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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: Bibliography H

… ►

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

… ►

Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.

ⓘ

External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.16.

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

… ►

Calculation of Coulomb wave functions for high energies. Phys. Rev. 54 (8), pp. 627–628.

ⓘ

External Links:

Document

Referenced by:

§33.7.

Encodings:

BibTeX

… ►

Coulomb Wave Functions. In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.), pp. 408–465.

ⓘ

External Links:

MathReview (G. Källén)

Referenced by:

§33.2(iii), §33.23(vii), §33.23(vii), §33.24, §33.5(ii), §33.7, Ch.33.

Encodings:

BibTeX

…

3: 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
…

►

… ►

Table 26.6.2: Motzkin numbers $M(n)$.

► ►►►

$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$
0	1	4	9	8	323	12	15511	16	8 53467
…

►

… ►

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

► ►►►

$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$
$0$	$1$	$4$	$90$	$8$	$41586$	$12$	$\mathrm{272\hspace{0.33em}97738}$	$16$	$\mathrm{2\hspace{0.33em}09271\hspace{0.33em}56706}$
…

►

…

4: 19.37 Tables

… ►

Functions $F(\varphi ,k)$ and $E(\varphi ,k)$

… ►($F(\varphi ,k)$ is presented as $\mathrm{\Pi }(\varphi ,0,k)$.) … ►Tabulated for $\varphi =0(5^{\circ }){90}^{\circ }$, $\arcsin k=0(1^{\circ }){90}^{\circ }$ to 6D by Byrd and Friedman (1971), for $\varphi =0(5^{\circ }){90}^{\circ }$, $\arcsin k=0(2^{\circ }){90}^{\circ }$ and $5^{\circ }({10}^{\circ }){85}^{\circ }$ to 8D by Abramowitz and Stegun (1964, Chapter 17), and for $\varphi =0({10}^{\circ }){90}^{\circ }$, $\arcsin k=0(5^{\circ }){90}^{\circ }$ to 9D by Zhang and Jin (1996, pp. 674–675). … ►Tabulated for $\varphi =5^{\circ }(5^{\circ }){80}^{\circ }({2.5}^{\circ }){90}^{\circ }$, ${\alpha }^2=-1(.1)-0.1,0.1(.1)1$, $k^2=0(.05)0.9(.02)1$ to 10D by Fettis and Caslin (1964) (and warns of inaccuracies in Selfridge and Maxfield (1958) and Paxton and Rollin (1959)). … ►

Function $R_F(a^2,b^2,c^2)$ with $abc=1$

…

5: 27.2 Functions

… ►($\nu \left(1\right)$ is defined to be 0.) …It can be expressed as a sum over all primes $p\le x$: … ►It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. …Note that ${\sigma }_0\left(n\right)=d\left(n\right)$. … ►Table 27.2.2 tabulates the Euler totient function $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. …

6: Bibliography

… ►

Evaluation of Coulomb wave functions along the transition line. Physical Rev. (2) 96, pp. 77–79.

ⓘ

External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§33.12(i).

Encodings:

BibTeX

… ►

Algorithm 610. A portable FORTRAN subroutine for derivatives of the psi function. ACM Trans. Math. Software 9 (4), pp. 494–502.

ⓘ

Notes:

Single and double precision, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, GAMS (module 610; package TOMS), MathReview Entry, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

ⓘ

External Links:

MathReview (N. G. de Bruijn), ZentralBlatt

Referenced by:

(25.11.41), (25.11.42), §25.11(xii).

Encodings:

BibTeX

… ►

Numerical Tables for Angular Correlation Computations in $\alpha$-, $\beta$- and $\gamma$-Spectroscopy: $3j$-, $6j$-, $9j$-Symbols, F- and $\mathrm{\Gamma }$-Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.

ⓘ

External Links:

ISBN 978-3-540-04218-1

Referenced by:

§34.14.

Encodings:

BibTeX

… ►

A new treatment of the ellipsoidal wave equation. Proc. London Math. Soc. (3) 9, pp. 21–50.

ⓘ

External Links:

ISSN 0024-6115, Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§29.11.

Encodings:

BibTeX

…

7: 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).

8: 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. …

9: 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 …

10: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions

…