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1: 34.6 Definition: $\mathit{9}j$ Symbol

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

►The $\mathit{9}j$ symbol may be defined either in terms of $\mathit{3}j$ symbols or equivalently in terms of $\mathit{6}j$ symbols: ►

34.6.1

ⓘ

Defines:

: $\mathit{9}j$ symbol

Symbols:

: $\mathit{3}j$ 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

ⓘ

Symbols:

: $\mathit{9}j$ symbol, : $\mathit{6}j$ 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

►The $\mathit{9}j$ symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

2: 19.2 Definitions

… ►The integral for $E(\varphi ,k)$ is well defined if $k^2={\sin }^2\varphi =1$, and the Cauchy principal value (§1.4(v)) of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is taken if $1-{\alpha }^2{\sin }^2\varphi$ vanishes at an interior point of the integration path. … ►

§19.2(iv) A Related Function: $R_C(x,y)$

… ►Formulas involving $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using $R_C(x,y)$. … ►When $x$ and $y$ are positive, $R_C(x,y)$ is an inverse circular function if and an inverse hyperbolic function (or logarithm) if $x>y$: …For the special cases of $R_C(x,x)$ and $R_C(0,y)$ see (19.6.15). …

3: Bibliography H

… ►

Integrals containing the Fresnel functions $S(x)$ and $C(x)$ . Bul. Akad. Štiince RSS Moldoven. 1975 (3), pp. 48–60, 93 (Russian).

ⓘ

External Links:

MathReview (J. Kaczmarski)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

►

Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

ⓘ

External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

►

Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

ⓘ

External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

… ►

The combination of $R$-matrix and complex coordinate methods: Application to the diamagnetic Rydberg spectra of Ba and Sr. J. Phys. B 26 (12), pp. 1775–1790.

ⓘ

External Links:

ISSN 0953-4075, Document, MathReview

Referenced by:

4th item.

Encodings:

BibTeX

… ►

Algorithm 56: Complete elliptic integral of the second kind. Comm. ACM 4 (4), pp. 180–181.

ⓘ

Notes:

Includes Algol program, maximum accuracy 8D. For certification see same journal 9 (1966), p. 12

External Links:

ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

Encodings:

BibTeX

…

4: 34.12 Physical Applications

§34.12 Physical Applications

►The angular momentum coupling coefficients ($\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols) are essential in the fields of nuclear, atomic, and molecular physics. …$\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).

5: 26.5 Lattice Paths: Catalan Numbers

… ► $C\left(n\right)$ is the Catalan number. …(Sixty-six equivalent definitions of $C\left(n\right)$ are given in Stanley (1999, pp. 219–229).) … ►

Table 26.5.1: Catalan numbers.

► ►►►

$n$	$C\left(n\right)$	$n$	$C\left(n\right)$	$n$	$C\left(n\right)$
…
2	2	9	4862	16	353 57670
…

►

… ►

26.5.4 $$C\left(n+1\right)=\frac{2(2n+1)}{n+2}C\left(n\right),$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iii), §26.5 and Ch.26

►

26.5.5 $$C\left(n+1\right)=\sum _{k=0}^{\lfloor n/2\rfloor }\left(\genfrac{}{}{0.0pt}{}{n}{2k}\right)2^{n-2k}C\left(k\right).$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iii), §26.5 and Ch.26

…

6: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions

…

7: 34.14 Tables

§34.14 Tables

►Tables of exact values of the squares of the $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols on pp. … ►Some selected $\mathit{9}j$ symbols are also given. … 16-17; for $\mathit{9}j$ symbols on p. … ► 310–332, and for the $\mathit{9}j$ symbols on pp. …

8: 34 3j, 6j, 9j Symbols

Chapter 34 $\mathit{3}j,\mathit{6}j,\mathit{9}j$ Symbols

…

9: 26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

► $D(m,n)$ is the number of paths from $(0,0)$ to $(m,n)$ that are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. … ►

Table 26.6.1: Delannoy numbers $D(m,n)$.

► ►►►

$m$	$n$
	0	1	2	3	4	5	6	7	8	9	10
…

►

… ► $M(n)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$ and are composed of directed line segments of the form $(2,0)$, $(0,2)$, or $(1,1)$. … ►

Table 26.6.4: Schröder numbers $r(n)$.

► ►►►

$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$
…
$1$	$2$	$5$	$394$	$9$	$\mathrm{2\hspace{0.33em}06098}$	$13$	$\mathrm{1420\hspace{0.33em}78746}$	$17$	$\mathrm{11\hspace{0.33em}18180\hspace{0.33em}26018}$
…

►

…

10: 1.12 Continued Fractions

… ►Define … ►Conversely, $C$ is called an extension of $C^{\prime }$. If $C_n^{\prime }=C_{2n}$, $n=0,1,2,\mathrm{\dots }$, then $C^{\prime }$ is called the even part of $C$. The even part of $C$ exists iff $b_{2k}\ne 0$, $k=1,2,\mathrm{\dots }$, and up to equivalence is given by …If $C_n^{\prime }=C_{2n+1}$, $n=0,1,2,\mathrm{\dots }$, then $C^{\prime }$ is called the odd part of $C$. …