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11: 26.8 Set Partitions: Stirling Numbers

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§26.8(vii) Asymptotic Approximations

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26.8.40 $$s(n+1,k+1)\sim {(-1)}^{n-k}\frac{n!}{k!}{(\gamma +\ln n)}^k,$$ $n\to \mathrm{\infty }$,

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Symbols:

$\gamma$: Euler’s constant, $s(n,k)$: Stirling number of the first kind, $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.8.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(vii), §26.8 and Ch.26

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26.8.41 $$s(n+k,k)\sim \frac{{(-1)}^n}{2^{n}n!}{k}^{2n},$$ $k\to \mathrm{\infty }$,

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Symbols:

$s(n,k)$: Stirling number of the first kind, $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.8.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(vii), §26.8 and Ch.26

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26.8.42 $$S(n,k)\sim \frac{k^n}{k!},$$ $n\to \mathrm{\infty }$,

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Symbols:

$S(n,k)$: Stirling number of the second kind, $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.8.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(vii), §26.8 and Ch.26

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26.8.43 $$S(n+k,k)\sim \frac{k^{2n}}{2^{n}n!},$$ $k\to \mathrm{\infty }$,

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Symbols:

$S(n,k)$: Stirling number of the second kind, $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§26.8(vii)

Permalink:

http://dlmf.nist.gov/26.8.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(vii), §26.8 and Ch.26

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12: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►These cases are treated in §§12.10(vii)–12.10(viii). … ►

12.10.23 $$\eta =\frac{1}{2}\arccos t-\frac{1}{2}t\sqrt{1-t^2},$$

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Defines:

$\eta$ (locally)

Symbols:

$\arccos z$: arccosine function

Referenced by:

§12.10(vii), §12.14(ix)

Permalink:

http://dlmf.nist.gov/12.10.E23

Encodings:

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See also:

Annotations for §12.10(iv), §12.10 and Ch.12

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§12.10(vii) Negative $a$, . Expansions in Terms of Airy Functions

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12.10.40 $$\varphi (\zeta )={\left(\frac{\zeta }{t^2-1}\right)}^{\frac{1}{4}}.$$

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Defines:

$\varphi (\zeta )$: function (locally)

Symbols:

$\zeta$: change of variable

Referenced by:

§12.10(vii)

Permalink:

http://dlmf.nist.gov/12.10.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vii), §12.10 and Ch.12

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12.10.41 $$t=1+w-\frac{1}{10}{w}^2+\frac{11}{350}{w}^3-\frac{823}{63000}{w}^4+\frac{\mathrm{1\hspace{0.33em}50653}}{\mathrm{242\hspace{0.33em}55000}}{w}^5+\mathrm{\cdots },$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $\zeta$: change of variable

Referenced by:

§12.11(iii), §12.11(iii)

Permalink:

http://dlmf.nist.gov/12.10.E41

Encodings:

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See also:

Annotations for §12.10(vii), §12.10 and Ch.12

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13: 25.21 Software

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§25.21(vii) Fermi–Dirac and Bose–Einstein Integrals

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14: 10.21 Zeros

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§10.21(vii) Asymptotic Expansions for Large Order

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10.21.24 $$\theta \left(-2^{\frac{1}{3}}\alpha \right)=\pi t,$$

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Defines:

$\alpha$: coefficients (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $\theta \left(z\right)$: Airy phase function

Permalink:

http://dlmf.nist.gov/10.21.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

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10.21.32 $$j_{\nu ,m}\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k}{{\nu }^{2k/3}},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$ , $m$: integer, $k$: nonnegative integer, $\nu$: complex parameter and $\alpha$: coefficients

Referenced by:

§10.21(vii), §10.21(vii)

Permalink:

http://dlmf.nist.gov/10.21.E32

Encodings:

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See also:

Annotations for §10.21(vii), §10.21 and Ch.10

… ►(There is an inaccuracy in Figures 11 and 14 in this reference. … ►This information includes asymptotic approximations analogous to those given in §§10.21(vi), 10.21(vii), and 10.21(x). …

15: 3.6 Linear Difference Equations

… ►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome. … ►

Table 3.6.1: Weber function $w_n={𝐄}_n\left(1\right)$ computed by Olver’s algorithm.

