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31: 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)): …

32: Bibliography W

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Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports 10 (5-6), pp. 189–371.

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External Links:

Document

Referenced by:

§2.11(vi), §3.9(v), §3.9(vi).

Encodings:

BibTeX

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33: 18.20 Hahn Class: Explicit Representations

… ►Here we use as convention for (16.2.1) with $b_q=-N$, $a_1=-n$, and $n=0,1,\mathrm{\dots },N$ that the summation on the right-hand side ends at $k=n$. …

34: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

… ►The double sums in (22.12.2)–(22.12.4) are convergent but not absolutely convergent, hence the order of the summations is important. …

35: 1.15 Summability Methods

… ►Methods of summation are regular if they are consistent with conventional summation. …

36: 11.10 Anger–Weber Functions

… ►where the prime on the second summation symbols means that the first term is to be halved. …

37: Bibliography

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Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

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External Links:

MathReview (R. N. Jain), ZentralBlatt

Referenced by:

§17.11.

Encodings:

BibTeX

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38: Bibliography K

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An extension of Saalschütz’s summation theorem for the series ${}_{r+3}{}^{}F_{r+2}^{}$ . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Christian Lavault), ZentralBlatt

Referenced by:

§16.4(ii), §16.4(ii).

Encodings:

BibTeX

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39: 8.12 Uniform Asymptotic Expansions for Large Parameter

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8.12.18 $$\begin{array}{c}Q(a,z)\\ P(a,z)\end{array}\}\sim \frac{z^{a-\frac{1}{2}}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a\right)}\left(d(\pm \chi )\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\chi )}{z^{k/2}}\mp \sum _{k=1}^{\mathrm{\infty }}\frac{B_k(\chi )}{z^{k/2}}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\chi$ , $d(\pm \chi )$ , $A_k(\chi )$ and $B_k(\chi )$

Referenced by:

§8.12, Erratum (V1.0.16) for Equation (8.12.18)

Permalink:

http://dlmf.nist.gov/8.12.E18

Encodings:

pMML, png, TeX

Errata (effective with 1.0.16):

The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.

Reported 2017-01-28 by Richard Paris

See also:

Annotations for §8.12 and Ch.8

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40: 19.21 Connection Formulas

… ►where both summations extend over the three cyclic permutations of $x,y,z$. …