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 .
It is defined by a Mellin–Barnes integral
:
H_^ \!\left \begin
( a_1 , A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\
( b_1 , B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end \right. \right= \frac\int_L
\frac

z^ \, ds,

where ''L'' is a certain contour separating the poles of the two factors in the numerator. Compare to the Meijer G-function:

:$G_^ \!\left( \left. \begin a_1, \dots, a_p \\ b_1, \dots, b_q \end \; \ \, z \right) = \frac \int_L \frac \,z^s \,ds.$

The special case for which the Fox H reduces to the Meijer G is ''A''_''j'' = ''B''_''k'' = ''C'', ''C'' > 0 for ''j'' = 1...''p'' and ''k'' = 1...''q'' :
:
H_^ \!\left \begin
( a_1 , C ) & ( a_2 , C ) & \ldots & ( a_p , C ) \\
( b_1 , C ) & ( b_2 , C ) & \ldots & ( b_q , C ) \end \right. \right= \frac
G_^ \!\left( \left. \begin a_1, \dots, a_p \\ b_1, \dots, b_q \end \; \ \, z^ \right).

A generalization of the Fox H-function is given by Ram Kishore Saxena Innayat Hussain . For a further generalization of this function, useful in physics and statistics was given by A.M.Mathai and Ram Kishore Saxena, see .

References

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External links

hypergeom
on GitLab

Use in solving $x+x^a=y$
on MathOverflow

Hypergeometric functions
Special functions

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