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$n$	$p_n$		$e_n$		$e_n/(p_n{p}_{n+1})$		$w_n$
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11	$\mathrm{0.29154\hspace{0.33em}738}$	$\mathrm{\times 10¹⁰}$	$\mathrm{0.37225\hspace{0.33em}201}$	$\mathrm{\times 10¹⁰}$	$\mathrm{0.19952\hspace{0.33em}026}$	$\mathrm{\times 10⁻¹⁰}$	$\mathrm{0.58373\hspace{0.33em}946}$	$\mathrm{\times 10⁻¹}$
…

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§3.6(vii) Linear Difference Equations of Other Orders

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3.6.17 $$a_n{w}_{n+1}-b_n{w}_n=d_n.$$

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Symbols:

$w_n$: sequence, $a_n$: coefficient, $b_n$: coefficient and $d_n$: coefficient

Permalink:

http://dlmf.nist.gov/3.6.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §3.6(vii), §3.6 and Ch.3

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3.6.18 $$a_{n,k}{w}_{n+k}+a_{n,k-1}{w}_{n+k-1}+\mathrm{\cdots }+a_{n,0}{w}_n=d_n,$$

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Symbols:

$w_n$: sequence, $a_n$: coefficient and $d_n$: coefficient

Permalink:

http://dlmf.nist.gov/3.6.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §3.6(vii), §3.6 and Ch.3

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16: 34.3 Basic Properties: $\mathit{3}j$ Symbol

… ►See Srinivasa Rao and Rajeswari (1993, pp. 44–47) and references given there. … ►

§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

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34.3.19

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, : $\mathit{3}j$ symbol and $l,l_r$: non-negative integers or non-negative integers plus one half.

Referenced by:

§14.18(iv), §14.30(iii), §34.3(vii)

Permalink:

http://dlmf.nist.gov/34.3.E19

Encodings:

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See also:

Annotations for §34.3(vii), §34.3 and Ch.34

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34.3.21

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, : $\mathit{3}j$ symbol and $l,l_r$: non-negative integers or non-negative integers plus one half.

Referenced by:

§14.17(iv)

Permalink:

http://dlmf.nist.gov/34.3.E21

Encodings:

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See also:

Annotations for §34.3(vii), §34.3 and Ch.34

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34.3.22

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, : $\mathit{3}j$ symbol and $l,l_r$: non-negative integers or non-negative integers plus one half.

Referenced by:

§14.30(ii), §34.3(vii)

Permalink:

http://dlmf.nist.gov/34.3.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §34.3(vii), §34.3 and Ch.34

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17: 9.18 Tables

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National Bureau of Standards (1958) tabulates $A_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$ and $-A_0^{\prime }(x)\equiv \pi {\mathrm{Hi}}^{\prime }\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$; ${\int }_0^x{A}_0(t)dt$ for $x=0.5,1(1)11$. Precision is 8D.

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Gil et al. (2003c) tabulates the only positive zero of ${\mathrm{Gi}}^{\prime }\left(z\right)$, the first 10 negative real zeros of $\mathrm{Gi}\left(z\right)$ and ${\mathrm{Gi}}^{\prime }\left(z\right)$, and the first 10 complex zeros of $\mathrm{Gi}\left(z\right)$, ${\mathrm{Gi}}^{\prime }\left(z\right)$, $\mathrm{Hi}\left(z\right)$, and ${\mathrm{Hi}}^{\prime }\left(z\right)$. Precision is 11 or 12S.

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§9.18(vii) Generalized Airy Functions

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18: 16.25 Methods of Computation

… ►See §§3.6(vii), 3.7(iii), Olde Daalhuis and Olver (1998), Lozier (1980), and Wimp (1984, Chapters 7, 8).

