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11: 34.2 Definition: $\mathit{3j}$ Symbol

… ►The quantities

j_{1},j_{2},j_{3}

in the

\mathit{3j}

symbol are called angular momenta. …The corresponding projective quantum numbers

m_{1},m_{2},m_{3}

are given by … ►

34.2.4

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{1}-j_{2}-m_{3}}}\Delta(j_{1}j_{2}j_{3% })\left((j_{1}+m_{1})!(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!(j_{3}+m_{3})!% (j_{3}-m_{3})!\right)^{\frac{1}{2}}\*\sum_{s}\frac{(-1)^{s}}{s!(j_{1}+j_{2}-j_% {3}-s)!(j_{1}-m_{1}-s)!(j_{2}+m_{2}-s)!(j_{3}-j_{2}+m_{1}+s)!(j_{3}-j_{1}-m_{2% }+s)!},

ⓘ

Defines:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol
Symbols:: $!$ : factorial (as in $n!$ ), $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $\Delta(j_{1}j_{2}j_{3})$ : product and $s$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

… ►where

{{}_{3}F_{2}}

is defined as in §16.2. ►For alternative expressions for the

\mathit{3j}

symbol, written either as a finite sum or as other terminating generalized hypergeometric series

{{}_{3}F_{2}}

of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).

12: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The cofactor

A_{jk}

a_{jk}

is … ►For real-valued

a_{jk}

, … ►where

\omega_{1},\omega_{2},\dots,\omega_{n}

are the

n

th roots of unity (1.11.21). … ►If

D_{n}[a_{j,k}]

tends to a limit

L

n\to\infty

, then we say that the infinite determinant

D_{\infty}[a_{j,k}]

converges and

D_{\infty}[a_{j,k}]=L

. … ►The corresponding eigenvectors

\mathbf{a}_{1},\dots,\mathbf{a}_{n}

can be chosen such that they form a complete orthonormal basis in

\mathbf{E}_{n}

. …

14: 26.9 Integer Partitions: Restricted Number and Part Size

… ►

p_{k}\left(n\right)

denotes the number of partitions of

n

into at most

k

parts. See Table 26.9.1. … ►It follows that

p_{k}\left(n\right)

also equals the number of partitions of

n

into parts that are less than or equal to

k

. ►

p_{k}\left(\leq m,n\right)

is the number of partitions of

n

into at most

k

parts, each less than or equal to

m

. …

15: 16.12 Products

… ►

16.12.1

{{}_{0}F_{1}}\left(-;a;z\right){{}_{0}F_{1}}\left(-;b;z\right)={{}_{2}F_{3}}% \left({\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)\atop a,b,a+b-1};4z\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: (10.8.3), §10.8, §10.8, §16.12, Erratum (V1.0.17) for Section 10.8
Permalink:: http://dlmf.nist.gov/16.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

… ►

16.12.2

\left({{}_{2}F_{1}}\left({a,b\atop a+b+\frac{1}{2}};z\right)\right)^{2}={{}_{3% }F_{2}}\left({2a,2b,a+b\atop a+b+\frac{1}{2},2a+2b};z\right).

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.12, §17.9(iii), Erratum (V1.1.12) for Subsection 17.9(iii)
Permalink:: http://dlmf.nist.gov/16.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

… ►

16.12.3

\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)\right)^{2}=\sum_{k=0}^{\infty}% \frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}}{\left(c-\frac{1}{2}\right)_{k% }}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}k!}{{}_{4}F_{3}}\left({-\frac{1% }{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}\atop a+\frac{1}{2},b+% \frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k},

|z|<1

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.12
Permalink:: http://dlmf.nist.gov/16.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

…

16: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►The generalized hypergeometric function

{{}_{p}F_{q}}

with matrix argument

\mathbf{T}\in\boldsymbol{\mathcal{S}}

, numerator parameters

a_{1},\dots,a_{p}

, and denominator parameters

b_{1},\dots,b_{q}

is … ►

§35.8(iii) ${{}_{3}F_{2}}$ Case

… ►Let

c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}

. … ►Let

a_{1}+a_{2}+a_{3}+\frac{1}{2}(m+1)=b_{1}+b_{2}

; one of the

a_{j}

be a negative integer;

\Re\left(b_{1}-a_{1}\right)

\Re\left(b_{1}-a_{2}\right)

\Re\left(b_{1}-a_{3}\right)

\Re\left(b_{1}-a_{1}-a_{2}-a_{3}\right)>\frac{1}{2}(m-1)

. … ►Again, let

c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}

. …

$p, q$	nonnegative integers.
…
$\rselection{a_{1},a_{2},\dots,a_{p}\\ b_{1},b_{2},\dots,b_{q}}$	real or complex parameters.
…
$\mathbf{a}$	vector $(a_{1},a_{2},\dots,a_{p})$ .
$\mathbf{b}$	vector $(b_{1},b_{2},\dots,b_{q})$ .
…

18: 16.18 Special Cases

… ►The

{{}_{1}F_{1}}

and

{{}_{2}F_{1}}

functions introduced in Chapters 13 and 15, as well as the more general

{{}_{p}F_{q}}

functions introduced in the present chapter, are all special cases of the Meijer

G

-function. … ►

16.18.1

{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\left({% \textstyle\ifrac{\prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_% {k=1}^{p}\Gamma\left(a_{k}\right)}}\right){G^{1,p}_{p,q+1}}\left(-z;{1-a_{1},% \dots,1-a_{p}\atop 0,1-b_{1},\dots,1-b_{q}}\right)=\left({\textstyle\ifrac{% \prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_{k=1}^{p}\Gamma% \left(a_{k}\right)}}\right){G^{p,1}_{q+1,p}}\left(-\frac{1}{z};{1,b_{1},\dots,% b_{q}\atop a_{1},\dots,a_{p}}\right).

