DLMF: Untitled Document

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Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta\left(s+1\right)$ and $\zeta\left(s+k\right)$ , $k=2,3,4,5,8$ , for $0\leq s\leq 1$ (23D).

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Morris (1979) gives rational approximations for $\operatorname{Li}_{2}\left(x\right)$ (§25.12(i)) for $0.5\leq x\leq 1$ . Precision is varied with a maximum of 24S.

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Antia (1993) gives minimax rational approximations for $\Gamma\left(s+1\right)F_{s}(x)$ , where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals $-\infty<x\leq 2$ and $2\leq x<\infty$ , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$ . For each $s$ there are three sets of approximations, with relative maximum errors $10^{-4},10^{-8},10^{-12}$ .

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B_{2n+1}=0

,

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24.2.4

B_{n}=B_{n}\left(0\right),

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials and $n$ : integer
A&S Ref:: 23.1.2
Referenced by:: (25.11.7)
Permalink:: http://dlmf.nist.gov/24.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.2(i), §24.2 and Ch.24

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Table 24.2.4: Euler numbers

E_{n}

.

► ►►

$n$	$E_{n}$
…

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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}

.

► ►►

	$k$
…

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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}

.

► ►►

	$k$
…

►

… ►Given numerical values of

w_{0}

and

w_{1}

, the solution

w_{n}

of the equation … ►beginning with

e_{0}=w_{0}

. … ►We apply the algorithm to compute

\mathbf{E}_{n}\left(1\right)

to 8S for the range

n=1,2,\dots,10

, beginning with the value

\mathbf{E}_{0}\left(1\right)=-0.56865\;\,663

obtained from the Maclaurin series expansion (§11.10(iii)). … ►The values of

w_{n}

for

n=1,2,\dots,10

are the wanted values of

\mathbf{E}_{n}\left(1\right)

. (It should be observed that for

n>10

, however, the

w_{n}

are progressively poorer approximations to

\mathbf{E}_{n}\left(1\right)

: the underlined digits are in error.) …

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Table 18.8.1: Classical OP’s: differential equations

A(x)f^{\prime\prime}(x)+B(x)f^{\prime}(x)+C(x)f(x)+\lambda_{n}f(x)=0

.

► ►►►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda_{n}$
…
8	$L^{(\alpha)}_{n}\left(x\right)$	$x$	$\alpha+1-x$	$0$	$n$
…
10	${\mathrm{e}}^{-\frac{1}{2}x}x^{\frac{1}{2}\alpha}L^{(\alpha)}_{n}\left(x\right)$	$x$	$1$	$-\frac{1}{4}x-\frac{1}{4}\alpha^{2}x^{-1}$	$n+\frac{1}{2}(\alpha+1)$
…
12	$H_{n}\left(x\right)$	$1$	$-2x$	$0$	$2n$
…
14	$\mathit{He}_{n}\left(x\right)$	$1$	$-x$	$0$	$n$

► …

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Abramowitz and Stegun (1964, Chapter 12) tabulates $\mathbf{H}_{n}\left(x\right)$ , $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ , and $I_{n}\left(x\right)-\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.1)5$ , $x^{-1}=0(.01)0.2$ to 6D or 7D.

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Barrett (1964) tabulates $\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0.2(.005)4(.05)10(.1)19.2$ to 5 or 6S, $x=6(.25)59.5(.5)100$ to 2S.

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Abramowitz and Stegun (1964, Chapter 12) tabulates $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t$ and $(2/\pi)\int_{x}^{\infty}t^{-1}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ for $x=0(.1)5$ to 5D or 7D; $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , and $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,% \mathrm{d}t$ for $x^{-1}=0(.01)0.2$ to 6D.

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Bernard and Ishimaru (1962) tabulates $\mathbf{J}_{\nu}\left(x\right)$ and $\mathbf{E}_{\nu}\left(x\right)$ for $\nu=-10(.1)10$ and $x=0(.1)10$ to 5D.

