.韩国兰芝岛世界杯公园_『网址:68707.vip』篮球世界杯竞猜_b5p6v3_pr7bnb1nb.com

SearchAdvancedHelp

(0.005 seconds)

21—30 of 765 matching pages

21: 16.19 Identities

… ►

16.19.1 $$G_{p,q}^{m,n}(\frac{1}{z};\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})=G_{q,p}^{n,m}(z;\genfrac{}{}{0pt}{}{1-b_1,\mathrm{\dots },1-b_q}{1-a_1,\mathrm{\dots },1-a_p}),$$

ⓘ

Symbols:

$G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: 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

►

16.19.2 $$z^{\mu }{G}_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})=G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1+\mu ,\mathrm{\dots },a_p+\mu }{b_1+\mu ,\mathrm{\dots },b_q+\mu }),$$

ⓘ

Symbols:

$G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.19 and Ch.16

►

16.19.3 $$G_{p+1,q+1}^{m,n+1}(z;\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q,a_0})=G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q}),$$

ⓘ

Symbols:

$G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: 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

… ►

16.19.5 $$\mathit{\vartheta }{G}_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})=G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1-1,a_2,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})+(a_1-1)G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q}),$$

ⓘ

Symbols:

$G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $\mathit{\vartheta }$: 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 $${\int }_0^1{t}^{-a_0}{(1-t)}^{a_0-b_{q+1}-1}{G}_{p,q}^{m,n}(zt;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})dt=\mathrm{\Gamma }\left(a_0-b_{q+1}\right)G_{p+1,q+1}^{m,n+1}(z;\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_{q+1}}),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function, $dx$: differential of $x$ , $\int$: integral, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: 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

…

22: 17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

§17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

… ►

17.5.1 $${}_0{}^{}\varphi _0^{}(-;-;q,z)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(q;q)}_n}={(z;q)}_{\mathrm{\infty }};$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§17.2(iii), Erratum (V1.1.11) for Equation (17.5.1)

Permalink:

http://dlmf.nist.gov/17.5.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.1.11):

The constraint is not necessary and was removed.

See also:

Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.2 $${}_1{}^{}\varphi _0^{}(a;-;q,z)=\frac{{(az;q)}_{\mathrm{\infty }}}{{(z;q)}_{\mathrm{\infty }}},$$ ;

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.3 $${}_1{}^{}\varphi _0^{}(q^{-n};-;q,z)={(z{q}^{-n};q)}_n.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.4 $${}_1{}^{}\varphi _0^{}(0;-;q,z)=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{{(q;q)}_n}=\frac{1}{{(z;q)}_{\mathrm{\infty }}},$$ ;

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§17.2(iii)

Permalink:

http://dlmf.nist.gov/17.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.5, §17.5 and Ch.17

…

23: 16.8 Differential Equations

… ►is a value $z_0$ of $z$ at which all the coefficients $f_j(z)$, $j=0,1,\mathrm{\dots },n-1$, are analytic. If $z_0$ is not an ordinary point but ${(z-z_0)}^{n-j}{f}_j(z)$, $j=0,1,\mathrm{\dots },n-1$, are analytic at $z=z_0$, then $z_0$ is a regular singularity. … ►where ${\alpha }_j$ and ${\beta }_j$ are constants. … ►where $\ast$ indicates that the entry $1+b_j-b_j$ is omitted. … ►where $\ast$ indicates that the entry $1-a_j+a_j$ is omitted. …

24: 16.2 Definition and Analytic Properties

… ►Throughout this chapter it is assumed that none of the bottom parameters $b_1$, $b_2$, $\mathrm{\dots }$, $b_q$ is a nonpositive integer, unless stated otherwise. Then formally …Equivalently, the function is denoted by ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$ or ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$, and sometimes, for brevity, by ${}_p{}^{}F_q^{}\left(z\right)$. … ►Suppose first one or more of the top parameters $a_j$ is a nonpositive integer. … ►See §16.5 for the definition of ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ as a contour integral when $p>q+1$ and none of the $a_k$ is a nonpositive integer. … ►When $p\le q+1$ and $z$ is fixed and not a branch point, any branch of ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ is an entire function of each of the parameters $a_1,\mathrm{\dots },a_p,b_1,\mathrm{\dots },b_q$.

