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Meijer G-function

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11: 16.1 Special Notation
The main functions treated in this chapter are the generalized hypergeometric function F q p ( a 1 , , a p b 1 , , b q ; z ) , the Appell (two-variable hypergeometric) functions F 1 ( α ; β , β ; γ ; x , y ) , F 2 ( α ; β , β ; γ , γ ; x , y ) , F 3 ( α , α ; β , β ; γ ; x , y ) , F 4 ( α , β ; γ , γ ; x , y ) , and the Meijer G -function G p , q m , n ( z ; a 1 , , a p b 1 , , b q ) . Alternative notations are F q p ( a b ; z ) , F q p ( a 1 , , a p ; b 1 , , b q ; z ) , and F q p ( a ; b ; z ) for the generalized hypergeometric function, F 1 ( α , β , β ; γ ; x , y ) , F 2 ( α , β , β ; γ , γ ; x , y ) , F 3 ( α , α , β , β ; γ ; x , y ) , F 4 ( α , β ; γ , γ ; x , y ) , for the Appell functions, and G p , q m , n ( z ; a ; b ) for the Meijer G -function.
12: Adri B. Olde Daalhuis
13: Adri B. Olde Daalhuis
14: 15.12 Asymptotic Approximations
where
15.12.13 G 0 ( ± β ) = ( 2 + e ± ζ ) c - b - ( 1 / 2 ) ( 1 + e ± ζ ) a - c + ( 1 / 2 ) ( z - 1 - e ± ζ ) - a + ( 1 / 2 ) β e ζ - e - ζ .
15: 10.17 Asymptotic Expansions for Large Argument
10.17.17 R ± ( ν , z ) = ( - 1 ) 2 cos ( ν π ) ( k = 0 m - 1 ( ± i ) k a k ( ν ) z k G - k ( 2 i z ) + R m , ± ( ν , z ) ) ,
16: 5.17 Barnes’ G -Function (Double Gamma Function)
When z in | ph z | π - δ ( < π ) , …
17: Richard A. Askey
18: 5.1 Special Notation
(For other notation see Notation for the Special Functions.) … The main functions treated in this chapter are the gamma function Γ ( z ) , the psi function (or digamma function) ψ ( z ) , the beta function B ( a , b ) , and the q -gamma function Γ q ( z ) . … Alternative notations for this function are: Π ( z - 1 ) (Gauss) and ( z - 1 ) ! . Alternative notations for the psi function are: Ψ ( z - 1 ) (Gauss) Jahnke and Emde (1945); Ψ ( z ) Davis (1933); F ( z - 1 ) Pairman (1919).
19: Bibliography F
  • J. L. Fields (1973) Uniform asymptotic expansions of certain classes of Meijer G -functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.
  • J. L. Fields (1983) Uniform asymptotic expansions of a class of Meijer G -functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.
  • 20: Bibliography M
  • O. I. Marichev (1984) On the Representation of Meijer’s G -Function in the Vicinity of Singular Unity. In Complex Analysis and Applications ’81 (Varna, 1981), pp. 383–398.
  • C. S. Meijer (1946) On the G -function. VII, VIII. Nederl. Akad. Wetensch., Proc. 49, pp. 1063–1072, 1165–1175 = Indagationes Math. 8, 661–670, 713–723 (1946).
  • J. W. Meijer and N. H. G. Baken (1987) The exponential integral distribution. Statist. Probab. Lett. 5 (3), pp. 209–211.