summation formulas

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1: 24.17 Mathematical Applications

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Euler–Maclaurin Summation Formula

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Boole Summation Formula

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2: 1.8 Fourier Series

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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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3: Bibliography V

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Certain summation formulae for $q$-series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).

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External Links:

ISSN 0019-5839, MathReview Entry, ZentralBlatt

Referenced by:

(17.6.4_5), §17.6(i), (17.8.8), §17.8.

Encodings:

BibTeX

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4: Bibliography B

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Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.

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External Links:

Document, MathReview (T. M. Apostol), ZentralBlatt

Referenced by:

§24.16(ii).

Encodings:

BibTeX

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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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5: 1.17 Integral and Series Representations of the Dirac Delta

… ►Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): …

6: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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17.7.11 $${}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},q^{n+1},c,-c}{e,c^{2}q/e,-q};q,q)=q^{\left(\genfrac{}{}{0.0pt}{}{n+1}{2}\right)}\frac{{(e{q}^{-n},e{q}^{n+1},c^2{q}^{1-n}/e,c^2{q}^{n+2}/e;q^2)}_{\mathrm{\infty }}}{{(e,c^{2}q/e;q)}_{\mathrm{\infty }}}.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Referenced by:

Erratum (V1.1.12) for Equation (17.7.11)

Permalink:

http://dlmf.nist.gov/17.7.E11

Encodings:

pMML, png, TeX

Correction (effective with 1.1.12):

The missing factor $q^{\left(\genfrac{}{}{0.0pt}{}{n+1}{2}\right)}$ was inserted on the right-hand side of this summationformula.

See also:

Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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7: 1.14 Integral Transforms

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Poisson’s Summation Formula

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8: 3.5 Quadrature

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

9: 2.10 Sums and Sequences

… ►The formula for summation by parts is …

10: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:

ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

Encodings:

BibTeX

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