These three transforms play key roles in this article:

a) For a function such that for some real value k, the Laplace transform of, , is

where r is a complex variable. Conversely, if is analytic, of order in some half-plane, with real and, then its inverse is, uniquely determined by:

, ,

evaluated over any line in the complex plane.

Two functions with same Laplace transform are identical. If is the density of X,

is the moment generating function of X.

b) The Fourier transform of, , s.t., for some real k is:

and its inverse is

.

c) The Mellin transform of, , where for some real k, is defined by:

Then its inverse Mellin transform is:

Equation (3) is valid under the condition that (2) exists as an analytic function of the complex variable s, for. The integral is independent of w.

In this section we consider only series and their limits. We have the series representation of the exponential function, which is a special case of the hypergeometric series:

, ,

where the ratio of two consecutive coefficients:

,.

One generalization of this notion is associated with the hypergeometric series, where this ratio is a rational expression of n. Then we should have:

in its decomposition into a rational form, i.e. depending on constants and 2 other constants r and s.

The corresponding series is then,

which becomes, after rearranging and change of scale:

The hypergeometric series has the above expression. For, , we have Gauss hypergeometric series in 3 parameters

where the Pochhammer symbol is,. Equation (4) reduces to the geometric series for, hence its name. a and b can be any real or complex value but c must be different from a negative integer. If a or b is zero or a negative integer the series becomes a polynomial.

The first work on hypergeometric function was made by Euler in 1687, when he studied series (4), as solution to Equation (21). Gauss (1812) and Riemann (1857) continued Euler’s work in the complex domain and solved the associated multivaluedness problem, presently known as monodromy problem.

The whole field of Special Functions is characterized by integral representations of various kinds (see e.g. Lebedev [19] ). We first recall the integral representation of the upper tail of the gamma distribution by a finite sum, well known in elementary statistics (Hogg and Craig [20] , p. 132):

.

Similarly, we have the integral representation of an infinite series. There are several advantages in dealing with an integral instead of a series, as already remarked by Carlson [10] . Continuity and even analyticity are usually provided by the integral, hence leading to a deeper study of its properties and extensions, and also faster convergence on a digital computer. The hypergeometric series (4), with its convergence region will be of limited interest if it cannot be extended to the whole complex plane. The principle of analytic continuation in complex analysis will permit us to precisely do that operation.

There are three integral representations of, of increasing complexity, that serve three different purposes, and propose three different ways in computing the values of a hypergeometric function:

In an effort to generalize and make sense of the case, we define the H-function, using Mellin-Barnes formula, and consider the ratio of two products of gamma functions as integrand. Fox’s H-function, is hence defined as the integral along the complex contour L, of the expression, i.e.

where

The Meijer function is a special case, when, , of. We notice that (15) is just one way to generalize the integrant in (13).

From (3) and (14) we can see that G and H-functions are Inverse Mellin Transforms of and that the Mellin-Barnes integral is now taken as the definition of the G-function, instead of a series, or a definite integral, as in preceding sections. But under some mild conditions on, the function can be expressed as a function and conversely:

The G-function converges when L is taken as one of the two paths encircling the right poles (related to), or the left poles (related to), defining for and respectively, depending on the values of p and q, or a third path can be taken as the vertical axis, separating them, for and, with, following Jordan’s lemma. For discussions on the G-function see Mathai and Saxena [23] , and on the H-function, see Springer [26] , which also treats some uses of these functions in Statistics, as well as some computational issues. We wish to mention the following points:

1) The three paths of integration are similar to those of, and the convergence of H and G now depends on, and also on.

2) There are numerous properties of the Meijer G-functions: Contiguity, relations with themselves, derivatives, integral transforms, etc., that we cannot list here, due to space limitation. They can be seen in Mathai and Saxena [23] .

3) The H-function can be brought to the G-function for computation, when all are positive rational numbers, by a simple change of variable and using the multiplication formula for gamma functions.

4) The Euler and Laplace representations of G involve other G-functions with lesser parameters, similarly to ((9) and (9’)):

,

.

The Laplace transforms pair of G:

,

and its inverse

(Taking we have the Laplace transform of).

Also, the relation permits the ana-

lytic continuation of from inside the unit disk to outside it, with an appropriate cut, if necessary, depending on the value of.

