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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: 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).

3: Bibliography H

… ►

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

►

Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

ⓘ

External Links:

MathReview (M. Lawrence Glasser)

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

… ►

Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§7.18(iii), §7.18(iv), §7.18(v), §7.18(vi).

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: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions

…

5: 34 3j, 6j, 9j Symbols

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

…

6: 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. …

7: 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 26.6.4: Schröder numbers $r(n)$.

► ►►►

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

►

… ►

26.6.12 $$C\left(n\right)=\sum _{k=1}^{n}N(n,k),$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number, $k$: nonnegative integer, $n$: nonnegative integer and $N(n,k)$: Narayana number

Permalink:

http://dlmf.nist.gov/26.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iv), §26.6 and Ch.26

►

26.6.13 $$M(n)=\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)C\left(n+1-k\right),$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer, $n$: nonnegative integer and $M(n)$: Motzkin number

Permalink:

http://dlmf.nist.gov/26.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iv), §26.6 and Ch.26

…

8: 1.12 Continued Fractions

… ► $C_n$ is called the $n$th approximant or convergent to $C$ . $A_n$ and $B_n$ are called the $n$th (canonical) numerator and denominator respectively. … ►Define … ►Conversely, $C$ is called an extension of $C^{\prime }$. … ►Then the convergents $C_n$ satisfy …

9: 16.26 Approximations

… ►For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).

10: 34.9 Graphical Method

§34.9 Graphical Method

… ►For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).