Fox H-function

Generalization of the Meijer G-function and the Fox–Wright function From Wikipedia, the free encyclopedia

Fox H-function

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

where L is a certain contour separating the poles of the two factors in the numerator.

Thumb
Plot of the Fox H function H((((a 1,α 1),...,(a n,α n)),((a n+1,α n+1),...,(a p,α p)),(((b 1,β 1),...,(b m,β m)),in ((b m+1,β m+1),...,(b q,β q))),z) with H(((),()),(((-1,1/2)),()),z)

Relation to other functions

Summarize
Perspective

Lambert W-function

A relation of the Fox H-Function to the -1 branch of the Lambert W-function is given by

where is the complex conjugate of .[1]

Meijer G-function

Compare to the Meijer G-function

The special case for which the Fox H reduces to the Meijer G is Aj = Bk = C, C > 0 for j = 1...p and k = 1...q :[2]

A generalization of the Fox H-function was given by Ram Kishore Saxena.[3][4] A further generalization of this function, useful in physics and statistics, was provided by A.M. Mathai and Ram Kishore Saxena.[5][6]

References

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