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11: 34.3 Basic Properties: $\mathit{3j}$ Symbol

… ►When any one of

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

is equal to

0,\tfrac{1}{2}

, or

1

, the

\mathit{3j}

symbol has a simple algebraic form. …For these and other results, and also cases in which any one of

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

\frac{3}{2}

2

, see Edmonds (1974, pp. 125–127). … ►Even permutations of columns of a

\mathit{3j}

symbol leave it unchanged; odd permutations of columns produce a phase factor

(-1)^{j_{1}+j_{2}+j_{3}}

, for example, … ►

34.3.15

(2j_{1}+1)\left((j_{2}(j_{2}+1)-j_{3}(j_{3}+1))m_{1}-j_{1}(j_{1}+1)(m_{3}-m_{2% })\right)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=(j_{1}+1)\left(j_{1}^{2}-(j_{2}-j_{3})^{2}% \right)^{\frac{1}{2}}\left((j_{2}+j_{3}+1)^{2}-j_{1}^{2}\right)^{\frac{1}{2}}% \left(j_{1}^{2}-m_{1}^{2}\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}-1&j_{2}&j_{% 3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}+j_{1}\left((j_{1}+1)^{2}-(j_{2}-j_{3})^{2}% \right)^{\frac{1}{2}}\left((j_{2}+j_{3}+1)^{2}-(j_{1}+1)^{2}\right)^{\frac{1}{% 2}}\left((j_{1}+1)^{2}-m_{1}^{2}\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}+1&j_% {2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.

ⓘ

Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.3.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(iii), §34.3 and Ch.34

… ►For the polynomials

P_{l}

see §18.3, and for the function

Y_{{l},{m}}

see §14.30. …

12: 22.7 Landen Transformations

… ►

22.7.1

k_{1}=\frac{1-k^{\prime}}{1+k^{\prime}},

ⓘ

Defines:: $k_{1}$ : change of variable (locally)
Symbols:: $k^{\prime}$ : complementary modulus
A&S Ref:: 16.12.1
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

►

22.7.2

\operatorname{sn}\left(z,k\right)=\frac{(1+k_{1})\operatorname{sn}\left(z/(1+k% _{1}),k_{1}\right)}{1+k_{1}{\operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}% \right)},

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.2
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

►

22.7.3

\operatorname{cn}\left(z,k\right)=\frac{\operatorname{cn}\left(z/(1+k_{1}),k_{% 1}\right)\operatorname{dn}\left(z/(1+k_{1}),k_{1}\right)}{1+k_{1}{% \operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}\right)},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.3
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

►

22.7.4

\operatorname{dn}\left(z,k\right)=\frac{{\operatorname{dn}}^{2}\left(z/(1+k_{1% }),k_{1}\right)-(1-k_{1})}{1+k_{1}-{\operatorname{dn}}^{2}\left(z/(1+k_{1}),k_% {1}\right)}.

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.4
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

… ►

k_{2}^{\prime}=\frac{1-k}{1+k},

…

13: 10.51 Recurrence Relations and Derivatives

… ►Let

f_{n}(z)

denote any of

\mathsf{j}_{n}\left(z\right)

\mathsf{y}_{n}\left(z\right)

{\mathsf{h}^{(1)}_{n}}\left(z\right)

, or

{\mathsf{h}^{(2)}_{n}}\left(z\right)

. … ►

nf_{n-1}(z)-(n+1)f_{n+1}(z)=(2n+1)f_{n}^{\prime}(z),

n=1,2,\dots

… ►Let

g_{n}(z)

denote

{\mathsf{i}^{(1)}_{n}}\left(z\right)

{\mathsf{i}^{(2)}_{n}}\left(z\right)

, or

(-1)^{n}

\mathsf{k}_{n}\left(z\right)

. Then … ►

ng_{n-1}(z)+(n+1)g_{n+1}(z)=(2n+1)g_{n}^{\prime}(z),

n=1,2,\dotsc

…

14: 26.3 Lattice Paths: Binomial Coefficients

… ►The number of lattice paths from

(0,0)

(m,n)

m\leq n

, that stay on or above the line

y=x

\genfrac{(}{)}{0.0pt}{}{m+n}{m}-\genfrac{(}{)}{0.0pt}{}{m+n}{m-1}.

