

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External Links:

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

Referenced by:

§20.7(v), §20.8.

Encodings:

BibTeX

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19: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

BibTeX

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

Encodings:

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Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.16(iii), §24.8(ii).

Encodings:

BibTeX

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On the Euler and Bernoulli polynomials. J. Reine Angew. Math. 234, pp. 45–64.

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External Links:

ISSN 0075-4102, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.12(iv).

Encodings:

BibTeX

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Bernoulli numbers and polynomials of arbitrary complex indices. Appl. Math. Lett. 5 (6), pp. 83–88.

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External Links:

ISSN 0893-9659, MathReview, ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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20: 25.5 Integral Representations

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25.5.7 $$\zeta \left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum _{m=1}^n\frac{B_{2m}}{(2m)!}{\left(s\right)}_{2m-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}-\sum _{m=1}^n\frac{B_{2m}}{(2m)!}{x}^{2m-1}\right)\frac{x^{s-1}}{{\mathrm{e}}^x}dx,$$ $\mathrm{\Re }s>-(2n+1)$, $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Proof sketch:

Derivable from (25.5.6) by adding and subtracting ${\sum }_{m=1}^n{B}_{2m}{x}^{2m-1}/(2m)!$ in the integrand, using (5.2.1), (5.2.5), and recognizing that ${({\mathrm{e}}^x-1)}^{-1}-x^{-1}+1/2-{\sum }_{m=1}^n{B}_{2m}{x}^{2m-1}/(2m)!=O\left(x^{2n+1}\right)$ as $x\to 0$, demonstrating the region of convergence.

Referenced by:

Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/25.5.E7

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

We have rewritten the original summation ${\sum }_{m=1}^n\frac{B_{2m}}{(2m)!}\frac{\mathrm{\Gamma }\left(s+2m-1\right)}{\mathrm{\Gamma }\left(s\right)}$ more concisely as ${\sum }_{m=1}^n\frac{B_{2m}}{(2m)!}{\left(s\right)}_{2m-1}$ using the Pochhammer symbol.

See also:

Annotations for §25.5(i), §25.5 and Ch.25

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