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11: Bibliography I

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IEEE (2018) IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017. The Institute of Electrical and Electronics Engineers, Inc..

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External Links:: ISBN 978-1-5044-4602-0, Document
Referenced by:: §3.1(ii), §3.1(ii), Erratum (V1.0.25) for Section 3.1.
Encodings:: BibTeX

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12: Bibliography C

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L. Carlitz (1960) Note on Nörlund’s polynomial

B_{n}^{(z)}

. Proc. Amer. Math. Soc. 11 (3), pp. 452–455.

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External Links:: FullText, MathReview (M. P. Drazin), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

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P. A. Clarkson and K. Jordaan (2018) Properties of generalized Freud polynomials. J. Approx. Theory 225, pp. 148–175.

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External Links:: ISSN 0021-9045, Document, Link, MathReview (Maria das Neves Rebocho)
Referenced by:: §18.32.
Encodings:: BibTeX

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P. A. Clarkson (2003b) The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

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J. A. Cochran (1963) Further formulas for calculating approximate values of the zeros of certain combinations of Bessel functions. IEEE Trans. Microwave Theory Tech. 11 (6), pp. 546–547.

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External Links:: ISSN 0018-9480, FullText
Referenced by:: §10.21(x).
Encodings:: BibTeX

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F. Cooper, A. Khare, and A. Saxena (2006) Exact elliptic compactons in generalized Korteweg-de Vries equations. Complexity 11 (6), pp. 30–34.

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External Links:: ISSN 1076-2787, Document, MathReview
Referenced by:: §19.6(i).
Encodings:: BibTeX

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13: Errata

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Paragraph Prime Number Theorem (in §27.12)

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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Version 1.0.21 (December 15, 2018)

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Version 1.0.20 (September 15, 2018)

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Version 1.0.19 (June 22, 2018)

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Version 1.0.18 (March 27, 2018)

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14: 11.11 Asymptotic Expansions of Anger–Weber Functions

… ►For sharp error bounds and exponentially-improved extensions, see Nemes (2018). …

15: 18.40 Methods of Computation

… ►Further, exponential convergence in

N

, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate

w(x)

for these OP systems on

x\in[-1,1]

and

(-\infty,\infty)

respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …

16: 18.14 Inequalities

… ►For further inequalities of this type see Koornwinder et al. (2018, §1) and references given there. …

17: 18.28 Askey–Wilson Class

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18.28.1_5

R_{n}(z)=R_{n}(z;a,b,c,d\,|\,q)=\frac{p_{n}\left(\frac{1}{2}(z+z^{-1});a,b,c,d% \,|\,q\right)}{a^{-n}\left(ab,ac,ad;q\right)_{n}}={{}_{4}\phi_{3}}\left({q^{-n% },abcdq^{n-1},az,az^{-1}\atop ab,ac,ad};q,q\right).

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Symbols:: $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\,|\,\NVar{q}\right)$ : Askey–Wilson polynomial, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $z$ : complex variable, $q$ : real variable and $n$ : nonnegative integer
Source:: Koornwinder and Mazzocco (2018, (1))
Referenced by:: (18.28.26), §18.28(ii), §18.28(ix), §18.28(x), §18.38(iii), Erratum (V1.2.0) Section 18.27
Permalink:: http://dlmf.nist.gov/18.28.E1_5
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.28(ii), §18.28 and Ch.18

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18: 26.2 Basic Definitions

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Table 26.2.1: Partitions

p\left(n\right)

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
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6	11	23	1255	40	37338
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9	30	26	2436	43	63261
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11	56	28	3718	45	89134
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13	101	30	5604	47	1 24754
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19: 24.2 Definitions and Generating Functions

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Table 24.2.3: Bernoulli numbers

B_{n}=N/D

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$n$	$N$	$D$	$B_{n}$
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$30$	$861\;58412\;76005$	$14322$	$6.01580\;8739$	$\times 10^{8}$
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Table 24.2.4: Euler numbers

E_{n}

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$n$	$E_{n}$
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$30$	$-44\;15438\;93249\;02310\;45536\;82821$
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Table 24.2.5: Coefficients

b_{n,k}

of the Bernoulli polynomials

B_{n}\left(x\right)=\sum_{k=0}^{n}b_{n,k}x^{k}

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	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
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$11$	$0$	$\tfrac{5}{6}$	$0$	$-\tfrac{11}{2}$	$0$	$11$	$0$	$-11$	$0$	$\tfrac{55}{6}$	$-\tfrac{11}{2}$	$1$
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Table 24.2.6: Coefficients

e_{n,k}

of the Euler polynomials

E_{n}\left(x\right)=\sum_{k=0}^{n}e_{n,k}x^{k}

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	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
…

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20: 11 Struve and Related Functions

Chapter 11 Struve and Related Functions

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