How are Fox-H functions useful in math?

The Fox H-function, as far as I know, is the most general families of functions - encompassing an even larger family of functions than the already very general Meijer G-function. While I've known about these functions for a while, I've never seen a good use of them - are they just generalizations for generalizations sake? Is there ever a time when the use of these functions has given insight into a problem that other assumptions (smooth, continuous, etc.) could not? What fields are they most used in?

Solution 1:

Fox–Wright function or even the Fox H-function are very useful on dealing the infinte series expressions whose its coefficient involves for example the gamma functions of linear expressions in the index $n$.

For example:

Definite Integral of $e^{ax+bx^c}$

Definite integration of a high order exponential function mixed with rational function

Can this series be expressed as a Hyper Geometric function

We can't say that the Fox H-function is the most general families of functions, since in fact there are no functions really in the most general families.

There are still some situations where Fox–Wright function or even the Fox H-function are still not enough to use, for example in Integral $\int_1^\infty\frac{dx}{1+2^x+3^x}$.

Solution 2:

This link http://www.dtic.mil/dtic/tr/fulltext/u2/a252517.pdf is a PhD-thesis using H-function in the study of statistical distributions. There are other papers with applications in this area, such as The Distribution of Products, Quotients and Powers of Independent H-Function Variates by Bradley D. Carter and Melvin D. Springer SIAM J. Appl. Math., 33(4), 542–558. (17 pages)

Read More: http://epubs.siam.org/doi/abs/10.1137/0133036

Search in google scholar will give you more papers in this area.

Read More: http://epubs.siam.org/doi/abs/10.1137/0133036