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11: 26.21 Tables

§26.21 Tables

►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

for

m

up to 50 and

n

up to 25; extends Table 26.4.1 to

n=10

; tabulates Stirling numbers of the first and second kinds,

s\left(n,k\right)

and

S\left(n,k\right)

, for

n

up to 25 and

k

up to

n

; tabulates partitions

p\left(n\right)

and partitions into distinct parts

p\left(\mathcal{D},n\right)

for

n

up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts

\not\equiv\pm 2\pmod{5}

, partitions into parts

\not\equiv\pm 1\pmod{5}

, and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to

\genfrac{[}{]}{0.0pt}{}{12}{6}_{q}

. ►Goldberg et al. (1976) contains tables of binomial coefficients to

n=100

and Stirling numbers to

n=40

12: 3.4 Differentiation

… ►For corresponding formulas for second, third, and fourth derivatives, with

t=0

, see Collatz (1960, Table III, pp. 538–539). … ►If

f

can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii)) …The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2). … ►The results in this subsection for the partial derivatives follow from Panow (1955, Table 10). … ►For additional formulas involving values of

\nabla^{2}u

and

\nabla^{4}u

on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546). …

13: 24.2 Definitions and Generating Functions

… ►

§24.2(iv) Tables

►

Table 24.2.1: Bernoulli and Euler numbers.

► ►►

$n$	$B_{n}$	$E_{n}$
…

► ►

Table 24.2.2: Bernoulli and Euler polynomials.

► ►►

$n$	$B_{n}\left(x\right)$	$E_{n}\left(x\right)$
…

► …

14: 34.9 Graphical Method

§34.9 Graphical Method

… ►For specific examples of the graphical method of representing sums involving the

\mathit{3j},\mathit{6j}

, and

\mathit{9j}

symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

15: 11.14 Tables

§11.14 Tables

… ►For tables before 1961 see Fletcher et al. (1962) and Lebedev and Fedorova (1960). Tables listed in these Indices are omitted from the subsections that follow. … ►

•

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.

… ►

§11.14(iv) Anger–Weber Functions

…

16: 26.9 Integer Partitions: Restricted Number and Part Size

… ►See Table 26.9.1. … ►The conjugate to the example in Figure 26.9.1 is

6+5+4+2+1+1+1

. … ►Equations (26.9.2)–(26.9.3) are examples of closed forms that can be computed explicitly for any positive integer

k

. … ►

p_{3}\left(n\right)=1+\left\lfloor\frac{n^{2}+6n}{12}\right\rfloor.

… ►equivalently, partitions into at most

k

parts either have exactly

k

parts, in which case we can subtract one from each part, or they have strictly fewer than

k

parts. …

17: 26.10 Integer Partitions: Other Restrictions

… ►

p\left(\mathcal{D}^{\prime}3,n\right)

denotes the number of partitions of

n

into parts with difference at least 3, except that multiples of 3 must differ by at least 6. …The set

\{2,3,4,\ldots\}

is denoted by

T

. …See Table 26.10.1. … ►Note that

p\left(\mathcal{D}^{\prime}3,n\right)\leq p\left(\mathcal{D}3,n\right)

, with strict inequality for

n\geq 9

. 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). …

18: 34.6 Definition: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►

34.6.2

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.13
Permalink:: http://dlmf.nist.gov/34.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

…

19: 26.3 Lattice Paths: Binomial Coefficients

… ►

\genfrac{(}{)}{0.0pt}{}{m}{n}

is the number of ways of choosing

n

objects from a collection of

m

distinct objects without regard to order. … ►For numerical values of

\genfrac{(}{)}{0.0pt}{}{m}{n}

and

\genfrac{(}{)}{0.0pt}{}{m+n}{n}

see Tables 26.3.1 and 26.3.2. ►

Table 26.3.1: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

► ►►

$m$	$n$
$m$	…

► ►

Table 26.3.2: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m+n}{m}

for lattice paths.

► ►►

$m$	$n$
$m$	…

► …

20: 28.6 Expansions for Small $q$

… ►Leading terms of the power series for

a_{m}\left(q\right)

and

b_{m}\left(q\right)

for

m\leq 6

are: … ►Numerical values of the radii of convergence

\rho_{n}^{(j)}

of the power series (28.6.1)–(28.6.14) for

n=0,1,\dots,9

are given in Table 28.6.1. …(Table 28.6.1 is reproduced from Meixner et al. (1980, §2.4).) … ►For

m=3,4,5,\dots

, … ►The radii of convergence of the series (28.6.21)–(28.6.26) are the same as the radii of the corresponding series for

a_{n}\left(q\right)

and

b_{n}\left(q\right)

; compare Table 28.6.1 and (28.6.20).

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11—20 of 764 matching pages

11: 26.21 Tables

§26.21 Tables

12: 3.4 Differentiation

13: 24.2 Definitions and Generating Functions

§24.2(iv) Tables

14: 34.9 Graphical Method

§34.9 Graphical Method

15: 11.14 Tables

§11.14 Tables

§11.14(iv) Anger–Weber Functions

16: 26.9 Integer Partitions: Restricted Number and Part Size

17: 26.10 Integer Partitions: Other Restrictions

18: 34.6 Definition: 9 ⁢ j Symbol

19: 26.3 Lattice Paths: Binomial Coefficients

20: 28.6 Expansions for Small q

18: 34.6 Definition: $\mathit{9j}$ Symbol

20: 28.6 Expansions for Small $q$