relation to Bernoulli numbers

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11: Bibliography

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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:

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Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

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External Links:

FullText, MathReview (D. P. Banerjee), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

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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.

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Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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External Links:

ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt

Referenced by:

(25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.

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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:

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12: 25.16 Mathematical Applications

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§25.16(i) Distribution of Primes

… ►which is related to the Riemann zeta function by … ►The prime number theorem (27.2.3) is equivalent to the statement … ► $H\left(s\right)$ has a simple pole with residue $\zeta \left(1-2r\right)$ ($=-B_{2r}/(2r)$) at each odd negative integer $s=1-2r$, $r=1,2,3,\mathrm{\dots }$. … ►Related results are: …

13: Bibliography Z

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On the Computation of Zeros of Bessel and Bessel-related Functions. In Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.), Utrecht, pp. 409–416.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

Encodings:

BibTeX

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The Dilogarithm Function in Geometry and Number Theory. In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.), Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.

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External Links:

MathReview (J. Browkin), ZentralBlatt

Referenced by:

§25.12(i).

Encodings:

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A modified Bernoulli number. Nieuw Arch. Wisk. (4) 16 (1-2), pp. 63–72.

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External Links:

ISSN 0028-9825, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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Generalized Watson Transforms and Applications to Group Representations. Ph.D. Thesis, University of Vermont, Burlington,VT.

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Referenced by:

§35.7(ii).

Encodings:

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Approximations to -, di- and tri-logarithms. J. Comput. Appl. Math. 202 (2), pp. 450–459.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview (F. Beukers), ZentralBlatt

Referenced by:

§25.18(i), §25.18(i).

Encodings:

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14: Software Index

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This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

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These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

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A cross index of mathematical software in use at NIST.

15: Bibliography B

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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:

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

1st item.

Encodings:

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Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

2nd item, 1st item.

Encodings:

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Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.

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External Links:

Document, MathReview (C. V. Coffman), ZentralBlatt

Referenced by:

§10.21(xi).

Encodings:

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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:

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16: 24.12 Zeros

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§24.12(i) Bernoulli Polynomials: Real Zeros

►In the interval $0\le x\le 1$ the only zeros of $B_{2n+1}\left(x\right)$, $n=1,2,\mathrm{\dots }$, are $0,\frac{1}{2},1$, and the only zeros of $B_{2n}\left(x\right)-B_{2n}$, $n=1,2,\mathrm{\dots }$, are $0,1$. … ►Let $R(n)$ be the total number of real zeros of $B_n\left(x\right)$. … ►

§24.12(iii) Complex Zeros

… ►A related topic is the irreducibility of Bernoulli and Euler polynomials. …

17: 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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Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

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External Links:

ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

Referenced by:

§35.9.

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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The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent. Proc. Roy. Soc. London Ser. A 361, pp. 265–275.

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External Links:

FullText, MathReview (A. D. Brjuno), ZentralBlatt

Referenced by:

§32.11(ii), §32.17, §32.3(ii).

Encodings:

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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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18: 5.17 Barnes’ $G$-Function (Double Gamma Function)

§5.17 Barnes’ $G$-Function (Double Gamma Function)

… ►When $z\to \mathrm{\infty }$ in , ►

5.17.5 $$\mathrm{Ln}G\left(z+1\right)\sim \frac{1}{4}{z}^2+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\ln z-\ln A+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.$$

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $z$: complex variable and $A$: Glaisher’s constant

Referenced by:

§5.17, §5.17, Erratum (V1.0.7) for Equation (5.17.5), Erratum (V1.1.11) for Equation (5.17.5)

Permalink:

http://dlmf.nist.gov/5.17.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.1.11):

For consistency we have replaced $\mathrm{Ln}z$ by $\ln z$.

Errata (effective with 1.0.7):

Originally the term $z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$ was incorrectly stated as $z\mathrm{\Gamma }\left(z+1\right)$. (This error was reported subsequently by Nick Jones on December 11, 2013.)

Reported 2013-08-01 by Gergő Nemes

See also:

Annotations for §5.17 and Ch.5

… ►Here $B_{2k+2}$ is the Bernoulli number (§24.2(i)), and $A$ is Glaisher’s constant, given by … ►

5.17.7 $$C=\underset{n\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^{n}k\ln k-\left(\frac{1}{2}{n}^2+\frac{1}{2}n+\frac{1}{12}\right)\ln n+\frac{1}{4}{n}^2\right)=\frac{\gamma +\ln \left(2\pi \right)}{12}-\frac{{\zeta }^{\prime }\left(2\right)}{2{\pi }^2}=\frac{1}{12}-{\zeta }^{\prime }\left(-1\right),$$

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer, $k$: nonnegative integer and $C$: log of Glaisher’s Constant

Referenced by:

§5.17

Permalink:

http://dlmf.nist.gov/5.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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19: Bibliography F

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A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.

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Notes:

Table erratum: Math. Comp. v. 36 (1981), no. 153, p. 318. Part II of this report, with the same date but numbered ARL 69-0173, again puts $k=R\exp (i\theta )$ but with $R$ as parameter and $\theta$ as variable instead of the other way around. Part III, dated May 1970 and numbered ARL 70-0081, contains tables of auxiliary functions to help interpolation.

External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(ii), §19.37(ii).

Encodings:

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On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

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External Links:

Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§19.7(i).

Encodings:

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Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

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Notes:

Fortran, minimum precision 6D.

External Links:

ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(ix).

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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Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond. In Recent Perspectives in Random Matrix Theory and Number Theory, London Math. Soc. Lecture Note Ser., Vol. 322, pp. 31–78.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§18.38(ii).

Encodings:

BibTeX

20: Errata

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Section: 15.9(v) Complete Elliptic Integrals. Equations: (11.11.9_5), (11.11.13_5), Intermediate equality in (15.4.27) which relates to $F(a,a;a+1;\frac{1}{2})$, (15.4.34), (19.5.4_1), (19.5.4_2) and (19.5.4_3).

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The largest known prime, which is a Mersenne prime, was updated from $2^{43,112,609}-1$ (2009) to $2^{82,589,933}-1$ (2018).

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The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.

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Following a suggestion from James McTavish on 2017-04-06, the recurrence relation $u_k=\frac{(6k-5)(6k-3)(6k-1)}{(2k-1)216k}{u}_{k-1}$ was added to Equation (9.7.2).

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A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

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