Fox H function - Alchetron, The Free Social Encyclopedia

In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function introduced by Charles Fox (1961). It is defined by a Mellin–Barnes integral

H p , q m , n [ z | ( a 1 , A 1 ) ( a 2 , A 2 ) … ( a p , A p ) ( b 1 , B 1 ) ( b 2 , B 2 ) … ( b q , B q ) ] = 1 2 π i ∫ L ( ∏ j = 1 m Γ ( b j + B j s ) ) ( ∏ j = 1 n Γ ( 1 − a j − A j s ) ) ( ∏ j = m + 1 q Γ ( 1 − b j − B j s ) ) ( ∏ j = n + 1 p Γ ( a j + A j s ) ) z − s d s

where L is a certain contour separating the poles of the two factors in the numerator. Another generalization of Fox H-function is given by Innayat Hussain AA (1987). For a further generalization of this function, useful in Physics and Statistics, see Rathie (1997).

The special case for which the Fox H-function reduces to the Meijer G-function is A_j = B_k = C, C > 0 for j = 1...p and k = 1...q (Srivastava 1984, p. 50):

H p , q m , n [ z | ( a 1 , C ) ( a 2 , C ) … ( a p , C ) ( b 1 , C ) ( b 2 , C ) … ( b q , C ) ] = 1 C G p , q m , n ( a 1 , … , a p b 1 , … , b q | z 1 / C ) .

References

Fox H-function Wikipedia

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