Generalizations of H-functions: We will not go beyond the H-function, but it is worth mentioning that generalized forms of H exist, e.g. the one in Rathie [27] , which depends on an additional set of parameters. It is defined by:

.

This function should not be confused with Carlson’s function [10] defined in section 6.3.

But the Fox-Wright function

can be expressed as a H-function, while the MacRobert E-function, defined below, can be expressed as a G-function.

,

,

,

.

a) Each of these functions can be expressed as an infinite series in x alone, with coefficients containing Gauss function. For example, we have:

,

and, similarly for other functions.

Also, and its generalization (see sect. 3.2), seem to be the most important among these functions, with numerous applications in several disciplines.

b) Other hypergeometric functions, 34 in total, have been defined by Jacob Horn. The main ones are, , and, , ,. They will not be treated here. Whittaker, Pandey, Srivastava, Wright, Macrobert, Kampé de Fériet, and Lauricella-Saran functions, as well as lesser-known functions, will not be treated either, due to space limitation, see Exton [28] .

c) Functions and of Humbert: These 7 confluent forms of the Appell series are denoted, , , , , , , and are limiting values of Appell functions. For example:

.

They have a particular role in the representation of Appell functions. For example, we have as a function of. The corresponding 13 confluent forms of the Horn series, denoted, will not be discussed in detail here. We refer to Srivastava and Karlsson [29] for these functions.

Lauricella functions are extensions of Appell functions to n variables, where, with, , and corresponding, respectively, to Appell functions, , , and in 2 variables.

And the Humbert function in n variables is defined as follows:

Partial and ordinary differential equations play an important role in Applied mathematics and to a lesser extent, in Statistics. They still constitute a major tool in the study of hypergeometric functions in pure and applied mathematics.

a) The basic hypergeometric equation (of Fuchsian type) in one variable is:

a solution of which, obtained under series form, is. Every second-order linear ODE with three regular singular points can be transformed into this equation. There is an extensive discussion in the literature (e.g. Lebedev [19] ) on values of this solution at regular singularities 0, 1 and, as well as when there are relations between coefficients containing integers. When c is not an integer the other solution independent of the first is:. The general solution of (21) is hence:, with constants.

Concerning other hypergeometric functions, the equation satisfied by G-functions is:

,

and, for partial differential systems, there is one for each Lauricella function and. For this last function, it is, for example:

.

The resolution of these systems is not simple and there are up to sixty solutions. Basically, there are several independent solutions which include the hypergeometric series obtained when using infinite series in searching for solutions. We invite the reader to consult Exton ( [28] , Chapter 5). We will again mention differential equations since these pde’s will be at the heart of A-hypergeometric systems presented later.

b) The differential equation satisfied by is

where. There are p more solutions if all are not integers. They are inde-

pendent, when the difference between any two of the values: is not an integer.

Differential equations for one-matrix hypergeometric functions can be considered. A short introduction to this topic is given by Muirhead ( [30] , chapter 7). Also

As for one variable, we use the Mellin-Barnes approach to define this function. Buschman [31] defined -functions of 2 variables as an integral in the complex planes of a ratio of two products, i.e.

where are curves in the two complex planes, and

But, as pointed out by Nguyen Thanh Hai and Yakubovich [32] , the representation as the residue sum still has difficulties. There are some results on the Cauchy integral formula for several complex variables but it is still unclear how the residues can be computed in the general case. Hence, like the univariate case, not all of these integrals can be expressed as double series. Euler and Laplace representations, in function of other functions, are quite complicated and are not given here. More information on can be obtained from Mathai and Saxena [33] . More advanced results on H are presented in [34] . We will not elaborate on these results, and neither on other definitions of encountered in the literature.

In going from a scalar variable to a matrix, there are several difficulties to define the hypergeometric function. First, functions of matrices, square or rectangular, can only be defined under certain conditions (Higham [35] ), and they can be scalar-valued, or matrix-valued. Secondly, for scalar-valued matrix functions, they are usually based on symmetric functions of the matrix entries, or of the eigenvalues of the input square matrices. A simple introduction to this topic is given by Pham-Gia and Turkkan [36] . We recall here some basic notions of calculus on matrices, that are not so obvious.