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.18.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.18 and Ch.16

►As a corollary, special cases of the

{{}_{1}F_{1}}

and

{{}_{2}F_{1}}

functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer

G

-function. …

19: 16.3 Derivatives and Contiguous Functions

… ►

16.3.1

\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p% }\atop b_{1},\dots,b_{q}};z\right)=\frac{{\left(\mathbf{a}\right)_{n}}}{{\left% (\mathbf{b}\right)_{n}}}{{}_{p}F_{q}}\left({a_{1}+n,\dots,a_{p}+n\atop b_{1}+n% ,\dots,b_{q}+n};z\right),

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(i), §16.3 and Ch.16

… ►

16.3.3

\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{\gamma-1}{{}_{p+1}F_% {q}}\left({\gamma,a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)\right)={% \left(\gamma\right)_{n}}z^{\gamma+n-1}{{}_{p+1}F_{q}}\left({\gamma+n,a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}};z\right),

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(i), §16.3 and Ch.16

►

16.3.4

\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{\gamma-1}{{}_{p}F_{q+1}}% \left({a_{1},\dots,a_{p}\atop\gamma,b_{1},\dots,b_{q}};z\right)\right)={\left(% \gamma-n\right)_{n}}z^{\gamma-n-1}{{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop% \gamma-n,b_{1},\dots,b_{q}};z\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(i), §16.3 and Ch.16

… ►Two generalized hypergeometric functions

{{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

are (generalized) contiguous if they have the same pair of values of

p

and

q

, and corresponding parameters differ by integers. … ►

16.3.7

{{}_{3}F_{2}}\left({a_{1}+2,a_{2},a_{3}\atop b_{1},b_{2}};z\right)a_{1}(a_{1}+% 1)(1-z)+{{}_{3}F_{2}}\left({a_{1}+1,a_{2},a_{3}\atop b_{1},b_{2}};z\right)a_{1% }\left(b_{1}+b_{2}-3a_{1}-2+z(2a_{1}-a_{2}-a_{3}+1)\right)+{{}_{3}F_{2}}\left(% {a_{1},a_{2},a_{3}\atop b_{1},b_{2}};z\right)\left((2a_{1}-b_{1})(2a_{1}-b_{2}% )+a_{1}-a_{1}^{2}-z(a_{1}-a_{2})(a_{1}-a_{3})\right)-{{}_{3}F_{2}}\left({a_{1}% -1,a_{2},a_{3}\atop b_{1},b_{2}};z\right)(a_{1}-b_{1})(a_{1}-b_{2})=0.

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.4(iii)
Permalink:: http://dlmf.nist.gov/16.3.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(ii), §16.3 and Ch.16

…

20: 3.6 Linear Difference Equations

… ►Given numerical values of

w_{0}

and

w_{1}

, the solution

w_{n}

of the equation …These errors have the effect of perturbing the solution by unwanted small multiples of

w_{n}

and of an independent solution

g_{n}

, say. … ►The unwanted multiples of

g_{n}

now decay in comparison with

w_{n}

, hence are of little consequence. … ►The latter method is usually superior when the true value of

w_{0}

is zero or pathologically small. … ►beginning with

e_{0}=w_{0}

. …

.2002年世界杯魔咒__『wn4.com_』_世界杯赛十六强产生情况_w6n2c9o_2022年11月29日5时25分54秒_l15ffl75v.com

11—20 of 785 matching pages

11: 34.2 Definition: $\mathit{3j}$ Symbol

12: 1.3 Determinants, Linear Operators, and Spectral Expansions

13: 1.12 Continued Fractions

14: 26.9 Integer Partitions: Restricted Number and Part Size

15: 16.12 Products

16: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8(iii) ${{}_{3}F_{2}}$ Case

17: 16.1 Special Notation

18: 16.18 Special Cases

19: 16.3 Derivatives and Contiguous Functions

20: 3.6 Linear Difference Equations

.2002年世界杯魔咒__『wn4.com_』_世界杯赛十六强产生情况_w6n2c9o_2022年11月29日5时25分54秒_l15ffl75v.com

11—20 of 785 matching pages

11: 34.2 Definition: 3 ⁢ j Symbol

12: 1.3 Determinants, Linear Operators, and Spectral Expansions

13: 1.12 Continued Fractions

14: 26.9 Integer Partitions: Restricted Number and Part Size

15: 16.12 Products

16: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8(iii) F 2 3 Case

17: 16.1 Special Notation

18: 16.18 Special Cases

19: 16.3 Derivatives and Contiguous Functions

20: 3.6 Linear Difference Equations

11: 34.2 Definition: $\mathit{3j}$ Symbol

§35.8(iii) ${{}_{3}F_{2}}$ Case