Meijer G-function

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1: 16.17 Definition

§16.17 Definition

… ►Then the Meijer $G$-function is defined via the Mellin–Barnes integral representation: … ►

…

Figure 16.17.1: s-plane. Path $L$ for the integral representation (16.17.1) of the Meijer $G$-function.

►When more than one of Cases (i), (ii), and (iii) is applicable the same value is obtained for the Meijer $G$-function. … ►Then …

2: 16.22 Asymptotic Expansions

§16.22 Asymptotic Expansions

►Asymptotic expansions of $G_{p,q}^{m,n}(z;𝐚;𝐛)$ for large $z$ are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer $G$-functions with large parameters see Fields (1973, 1983).

3: 16.20 Integrals and Series

§16.20 Integrals and Series

►Integrals of the Meijer $G$-function are given in Apelblat (1983, §19), Erdélyi et al. (1953a, §5.5.2), Erdélyi et al. (1954a, §§6.9 and 7.5), Luke (1969a, §3.6), Luke (1975, §5.6), Mathai (1993, §3.10), and Prudnikov et al. (1990, §2.24). Extensive lists of Laplace transforms and inverse Laplace transforms of the Meijer $G$-function are given in Prudnikov et al. (1992a, §3.40) and Prudnikov et al. (1992b, §3.38). ►Series of the Meijer $G$-function are given in Erdélyi et al. (1953a, §5.5.1), Luke (1975, §5.8), and Prudnikov et al. (1990, §6.11).

4: 16.26 Approximations

§16.26 Approximations

►For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).

5: 16.19 Identities

§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

…

6: 16.18 Special Cases

§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. …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. Representations of special functions in terms of the Meijer $G$-function are given in Erdélyi et al. (1953a, §5.6), Luke (1969a, §§6.4–6.5), and Mathai (1993, §3.10).

7: 15.12 Asymptotic Approximations

… ►where ►

15.12.13 $$G_0(\pm \beta )={\left(2+{\mathrm{e}}^{\pm \zeta }\right)}^{c-b-(1/2)}{\left(1+{\mathrm{e}}^{\pm \zeta }\right)}^{a-c+(1/2)}{\left(z-1-{\mathrm{e}}^{\pm \zeta }\right)}^{-a+(1/2)}\sqrt{\frac{\beta }{{\mathrm{e}}^{\zeta }-{\mathrm{e}}^{-\zeta }}}.$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\zeta$: change of variable, $\beta$ and $G_0(\pm \zeta )$: function

Permalink:

http://dlmf.nist.gov/15.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

…

8: 16.21 Differential Equation

§16.21 Differential Equation

► $w=G_{p,q}^{m,n}(z;𝐚;𝐛)$ satisfies the differential equation … ►

16.21.2 $$G_{p,q}^{1,p}(z{\mathrm{e}}^{(p-m-n-1)\pi \mathrm{i}};\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_j,b_1,\mathrm{\dots },b_{j-1},b_{j+1},\mathrm{\dots },b_q}),$$ $j=1,\mathrm{\dots },q$.

ⓘ

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, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $p$: 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/16.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.21 and Ch.16

…

9: 16 Generalized Hypergeometric Functions & Meijer G-Function

Chapter 16 Generalized Hypergeometric Functions and Meijer $G$-Function

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10: 16.1 Special Notation

… ►The main functions treated in this chapter are the generalized hypergeometric function ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$, the Appell (two-variable hypergeometric) functions $F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, and the Meijer $G$-function $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$. Alternative notations are ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$, and ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ for the generalized hypergeometric function, $F_1(\alpha ,\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ,\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime },\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, for the Appell functions, and $G_{p,q}^{m,n}(z;𝐚;𝐛)$ for the Meijer $G$-function.