%E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E5%B9%B3%E5%8F%B0,%E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E7%BD%91%E7%AB%99,%E3%80%90%E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E5%9C%B0%E5%9D%80%E2%88%B622kk33.com%E3%80%91%E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E8%AE%BA%E5%9D%9B,%E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E5%85%AC%E5%8F%B8,%E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E8%AE%BA%E5%9D%9B,%E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E8%B5%84%E8%AE%AF,%E7%9C%9F%E4%BA%BA%E5%8D%9A%E5%BD%A9%E6%B8%B8%E6%88%8F,%E3%80%90%E5%8D%9A%E5%BD%A9%E7%BD%91%E5%9D%80%E2%88%B622kk33.com%E3%80%91%E7%BD%91%E5%9D%80Zg0ngCkEEkEAkfn

SearchAdvancedHelp

Did you mean %E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E5%B9%B3%E5%8F%B0,%E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E7%BD%91%E7%AB%99,%E3%80%90%E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E5%9C%B0%E5%9D%80%E2%88%gcn.com%E3%80%91%E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E8%AE%BA%E5%9D%9B,%E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E5%85%AC%E5%8F%B8,%E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E8%AE%BA%E5%9D%9B,%E7%BD%91%E4%B8%8A%E5%8D%9A%E5%BD%A9%E8%B5%84%E8%AE%AF,%E7%9C%9F%E4%BA%BA%E5%8D%9A%E5%BD%A9%E6%B8%B8%E6%88%8F,%E3%80%90%E5%8D%9A%E5%BD%A9%E7%BD%91%E5%9D%80%E2%88%gcn.com%E3%80%91%E7%BD%91%E5%9D%80Zg0ngCkEEkEAkfn ?

(0.068 seconds)

1—10 of 582 matching pages

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

… ►

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

… ►

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

… ►

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

►

Variable precision exponential function. ACM Trans. Math. Software 12 (2), pp. 79–91.

ⓘ

External Links:

ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

§4.48(iii).

Encodings:

BibTeX

…

3: 27.2 Functions

… ►

27.2.9 $$d\left(n\right)=\sum _{d|n}1$$

ⓘ

Symbols:

$d_k\left(n\right)$: divisor function, $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.3 I.A

Permalink:

http://dlmf.nist.gov/27.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►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 $n=1(1)52$. … ►

Table 27.2.2: Functions related to division.

► ►►►

$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
…
10	4	4	18	23	22	2	24	36	12	9	91	49	42	3	57
…

►

4: 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 $D(m,n)$.

► ►►

$m$	$n$
	…

►

… ►

26.6.4 $$r(n)=D(n,n)-D(n+1,n-1),$$ $n\ge 1$.

ⓘ

Defines:

$r(n)$: Schröder number (locally)

Symbols:

$n$: nonnegative integer and $D(m,n)$: Delannoy number

Referenced by:

§26.6(i)

Permalink:

http://dlmf.nist.gov/26.6.E4

Encodings:

pMML, png, TeX

See also:

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

… ►

26.6.10 $$D(m,n)=D(m,n-1)+D(m-1,n)+D(m-1,n-1),$$ $m,n\ge 1$,

ⓘ

Symbols:

$m$: nonnegative integer, $n$: nonnegative integer and $D(m,n)$: Delannoy number

Referenced by:

§26.6(iii)

Permalink:

http://dlmf.nist.gov/26.6.E10

Encodings:

pMML, png, TeX

See also:

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

…

5: Bibliography D

… ►

Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

ⓘ

External Links:

ISSN 0022-314X, Document, MathReview (T. Metsänkylä)

Referenced by:

§24.12(iii).

Encodings:

BibTeX

… ►

Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

ⓘ

External Links:

MathReview (I. I. Hirschman), ZentralBlatt

Referenced by:

§2.5(i).

Encodings:

BibTeX

… ►

Lamé instantons. J. High Energy Phys. 2000 (1), pp. Paper 19, 8.

ⓘ

Notes:

(electronic only, article id. JHEP01(2000)019, 8 pp.).

External Links:

ISSN 1029-8479, Document, MathReview, ZentralBlatt

Referenced by:

§22.11.

Encodings:

BibTeX

… ►

Error analysis in a uniform asymptotic expansion for the generalised exponential integral. J. Comput. Appl. Math. 80 (1), pp. 127–161.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§2.11(iii), §8.20(ii).

Encodings:

BibTeX

… ►

Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.

ⓘ

External Links:

Document, MathReview (R. C. Varma), ZentralBlatt

Referenced by:

§18.17(iii).

Encodings:

BibTeX

…

6: Bibliography

… ►

Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

ⓘ

External Links:

FullText, MathReview (D. P. Banerjee), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

… ►

On basic hypergeometric series, mock theta functions, and partitions. II. Quart. J. Math. Oxford Ser. (2) 17, pp. 132–143.

ⓘ

External Links:

Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§17.6(ii).

Encodings:

BibTeX

… ►

Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.

ⓘ

Notes:

In memory of Gian-Carlo Rota

External Links:

ISSN 0097-3165, Document, MathReview (Alessandro Di Bucchianico), ZentralBlatt

Referenced by:

§17.12.

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

… ►

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: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions

…

9: 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. …

10: 24.2 Definitions and Generating Functions

… ►

24.2.1 $$\frac{t}{{\mathrm{e}}^t-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_n\frac{t^n}{n!},$$ .

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer and $t$: real or complex

Referenced by:

§24.19(ii), §26.14(iii)

Permalink:

http://dlmf.nist.gov/24.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.2(i), §24.2 and Ch.24

►

$B_{2n+1}=0$ ,

►

${(-1)}^{n+1}{B}_{2n}>0$ , $n=1,2,\mathrm{\dots }$.

… ►

Table 24.2.3: Bernoulli numbers $B_n=N/D$.

► ►►

$n$	$N$	$D$	$B_n$
…

►

… ►

Table 24.2.6: Coefficients $e_{n,k}$ of the Euler polynomials $E_n\left(x\right)={\sum }_{k=0}^n{e}_{n,k}{x}^k$.

► ►►►

	$k$
…
$14$	$0$	$-38227$	$0$	$62881$	$0$	$-31031$	$0$	$7293$	$0$	$-1001$	$0$	$91$	$0$	$-7$	$1$
…

►