# 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?

• It is usually very rare that general results can be proven using them but it does happen. – Ali Caglayan Oct 6 '14 at 18:50
• Yes, I am looking for examples where this is the case. – Danny W. Oct 6 '14 at 19:10
• Apparently there are applications in fractional diffusion – Ali Caglayan Oct 6 '14 at 19:13
• And finally here is a book discussing the H-Function in depth. – Ali Caglayan Oct 6 '14 at 19:14
• There are functions more general than Fox's H-function. The H-function has uses in fluids, special functions, Mellin transforms, fractional calculus, etc.. There are cases where it is better to use the H-function, rather than the G-function, due to the structure of the poles in the contour. The pole location is what makes the A, A*, and I-functions arise. – Leucippus Oct 6 '14 at 20:42

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 $\int_1^\infty\frac{dx}{1+2^x+3^x}$.

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)

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