25: 16.6 Transformations of Variable

… ►

16.6.1 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,b,c}{a-b+1,a-c+1};z)={(1-z)}^{-a}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a-b-c+1,\frac{1}{2}a,\frac{1}{2}(a+1)}{a-b+1,a-c+1};\frac{-4z}{{(1-z)}^2}).$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.6

Permalink:

http://dlmf.nist.gov/16.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.6, §16.6 and Ch.16

… ►

16.6.2 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,2b-a-1,2-2b+a}{b,a-b+\frac{3}{2}};\frac{z}{4})={(1-z)}^{-a}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{3}a+\frac{2}{3}}{b,a-b+\frac{3}{2}};\frac{-27z}{4{(1-z)}^3}).$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.6

Permalink:

http://dlmf.nist.gov/16.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.6, §16.6 and Ch.16

►For Kummer-type transformations of ${}_2{}^{}F_2^{}$ functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).

26: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

… ►

§17.9(i) ${}_2{}^{}\varphi _1^{}\to {}_2{}^{}\varphi _2^{}$, ${}_3{}^{}\varphi _1^{}$, or ${}_3{}^{}\varphi _2^{}$

… ►

§17.9(ii) ${}_3{}^{}\varphi _2^{}\to {}_3{}^{}\varphi _2^{}$

►

Transformations of ${}_3{}^{}\varphi _2^{}$-Series

… ►

§17.9(iii) Further ${}_r{}^{}\varphi _s^{}$ Functions

►

Sears’ Balanced ${}_4{}^{}\varphi _3^{}$ Transformations

…

27: 10.63 Recurrence Relations and Derivatives

… ►

§10.63(i) ${\mathrm{ber}}_{\nu }x$, ${\mathrm{bei}}_{\nu }x$, ${\ker }_{\nu }x$, ${\mathrm{kei}}_{\nu }x$

►Let $f_{\nu }(x)$, $g_{\nu }(x)$ denote any one of the ordered pairs: ►

${\mathrm{ber}}_{\nu }x,{\mathrm{bei}}_{\nu }x;$

►

${\mathrm{bei}}_{\nu }x,-{\mathrm{ber}}_{\nu }x;$

►

${\ker }_{\nu }x,{\mathrm{kei}}_{\nu }x;$

…

28: 17.12 Bailey Pairs

… ►

${\beta }_n={\displaystyle \sum _{j=0}^n}{\alpha }_j{u}_{n-j}{v}_{n+j},$

►

${\gamma }_n={\displaystyle \sum _{j=n}^{\mathrm{\infty }}}{\delta }_j{u}_{j-n}{v}_{j+n}.$

… ►A sequence of pairs of rational functions of several variables $({\alpha }_n,{\beta }_n)$, $n=0,1,2,\mathrm{\dots }$, is called a Bailey pair provided that for each $n\geqq 0$ … ►If $({\alpha }_n,{\beta }_n)$ is a Bailey pair, then … ►If $({\alpha }_n,{\beta }_n)$ is a Bailey pair, then so is $({\alpha }_n^{\prime },{\beta }_n^{\prime })$, where …

29: 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 $e_0=w_0$. …

30: 23.7 Quarter Periods

… ►

23.7.1 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_1\right)=e_1+\sqrt{(e_1-e_3)(e_1-e_2)}=e_1+{\omega }_1^{-2}{(K\left(k\right))}^2{k}^{\prime },$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

►

23.7.2 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_2\right)=e_2-\mathrm{i}\sqrt{(e_1-e_2)(e_2-e_3)}=e_2-\mathrm{i}{\omega }_1^{-2}{(K\left(k\right))}^{2}k{k}^{\prime },$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

►

23.7.3 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_3\right)=e_3-\sqrt{(e_1-e_3)(e_2-e_3)}=e_3-{\omega }_1^{-2}{(K\left(k\right))}^{2}k,$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $k$: modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

…