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21: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

22: 23 Weierstrass Elliptic and Modular
Functions

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23: 26.7 Set Partitions: Bell Numbers

§26.7 Set Partitions: Bell Numbers

… ► $B\left(n\right)$ is the number of partitions of $\{1,2,\mathrm{\dots },n\}$. … ►

26.7.3 $$B\left(n\right)=\sum _{k=1}^m\frac{k^n}{k!}\sum _{j=0}^{m-k}\frac{{(-1)}^j}{j!},$$ $m\ge n$,

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Symbols:

$B\left(n\right)$: Bell number, $!$: factorial (as in $n!$), $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§26.7(i)

Permalink:

http://dlmf.nist.gov/26.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.7(i), §26.7 and Ch.26

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26.7.4 $$B\left(n\right)={\mathrm{e}}^{-1}\sum _{k=1}^{\mathrm{\infty }}\frac{k^n}{k!}=1+\lfloor {\mathrm{e}}^{-1}\sum _{k=1}^{2n}\frac{k^n}{k!}\rfloor .$$

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Symbols:

$B\left(n\right)$: Bell number, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.7(i), §26.7 and Ch.26

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26.7.5 $$\sum _{n=0}^{\mathrm{\infty }}B\left(n\right)\frac{x^n}{n!}=\exp ({\mathrm{e}}^x-1).$$

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Symbols:

$B\left(n\right)$: Bell number, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $x$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.7(ii), §26.7 and Ch.26

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24: Bibliography R

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On the partition function p(n). Proc. London Math. Soc. (2) 43 (4), pp. 241–254.

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External Links:

Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(iii).

Encodings:

BibTeX

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Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.

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External Links:

ISSN 0025-5874, ISSN 0025-5874, Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(v).

Encodings:

BibTeX

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

BibTeX

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Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.

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Notes:

Software packages for hypergeometric and $q$-hypergeometric summation, sequences and power series, permutation groups, partition analysis, and difference equations.

External Links:

RISC Combinatorics Group

Referenced by:

6th item.

Encodings:

BibTeX

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25: Bibliography G

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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Tables of Partitions. Royal Society Math. Tables, Vol. 4, Cambridge University Press.

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External Links:

ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

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A table of partitions. Proc. London Math. Soc. (2) 39, pp. 142–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

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A table of partitions (II). Proc. London Math. Soc. (2) 42, pp. 546–549.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

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26: 36 Integrals with Coalescing Saddles

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27: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

28: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

29: Bibliography O

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Inverse factorial-series solutions of difference equations. Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.

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External Links:

ISSN 0013-0915, Document, MathReview (Niro Yanagihara), ZentralBlatt

Referenced by:

§2.9(i).

Encodings:

BibTeX

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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Distribution of the partition function modulo $m$ . Ann. of Math. (2) 151 (1), pp. 293–307.

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External Links:

ISSN 0003-486X, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§27.14(v).

Encodings:

BibTeX

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30: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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