Boole summation formula

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1: 24.17 Mathematical Applications

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§24.17(i) Summation

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Euler–Maclaurin Summation Formula

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Boole Summation Formula

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2: Bibliography H

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Formulae for growth factors in expanding universes containing matter and a cosmological constant. Monthly Notices Roy. Astronom. Soc. 322 (2), pp. 419–425.

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External Links:

Document

Referenced by:

§8.24(ii).

Encodings:

BibTeX

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Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. (2) 17, pp. 75–115.

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Notes:

Reprinted in Ramanujan (1962, pp. 276–309).

External Links:

Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(iii).

Encodings:

BibTeX

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A Boole-type Formula involving Conjugate Euler Polynomials. In Charlemagne and his Heritage. 1200 Years of Civilization and Science in Europe, Vol. 2 (Aachen, 1995), P.L. Butzer, H. Th. Jongen, and W. Oberschelp (Eds.), pp. 361–375.

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External Links:

ISBN 2-503-50674-7, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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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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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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3: 5.19 Mathematical Applications

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§5.19(i) Summation of Rational Functions

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4: 1.8 Fourier Series

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Parseval’s Formula

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§1.8(iv) Poisson’s Summation Formula

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1.8.16 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-{(n+x)}^2\omega }=\sqrt{\frac{\pi }{\omega }}\left(1+2\sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-n^2{\pi }^2/\omega }\cos \left(2n\pi x\right)\right),$$ $\mathrm{\Re }\omega >0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $n$: nonnegative integer

Referenced by:

§1.8(iv)

Permalink:

http://dlmf.nist.gov/1.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(iv), §1.8 and Ch.1

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5: 28.34 Methods of Computation

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Representations for $w_{\text{I}}(\pi ;a,\pm q)$ with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed $a$ and $q$; see Schäfke (1961a).

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Summation of the power series in §§28.6(i) and 28.15(i) when $\left|q\right|$ is small.

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Summation of the power series in §§28.6(ii) and 28.15(ii) when $\left|q\right|$ is small.

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Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of $q$ and $z$.

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6: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

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Ramanujan’s ${}_1{}^{}\psi _1^{}$ Summation

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Bailey’s Bilateral Summations

… ►For similar formulas see Verma and Jain (1983).

7: 2.10 Sums and Sequences

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§2.10(i) Euler–Maclaurin Formula

… ►This is the Euler–Maclaurin formula. Another version is the Abel–Plana formula: … ►

§2.10(ii) Summation by Parts

►The formula for summation by parts is …

8: Bibliography Z

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Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

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Notes:

Includes hypergeometric and $q$-hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).

Encodings:

BibTeX

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The summation of series of hyperbolic functions. SIAM J. Math. Anal. 10 (1), pp. 192–206.

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External Links:

ISSN 0036-1410, Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§4.41.

Encodings:

BibTeX

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Tables and formulae for the spherical functions $P_{-1/2+i\tau }^m(z)$ . Translated by E. L. Albasiny, Pergamon Press, Oxford.

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Notes:

Translation of Žurina and Karmazina, Tablitsy i formuly dlya sfericheskikh funktsii $P_{-1/2+i\tau }^m(z)$ , Vyčisl. Centr Akad Nauk SSSR, Moscow. 1962

External Links:

MathReview, ZentralBlatt

Referenced by:

§14.20(vii).

Encodings:

BibTeX

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9: 3.5 Quadrature

… ►See also Poisson’s summation formula (§1.8(iv)). … ► … ►

Gauss–Legendre Formula

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Gauss–Laguerre Formula

… ►a complex Gauss quadrature formula is available. …

10: 18.3 Definitions

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As given by a Rodrigues formula (18.5.5).

… ►For representations of the polynomials in Table 18.3.1 by Rodrigues formulas, see §18.5(ii). … ►In this chapter, formulas for the Chebyshev polynomials of the second, third, and fourth kinds will not be given as extensively as those of the first kind. However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6). … ►Formula (18.3.1) can be understood as a Gauss-Chebyshev quadrature, see (3.5.22), (3.5.23). …