Domain of integration: Let be a scalar function of the matrix X. Then is the iterated integral of for each entry of X separately, over the region located within the space defined by the simplex bounding the ranges of the entries of X.

Since it is usually very difficult to carry out direct integration over a complex region, integration on simple regions are frequently done by changes of variables, matrix decompositions, and finally identification with known expressions.

We have also the region as the set of all square matrices such that X and are positive definite, which reduces to the continuous variable x being between 0 and 1 in a unidimensional space.

Jacobian and Exterior product: In carrying out the required changes of variables mentioned above we have to use jacobians, and using wedge products

and exterior forms would be helpful. We have, for example, for

and transforms, the result, with where the jacobian of the transformation is the absolute value of the determinant.

The multigamma function: Let, where is the exponential of the trace of X, with the domain of positive definite matrices being, we have the multivariate gamma function

. Carrying out integration as explained above, we obtain a product of m ordinary gamma functions.

The Matrix Laplace Transform: Let be a scalar function of the positive definite symmetric matrix S. Its Laplace transform is defined by symmetric.

We assume that the integral converges in the half-plane, for some positive definite matrix. Then is analytic in Z in the half-plane. If and, then the inverse Laplace transform is:.

Gupta and Nagar [37] can be consulted for several notions on matrix variate distributions.

To define hypergeometric functions in one matrix argument, there are three approaches offered in the literature.

Hypergeometric function in two matrix variates is present in a basic result of multivariate analysis (Muirhead ( [30] , p.259)), defined with zonal polynomials, since it does not seem convenient, although possible, to define using either Laplace transform, or M-transform method.

.

Here, is the normalized invariant measure on, X and Y are symmetric matrices, and

.

It is straightforward to extend the number of matrices to (Mathai and Pederzoli [46] ), even when using H and G functions.

In the past several serious efforts were made to find so-called computable forms for H and G-functions, with some success since the formulas obtained are extremely complicated (see e.g. Mathai and Saxena [23] ). Classical hypergeometric functions and G-functions, in one scalar variable, are now found in most commercial software (Maple, Mathematica, Matlab, etc.). In determining the numerical value of G by Mellin- Barnes method, the number of poles can influence its accuracy, since this value is computed from the numerical values of residues at regular poles, as presented in Springer [26] . Pearson’s thesis [47] discusses several points on the computation of. Table 17 there makes some recommendations on methods to be used. It is interesting to note here that, usually, the series converges very slowly while the integral (5), or (5’), converges quite rapidly. Also, (5) and/or (5’) remains valid when parameters differ by integers while (7) has to be adjusted. Section 2.3.1 above can be consulted for these questions.

G-functions are used lately to carry out difficult definite integrals computations (Adamchik [48] ) because of various relations that exist between transforms of G-functions, and between products of G-functions. For example,

with the values of the parameters on the RHS obtainable from those of the LHS.

The two integral representations of G below are also used to deal with definite integrals:

and

.

These properties have been used in the software on integration, called REDUCE (Gaskell [49] , http://www.reduce-algebra.com/).

There are serious difficulties, however, in carrying out computations for hypergeometric functions in one or several matrix arguments, beginning with difficulties associated with zonal polynomials. Gutiérrez, Rodriguez and Saéz [50] are the early authors who reported results on this topic. Their work was limited to and and the values obtained from truncated series are quite good. However, there are already 627 zonal polynomials to be computed when, demanding a lot of computer time. Koev and Edelman [51] have succeeded to have better accuracy and a much shorter computer time, by using Jack polynomials (which are generalizations of zonal polynomials), with an updating strategy to compute them. Butler and Wood [52] , using the same Laplace approximation approach applied to one matrix argument in an earlier paper, reported fair to excellent accuracies in approximating, for equal 0 or 1.

The theory of Grobner basis has great influence on computations lately, in several domains of mathematics and algebraic statistics. Saito, Sturmfels and Nakayama [12] used it to study and approximate hypergeometric integrals belonging to the GKZ family. They also used it to study systems of multidimensional hypergeometric partial differential equations. This approach can be compared to the Perturbations approach to solve a problem in classical mathematics. There are several important results in [12] but they lie outside the scope of this survey.