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Agrest and Maksimov (1971, Chapter 11) defines incomplete Struve, Anger, and Weber functions and includes tables of an incomplete Struve function $\mathbf{H}_{n}\left(x,\alpha\right)$ for $n=0,1$ , $x=0(.2)10$ , and $\alpha=0(.2)1.4,\tfrac{1}{2}\pi$ , together with surface plots.

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p_{k}\left(n\right)

denotes the number of partitions of

n

into at most

k

parts. See Table 26.9.1. … ►It follows that

p_{k}\left(n\right)

also equals the number of partitions of

n

into parts that are less than or equal to

k

. ►

p_{k}\left(\leq m,n\right)

is the number of partitions of

n

into at most

k

parts, each less than or equal to

m

. …

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•

Zhang and Jin (1996, Table 3.9) tabulates $I_{x}\left(a,b\right)$ for $x=0(.05)1$ , $a=0.5,1,3,5,10$ , $b=1,10$ to 8D.

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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 $(x+n)e^{x}E_{n}\left(x\right)$ 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; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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•

Stankiewicz (1968) tabulates $E_{n}\left(x\right)$ for $n=1(1)10$ , $x=0.01(.01)5$ to 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.

… ►A transformation of a convergent sequence

\{s_{n}\}

with limit

\sigma

into a sequence

\{t_{n}\}

is called limit-preserving if

\{t_{n}\}

converges to the same limit

\sigma

. … ►Then the transformation of the sequence

\{s_{n}\}

into a sequence

\{t_{n,2k}\}

is given by … ►Then

t_{n,2k}=\varepsilon_{2k}^{(n)}

. … ►with

s_{\infty}=\frac{1}{12}{\pi}^{2}=0.82246\;70334\;24\dotsc

. … ►We give a special form of Levin’s transformation in which the sequence

s=\{s_{n}\}

of partial sums

s_{n}=\sum_{j=0}^{n}a_{j}

is transformed into: …

… ►

16.12.1

{{}_{0}F_{1}}\left(-;a;z\right){{}_{0}F_{1}}\left(-;b;z\right)={{}_{2}F_{3}}% \left({\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)\atop a,b,a+b-1};4z\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: (10.8.3), §10.8, §10.8, §16.12, Erratum (V1.0.17) for Section 10.8
Permalink:: http://dlmf.nist.gov/16.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

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16.12.2

\left({{}_{2}F_{1}}\left({a,b\atop a+b+\frac{1}{2}};z\right)\right)^{2}={{}_{3% }F_{2}}\left({2a,2b,a+b\atop a+b+\frac{1}{2},2a+2b};z\right).

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.12, §17.9(iii), Erratum (V1.1.12) for Subsection 17.9(iii)
Permalink:: http://dlmf.nist.gov/16.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

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16.12.3

\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)\right)^{2}=\sum_{k=0}^{\infty}% \frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}}{\left(c-\frac{1}{2}\right)_{k% }}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}k!}{{}_{4}F_{3}}\left({-\frac{1% }{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}\atop a+\frac{1}{2},b+% \frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k},

|z|<1

.

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.12
Permalink:: http://dlmf.nist.gov/16.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

…

… ►

p_{m}\left(\mathcal{D},n\right)

denotes the number of partitions of

n

into at most

m

distinct parts. …The set

\{n\geq 1\>|\>n\equiv\pm j\ \pmod{k}\}

is denoted by

A_{j,k}

. … ►It is known that for

k>3

,

p\left(\mathcal{D}k,n\right)\geq p\left(\in\!A_{1,k+3},n\right)

, with strict inequality for

n

sufficiently large, provided that

k=2^{m}-1,m=3,4,5

, or

k\geq 32

; see Yee (2004). … ►where

I_{1}\left(x\right)

is the modified Bessel function (§10.25(ii)), and …The quantity

A_{k}(n)

is real-valued. …

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11—20 of 815 matching pages

11: 25.20 Approximations

12: 24.2 Definitions and Generating Functions

13: 3.6 Linear Difference Equations

14: 18.8 Differential Equations

15: 11.14 Tables

16: 26.9 Integer Partitions: Restricted Number and Part Size

17: 8.26 Tables

18: 3.9 Acceleration of Convergence

19: 16.12 Products

20: 26.10 Integer Partitions: Other Restrictions