… ►

26.3.3

\sum_{n=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{n}x^{n}=(1+x)^{m},

m=0,1,\dotsc

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.1
Permalink:: http://dlmf.nist.gov/26.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §26.3(ii), §26.3 and Ch.26

►

26.3.4

\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt}{}{m+n}{m}x^{m}=\frac{1}{(1-x)^{n+1}},

|x|<1

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.1 (in slightly different form)
Permalink:: http://dlmf.nist.gov/26.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.3(ii), §26.3 and Ch.26

… ►

26.3.7

\genfrac{(}{)}{0.0pt}{}{m+1}{n+1}=\sum_{k=n}^{m}\genfrac{(}{)}{0.0pt}{}{k}{n},

m\geq n\geq 0

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : nonnegative integer, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.3.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.3(iii), §26.3 and Ch.26

… ►

26.3.10

\genfrac{(}{)}{0.0pt}{}{m}{n}=\sum_{k=0}^{n}(-1)^{n-k}\genfrac{(}{)}{0.0pt}{}{% m+1}{k},

m\geq n\geq 0

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : nonnegative integer, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.3.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §26.3(iv), §26.3 and Ch.26

…

15: 24.6 Explicit Formulas

… ►The identities in this section hold for

n=1,2,\dotsc

. … ►

24.6.1

B_{2n}=\sum_{k=2}^{2n+1}\frac{(-1)^{k-1}}{k}{2n+1\choose k}\sum_{j=1}^{k-1}j^{% 2n},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

►

24.6.2

B_{n}=\frac{1}{n+1}\sum_{k=1}^{n}\sum_{j=1}^{k}(-1)^{j}j^{n}{\genfrac{(}{)}{0.% 0pt}{}{n+1}{k-j}}\Bigg{/}{\genfrac{(}{)}{0.0pt}{}{n}{k}},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

… ►

24.6.4

E_{2n}=\sum_{k=1}^{n}\frac{1}{2^{k-1}}\sum_{j=1}^{k}(-1)^{j}{2k\choose k-j}j^{% 2n},

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

… ►

24.6.10

E_{n}=\frac{1}{2^{n}}\sum_{k=1}^{n+1}{n+1\choose k}\sum_{j=0}^{k-1}(-1)^{j}(2j% +1)^{n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6, §24.6
Permalink:: http://dlmf.nist.gov/24.6.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

…

16: 34.4 Definition: $\mathit{6j}$ Symbol

… ►

34.4.1

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\sum_{m_{r}m^{\prime}_{s}}(-1)^{l_{1}+m^{\prime% }_{1}+l_{2}+m^{\prime}_{2}+l_{3}+m^{\prime}_{3}}\*\begin{pmatrix}j_{1}&j_{2}&j% _{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&l_{2}&l_{3}\\ m_{1}&m^{\prime}_{2}&-m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&j_{2}&l_% {3}\\ -m^{\prime}_{1}&m_{2}&m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&j_% {3}\\ m^{\prime}_{1}&-m^{\prime}_{2}&m_{3}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $r$ : nonnegative integer
Referenced by:: §34.10, §34.4, §34.9
Permalink:: http://dlmf.nist.gov/34.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.4 and Ch.34

… ►Except in degenerate cases the combination of the triangle inequalities for the four

\mathit{3j}

symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths

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

; see Figure 34.4.1. … ►

34.4.3

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}={(-1)^{j_{1}+j_{3}+l_{1}+l_{3}}}\frac{\Delta(j_% {1}j_{2}j_{3})\Delta(j_{2}l_{1}l_{3})(j_{1}-j_{2}+l_{1}+l_{2})!(-j_{2}+j_{3}+l% _{2}+l_{3})!(j_{1}+j_{3}+l_{1}+l_{3}+1)!}{\Delta(j_{1}l_{2}l_{3})\Delta(j_{3}l% _{1}l_{2})(j_{1}-j_{2}+j_{3})!(-j_{2}+l_{1}+l_{3})!(j_{1}+l_{2}+l_{3}+1)!(j_{3% }+l_{1}+l_{2}+1)!}\*{{}_{4}F_{3}}\left({-j_{1}+j_{2}-j_{3},j_{2}-l_{1}-l_{3},-% j_{1}-l_{2}-l_{3}-1,-j_{3}-l_{1}-l_{2}-1\atop-j_{1}+j_{2}-l_{1}-l_{2},j_{2}-j_% {3}-l_{2}-l_{3},-j_{1}-j_{3}-l_{1}-l_{3}-1};1\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, $!$ : factorial (as in $n!$ ), $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product
Referenced by:: §18.38(iii)
Permalink:: http://dlmf.nist.gov/34.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §34.4 and Ch.34

►where

{{}_{4}F_{3}}

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

\mathit{6j}

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

{{}_{4}F_{3}}

of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).

17: 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. …

18: 10.53 Power Series

… ►

10.53.2

\mathsf{y}_{n}\left(z\right)=-\frac{1}{z^{n+1}}\sum_{k=0}^{n}\frac{(2n-2k-1)!!% (\frac{1}{2}z^{2})^{k}}{k!}+\frac{(-1)^{n+1}}{z^{n+1}}\sum_{k=n+1}^{\infty}% \frac{(-\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}.

