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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 ( 1 z ; a 1 , , a p b 1 , , b q ) = G q , p n , m ( z ; 1 - b 1 , , 1 - b q 1 - a 1 , , 1 - a p ) ,
16.19.2 z μ G p , q m , n ( z ; a 1 , , a p b 1 , , b q ) = G p , q m , n ( z ; a 1 + μ , , a p + μ b 1 + μ , , b q + μ ) ,
16.19.3 G p + 1 , q + 1 m , n + 1 ( z ; a 0 , , a p b 1 , , b q , a 0 ) = G p , q m , n ( z ; a 1 , , a p b 1 , , b q ) ,
16.19.5 ϑ G p , q m , n ( z ; a 1 , , a p b 1 , , b q ) = G p , q m , n ( z ; a 1 - 1 , a 2 , , a p b 1 , , b q ) + ( a 1 - 1 ) G p , q m , n ( z ; a 1 , , a p b 1 , , b q ) ,
6: 16.18 Special Cases
§16.18 Special Cases
The F 1 1 and F 1 2 functions introduced in Chapters 13 and 15, as well as the more general F q p functions introduced in the present chapter, are all special cases of the Meijer G -function. …As a corollary, special cases of the F 1 1 and F 1 2 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: 16.21 Differential Equation
§16.21 Differential Equation
w = G p , q m , n ( z ; a ; b ) satisfies the differential equation …
8: 16 Generalized Hypergeometric Functions & Meijer G-Function
Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function
9: 9.7 Asymptotic Expansions
9.7.20 R n ( z ) = ( - 1 ) n k = 0 m - 1 ( - 1 ) k u k G n - k ( 2 ζ ) ζ k + R m , n ( z ) ,
9.7.21 S n ( z ) = ( - 1 ) n - 1 k = 0 m - 1 ( - 1 ) k v k G n - k ( 2 ζ ) ζ k + S m , n ( z ) ,
9.7.22 G p ( z ) = e z 2 π Γ ( p ) Γ ( 1 - p , z ) .
10: 10.40 Asymptotic Expansions for Large Argument
10.40.13 R ( ν , z ) = ( - 1 ) 2 cos ( ν π ) ( k = 0 m - 1 a k ( ν ) z k G - k ( 2 z ) + R m , ( ν , z ) ) ,