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21: 26.8 Set Partitions: Stirling Numbers

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Table 26.8.1: Stirling numbers of the first kind $s(n,k)$.

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$n$	$k$
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10	$0$	$-\mathrm{3\hspace{0.17em}62880}$	$\mathrm{10\hspace{0.17em}26576}$	$-\mathrm{11\hspace{0.17em}72700}$	$\mathrm{7\hspace{0.17em}23680}$	$-\mathrm{2\hspace{0.17em}69325}$	$63273$	$-9450$	$870$	$-45$	$1$

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Table 26.8.2: Stirling numbers of the second kind $S(n,k)$.

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$n$	$k$
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10	0	1	511	9330	34105	42525	22827	5880	750	45	1

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22: Bibliography F

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A note on the asymptotic expansion of a ratio of gamma functions. Proc. Edinburgh Math. Soc. (2) 15, pp. 43–45.

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External Links:

ISSN 0013-0915, Document, Link, MathReview (H. A. Lauwerier), ZentralBlatt

Referenced by:

(5.11.14), §5.11(iii), Erratum (V1.0.21) for Equation (5.11.14).

Encodings:

BibTeX

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A note on asymptotic evaluation of some Hankel transforms. Math. Comp. 45 (172), pp. 537–548.

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External Links:

ISSN 0025-5718, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.22(v), §18.15(i), §18.16(ii).

Encodings:

BibTeX

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23: Bibliography H

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Definitions of Stieltjes Integrals of the Riemann Type. Amer. Math. Monthly 45 (5), pp. 265–278.

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External Links:

ISSN 0002-9890, Document, Link, MathReview Entry

Referenced by:

§1.16(ix).

Encodings:

BibTeX

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Solutions of Poisson’s equation in channel-like geometries. Comput. Phys. Comm. 115 (1), pp. 45–68.

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External Links:

ISSN 0010-4655, Document, Code, MathReview Entry, ZentralBlatt

Referenced by:

§14.31(i).

Encodings:

BibTeX

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24: Bibliography W

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Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules. Ann. Inst. Statist. Math. 45 (3), pp. 467–475.

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External Links:

ISSN 0020-3157, Document, MathReview (N. K. Thakare), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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Hill’s equation with a large potential. SIAM J. Appl. Math. 45 (2), pp. 200–214.

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External Links:

ISSN 0036-1399, Document, MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§29.7(ii).

Encodings:

BibTeX

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25: 4.21 Identities

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4.21.8 $$\cos u+\cos v=2\cos \left(\frac{u+v}{2}\right)\cos \left(\frac{u-v}{2}\right),$$

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Symbols:

$\cos z$: cosine function

A&S Ref:

4.3.36

Permalink:

http://dlmf.nist.gov/4.21.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(i), §4.21 and Ch.4

… ►If $t=\tan \left(\frac{1}{2}z\right)$, then ►

$\sin z={\displaystyle \frac{2t}{1+t^2}},$

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$\cos z={\displaystyle \frac{1-t^2}{1+t^2}},$

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$dz={\displaystyle \frac{2}{1+t^2}}dt.$

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26: 10.2 Definitions

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10.2.1 $$z^2\frac{d^{2}w}{{dz}^2}+z\frac{dw}{dz}+(z^2-{\nu }^2)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.1

Referenced by:

§10.13, §10.15, §10.2(ii), §10.2(ii), §10.2(iii), §10.24, §10.25(i), §10.27, §10.4, §10.61(ii), §10.74(ii)

Permalink:

http://dlmf.nist.gov/10.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.2(i), §10.2 and Ch.10

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10.2.2 $$J_{\nu }\left(z\right)={(\frac{1}{2}z)}^{\nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{{(\frac{1}{4}{z}^2)}^k}{k!\mathrm{\Gamma }\left(\nu +k+1\right)}.$$

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Defines:

$J_{\nu }\left(z\right)$: Bessel function of the first kind

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.10

Referenced by:

§10.11, §10.16, §10.19(i), §10.22(ii), §10.22(iii), §10.24, §10.53, §10.65(i), §10.7(i), §10.74(ii), §10.8, §18.38(iii), §8.6(i)

Permalink:

http://dlmf.nist.gov/10.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

… ►The principal branch of $J_{\nu }\left(z\right)$ corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$ (§4.2(iv)) and is analytic in the $z$-plane cut along the interval $(-\mathrm{\infty },0]$. … ►

10.2.4 $$Y_n\left(z\right)={\frac{1}{\pi }\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}+{\frac{{(-1)}^n}{\pi }\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =-n},$$ $n=0,\pm 1,\pm 2,\mathrm{\dots }$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.15, §10.2(ii), §10.22(iv), §10.4

Permalink:

http://dlmf.nist.gov/10.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

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10.2.5 $$H_{\nu }^{(1)}\left(z\right)\sim \sqrt{2/(\pi z)}{\mathrm{e}}^{\mathrm{i}(z-\frac{1}{2}\nu \pi -\frac{1}{4}\pi )}$$

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Defines:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function)

Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.2.3

Referenced by:

§10.2(ii), §10.7(ii)

Permalink:

http://dlmf.nist.gov/10.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

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27: 10.6 Recurrence Relations and Derivatives

… ►For results on modified quotients of the form $z{𝒞}_{\nu \pm 1}\left(z\right)/{𝒞}_{\nu }\left(z\right)$ see Onoe (1955) and Onoe (1956). … ►

$p_{\nu +1}-p_{\nu -1}=-{\displaystyle \frac{2\nu }{a}}{q}_{\nu }-{\displaystyle \frac{2\nu }{b}}{r}_{\nu },$

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$q_{\nu +1}+r_{\nu }={\displaystyle \frac{\nu }{a}}{p}_{\nu }-{\displaystyle \frac{\nu +1}{b}}{p}_{\nu +1},$

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$r_{\nu +1}+q_{\nu }={\displaystyle \frac{\nu }{b}}{p}_{\nu }-{\displaystyle \frac{\nu +1}{a}}{p}_{\nu +1},$

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$s_{\nu }=\frac{1}{2}{p}_{\nu +1}+\frac{1}{2}{p}_{\nu -1}-{\displaystyle \frac{{\nu }^2}{ab}}{p}_{\nu },$

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28: 10.60 Sums

… ►For collections of sums of series relevant to spherical Bessel functions or Bessel functions of half odd integer order see Erdélyi et al. (1953b, pp. 43–45 and 98–105), Gradshteyn and Ryzhik (2000, §§8.51, 8.53), Hansen (1975), Magnus et al. (1966, pp. 106–108 and 123–138), and Prudnikov et al. (1986b, pp. 635–637 and 651–700). …

29: 11.6 Asymptotic Expansions

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$c_4(\lambda )=70{\lambda }^{-8}-\frac{45}{2}{\lambda }^{-6}+\frac{3}{8}{\lambda }^{-4}.$

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30: 27.2 Functions

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Table 27.2.2: Functions related to division.

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$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
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6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
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