19: 24.2 Definitions and Generating Functions

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24.2.3 $$\frac{t{\mathrm{e}}^{xt}}{{\mathrm{e}}^t-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_n\left(x\right)\frac{t^n}{n!},$$ .

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $t$: real or complex and $x$: real or complex

A&S Ref:

23.1.1

Referenced by:

§18.2(xii), §24.4(iv), §24.4(vii)

Permalink:

http://dlmf.nist.gov/24.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.2(i), §24.2 and Ch.24

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24.2.8 $$\frac{2{\mathrm{e}}^{xt}}{{\mathrm{e}}^t+1}=\sum _{n=0}^{\mathrm{\infty }}{E}_n\left(x\right)\frac{t^n}{n!},$$ ,

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $t$: real or complex and $x$: real or complex

A&S Ref:

23.1.1

Referenced by:

§18.2(xii), §24.4(iv), §24.4(vii)

Permalink:

http://dlmf.nist.gov/24.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.2(ii), §24.2 and Ch.24

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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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$28$	$-\mathrm{2\hspace{0.33em}37494\hspace{0.33em}61029}$	$870$	$-\mathrm{2.72982\hspace{0.33em}3107}$	$\mathrm{\times 10⁷}$
…

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Table 24.2.4: Euler numbers $E_n$.

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$n$	$E_n$
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$28$	$\mathrm{12522\hspace{0.33em}59641\hspace{0.33em}40362\hspace{0.33em}98654\hspace{0.33em}68285}$
…

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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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	$k$
…
$11$	$0$	$\frac{5}{6}$	$0$	$-\frac{11}{2}$	$0$	$11$	$0$	$-11$	$0$	$\frac{55}{6}$	$-\frac{11}{2}$	$1$
…

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20: Bibliography W

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The Algebraic Eigenvalue Problem. Monographs on Numerical Analysis. Oxford Science Publications, The Clarendon Press, Oxford University Press, Oxford.

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Notes:

Paperback reprint of original edition, published in 1965 by Clarendon Press, Oxford.

External Links:

ISBN 0-19-853418-3, MathReview, ZentralBlatt

Referenced by:

§3.2(v), §3.2(vi), §3.2(vii).

Encodings:

BibTeX

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Computation with Recurrence Relations. Pitman, Boston, MA.

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External Links:

ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt

Referenced by:

§13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).

Encodings:

BibTeX

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On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.

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External Links:

ISSN 1044-677X, FullText

Referenced by:

§28.4(vii).

Encodings:

BibTeX

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Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.

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External Links:

ISSN 0029-599X, Document, MathReview (A.J. Rodrigues), ZentralBlatt

Referenced by:

§10.74(vii), §3.5(vii).

Encodings:

BibTeX

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Asymptotic Approximations of Integrals. Academic Press Inc., Boston-New York.

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Notes:

Reprinted with corrections by SIAM, Philadelphia, PA, 2001.

External Links:

ISBN 0-12-762535-6, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§1.14(iv), §1.14(vii), §1.15(iv), §1.16(i), §1.16(ii), §1.16(iii), §1.16(iv), §1.16(v), §1.16(vi), §1.16(vii), §1.16(viii), §14.26, Ch.14, §2.10(iv), §2.3(i), §2.3(ii), §2.3(iii), §2.3(v), §2.4(i), §2.4(ii), §2.4(iv), §2.4(v), §2.4(vi), §2.5(iii), §2.5(i), §2.5(ii), §2.5(ii), §2.5(ii), §2.5(iii), §2.6(iii), §2.6(i), §2.6(i), §2.6(ii), §2.6(ii), §2.6(iii), §2.6(iv), §2.6(iv), Ch.2, §7.20(i), Ch.8, Ch.9, Profile Roderick S. C. Wong.

Encodings:

BibTeX

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