summation by parts

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11: 2.11 Remainder Terms; Stokes Phenomenon

… ►By integration by parts (§2.3(i)) … ►However, regardless whether we can bound the remainder, the accuracy achievable by direct numerical summation of a divergent asymptotic series is always limited. … ►when $\mathrm{\Re }p>0$ and , and by analytic continuation for other values of $p$ and $z$. …

12: Bibliography M

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Numerical Methods for Roots of Polynomials. Part I. Studies in Computational Mathematics, Vol. 14, Elsevier, Amsterdam.

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External Links:

ISBN 978-0-444-52729-5; 0-444-52729-X, MathReview Entry, ZentralBlatt

Referenced by:

§3.8(iv).

Encodings:

BibTeX

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A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§16.10.

Encodings:

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A $q$-analog of the ${}_5{}^{}F_4^{}(1)$ summation theorem for hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ . Adv. in Math. 57 (1), pp. 14–33.

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External Links:

ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§17.15.

Encodings:

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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External Links:

ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.15.

Encodings:

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Balanced ${}_3{}^{}\Theta _2^{}$ summation theorems for $U(n)$ basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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External Links:

ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt

Referenced by:

§17.15.

Encodings:

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13: Errata

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The general constraint $\mathrm{\Re }s>2$ has been extended to $\mathrm{\Re }s>1$ for (20.10.1), (20.10.2) and to $\mathrm{\Re }s>0$ for (20.10.3).

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The constraint which originally read “$\mathrm{\Re }s>1$, ” has been extended to be “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $a=1$”.

Reported by Gergő Nemes on 2021-08-23

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The constraint which originally read “, $\mathrm{\Re }s>0$” has been extended to be “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$”.

Reported by Gergő Nemes on 2021-09-14

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23.18.7 $$s(d,c)=\sum _{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\lfloor \frac{dr}{c}\rfloor -\frac{1}{2}\right),$$ $c>0$

Originally the sum ${\sum }_{r=1}^{c-1}$ was written with an additional condition on the summation, that $\left(r,c\right)=1$. This additional condition was incorrect and has been removed.

Reported 2015-10-05 by Howard Cohl and Tanay Wakhare.

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26.7.6 $$B\left(n+1\right)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B\left(k\right)$$

Originally this equation appeared with $B\left(n\right)$ in the summation, instead of $B\left(k\right)$.

Reported 2010-11-07 by Layne Watson.

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14: 3.11 Approximation Techniques

… ►When $n>0$ and $0\le j\le n$, $0\le k\le n$, … ►Here the single prime on the summation symbol means that the first term is to be halved. … ►

Summation of Chebyshev Series: Clenshaw’s Algorithm

… ►Now suppose that $X_{k\mathrm{\ell }}=0$ when $k\ne \mathrm{\ell }$, that is, the functions ${\varphi }_k(x)$ are orthogonal with respect to weighted summation on the discrete set $x_1,x_2,\mathrm{\dots },x_J$. … …

15: 1.17 Integral and Series Representations of the Dirac Delta

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1.17.13 $$\delta \left(x-a\right)=x{\int }_0^{\mathrm{\infty }}t{J}_{\nu }\left(xt\right)J_{\nu }\left(at\right)dt,$$ $\mathrm{\Re }\nu >-1$, $x>0$, $a>0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Referenced by:

§1.18(vi)

Permalink:

http://dlmf.nist.gov/1.17.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(ii), §1.17(ii), §1.17 and Ch.1

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

16: Bibliography W

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Surface Waves. In Handbuch der Physik, Vol. 9, Part 3, pp. 446–778.

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External Links:

Online edition, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§11.12.

Encodings:

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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:

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17: 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 $k=n$. … ►

18.20.9 $$p_n(x;a,b,\overline{a},\overline{b})=\frac{{\mathrm{i}}^n{\left(a+\overline{a}\right)}_n{\left(a+\overline{b}\right)}_n}{n!}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,n+2\mathrm{\Re }\left(a+b\right)-1,a+\mathrm{i}x}{a+\overline{a},a+\overline{b}};1).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.19, §18.20(ii), §18.21(ii), §18.22(iii), §18.26(ii)

Permalink:

http://dlmf.nist.gov/18.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(ii), §18.20 and Ch.18

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