ⓘ

Symbols:: $!!$ : double factorial, $!$ : factorial (as in $n!$ ), $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 10.1.3
Referenced by:: §10.53
Permalink:: http://dlmf.nist.gov/10.53.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.53 and Ch.10

►

10.53.3

{\mathsf{i}^{(1)}_{n}}\left(z\right)=z^{n}\sum_{k=0}^{\infty}\frac{(\frac{1}{2% }z^{2})^{k}}{k!(2n+2k+1)!!},

ⓘ

Symbols:: $!!$ : double factorial, $!$ : factorial (as in $n!$ ), ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 10.2.5
Referenced by:: §10.53
Permalink:: http://dlmf.nist.gov/10.53.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §10.53 and Ch.10

►

10.53.4

{\mathsf{i}^{(2)}_{n}}\left(z\right)=\frac{(-1)^{n}}{z^{n+1}}\sum_{k=0}^{n}% \frac{(2n-2k-1)!!(-\frac{1}{2}z^{2})^{k}}{k!}+\frac{1}{z^{n+1}}\sum_{k=n+1}^{% \infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}.

ⓘ

Symbols:: $!!$ : double factorial, $!$ : factorial (as in $n!$ ), ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 10.2.6
Referenced by:: §10.53
Permalink:: http://dlmf.nist.gov/10.53.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §10.53 and Ch.10

►For

{\mathsf{h}^{(1)}_{n}}\left(z\right)

and

{\mathsf{h}^{(2)}_{n}}\left(z\right)

combine (10.47.10), (10.53.1), and (10.53.2). For

\mathsf{k}_{n}\left(z\right)

combine (10.47.11), (10.53.3), and (10.53.4).

19: 16.19 Identities

… ►

16.19.1

{G^{m,n}_{p,q}}\left(\frac{1}{z};{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}% \right)={G^{n,m}_{q,p}}\left(z;{1-b_{1},\dots,1-b_{q}\atop 1-a_{1},\dots,1-a_{% p}}\right),

ⓘ

Symbols:: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ 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
Referenced by:: §16.21
Permalink:: http://dlmf.nist.gov/16.19.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.19 and Ch.16

… ►

16.19.3

{G^{m,n+1}_{p+1,q+1}}\left(z;{a_{0},\dots,a_{p}\atop b_{1},\dots,b_{q},a_{0}}% \right)={G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}% \right),

ⓘ

Symbols:: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ 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/16.19.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.19 and Ch.16

►

16.19.4

{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=\frac% {2^{p+1+b_{1}+\dots+b_{q}-m-n-a_{1}-\dots-a_{p}}}{\pi^{m+n-\frac{1}{2}(p+q)}}{% G^{2m,2n}_{2p,2q}}\left(2^{2p-2q}z^{2};{\frac{1}{2}a_{1},\frac{1}{2}a_{1}+% \frac{1}{2},\dots,\frac{1}{2}a_{p},\frac{1}{2}a_{p}+\frac{1}{2}\atop\frac{1}{2% }b_{1},\frac{1}{2}b_{1}+\frac{1}{2},\dots,\frac{1}{2}b_{q},\frac{1}{2}b_{q}+% \frac{1}{2}}\right),

ⓘ

Symbols:: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $\pi$ : the ratio of the circumference of a circle to its diameter, $p$ : 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/16.19.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.19 and Ch.16

►

16.19.5

\vartheta{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}% \right)={G^{m,n}_{p,q}}\left(z;{a_{1}-1,a_{2},\dots,a_{p}\atop b_{1},\dots,b_{% q}}\right)+(a_{1}-1){G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots% ,b_{q}}\right),

ⓘ

Symbols:: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ 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, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $\vartheta$ : differential operator
Permalink:: http://dlmf.nist.gov/16.19.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §16.19 and Ch.16

►

16.19.6

\int_{0}^{1}t^{-a_{0}}(1-t)^{a_{0}-b_{q+1}-1}{G^{m,n}_{p,q}}\left(zt;{a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}}\right)\,\mathrm{d}t=\Gamma\left(a_{0}-b_{q% +1}\right){G^{m,n+1}_{p+1,q+1}}\left(z;{a_{0},\dots,a_{p}\atop b_{1},\dots,b_{% q+1}}\right),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $p$ : 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
Referenced by:: §16.19
Permalink:: http://dlmf.nist.gov/16.19.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §16.19 and Ch.16

…

20: 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

…

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11—20 of 848 matching pages

11: 34.3 Basic Properties: 3 ⁢ j Symbol

12: 22.7 Landen Transformations

13: 10.51 Recurrence Relations and Derivatives

14: 26.3 Lattice Paths: Binomial Coefficients

15: 24.6 Explicit Formulas

16: 34.4 Definition: 6 ⁢ j Symbol

17: 16.18 Special Cases

18: 10.53 Power Series

19: 16.19 Identities

20: 16.12 Products

11: 34.3 Basic Properties: $\mathit{3j}$ Symbol

16: 34.4 Definition: $\mathit{6j}$ Symbol