As long as the computation of results cannot be made, progresses in that area are hampered. This is the case of zonal polynomials, which looked promising when they were first introduced, but there is now a high volume of highly complex theoretical results, and formulas, in need of confirmation by computation. Fortunately, fractional calculus applied to hypergeometric functions has some recent software and numerical methods recently made available (see Baleanu et al. [53] , and the list therein of six hundred references).

New statistical technics are required in face of the data evolution. Now, the number of variables can be much larger than the sample size, as is frequently encountered in data sets in some statistical/biometric problems. Ledoit and Wolf ’s results [54] on estimating the covariance matrix in that case, are of interest. Similar approaches, related to other problems, are proposed by Fujikoshi and Ulyanov [55] in their joint work.

It should be mentioned that NIST, the National Institute of Standards and Technology (GB) maintains an on-line public library (Digital Library of Mathematical Functions at http:dlmf.nist.gov) with a special section on Functions of Matrix Argument.

It is understandable that the huge volume of relations between hypergeometric functions of all types presented in the literature, and new ones regularly introduced in journals, raise various pertinent questions: Are they correct? How can we recognize a series as being of hypergeometric type? Can some of them be merely modified versions of existing ones? What are the mechanisms to derive new results from existing fundamental ones? Can we identify those which are really basic?

Instead of manually consulting huge data bases of published results, different computer algorithms have been introduced, and run, to provide answers to the above questions. For example, Milgram [56] used computer algorithms to numerically test all closed forms identities given in Prudnikov et al. [57] . He could then omit some equations and amended others, as well as introduce a few new ones. By repeating this process he obtained a final of 89 identities, only 23 of them were in the original set (see Hannah [58] for other similar concepts and approaches).

Fractional calculus starts from the principle that a derivative can be of any order, unlike in classical calculus where these orders must be integers. Derivatives and integrals can then be unified into a single operation, called the differintegral: There are several approaches in defining a fractional derivative D or integral I, the most popular one being the Riemann-Liouville integral,

,

which leads to:

,

with n being the nearest integer larger than. With we have a fractional derivative, and, a fractional integral (Kiryakova [59] ).

Lavoie et al. [60] gives a simple survey of these approaches, mostly oriented toward special functions, which include Cauchy integral, Euler and Pochhammer contour integrals, etc. Leibnitz rule for derivatives of products becomes:

.

The generalized hypergeometric function, expressed as a fractional derivative, is as follows:

,

and the more general relation is:

.

Using fractional calculus, Kiryakova [59] shows that any special function is a differintegral of an elementary function. More precisely, we have 3 cases for:

a): is the differintegral of the generalized cosine function .

b): is the differintegral of the elementary function.

c): is the differintegral of the elementary function.

Kiryakova [59] uses the Kober?Erdelyi transform with kernel the G-function, which is then shifted backwards from to, and progressively to, to, or to respectively. For example in the first case we have:

.

Using Poisson type representation we obtain the cosine function.

is, however, a complicated generalized operator of fractional integration of Riemann-Liouville type,

The generalized m-tuple fractional derivative is then:

,

where

.

We have, as expected,.

Using the composition of m-tuple and n-tuple integrals as -tuple integral

, ,

and considering separately each of the three above cases, we obtain the above results.

NOTE: 1) This interesting result has to be interpreted with care however, since the special function G is used as kernel in the operator.

2) The idea of averaging, using simple functions, is similar to the one carried out by Carlson in (sect. 6.3) and other authors. Following the same idea, Pham-Gia [61] used the limit of an iterative convolution process to obtain interesting functions in quasi-analyticity.

There are several convincing applications of Fractional calculus in Engineering and Applied Probability. In Theoretical Statistics, several recent research results on hypergeometric functions use fractional calculus (Mathai [44] ), associated with functions of matrix arguments (Mathai and Haubold [62] ). But it is still too early to appraise the impact of this notion on Statistics.

Group theory has had important influence on Statistics. As stated by Giri [63] , by introducing the group invariance principle and restricting attention to invariant decision rules a reduction of the dimension of the parametric space is possible. He also provides several examples where the hypergeometric functions are present. Group representation is another well-used concept in multivariate statistics, as seen in zonal polynomials. Wijsman [64] gives a simple example of how the distribution of can be obtained using this approach, and also some statistical problems to which a special group structure applies, called Type I.

It can be proved that, starting from the structure of an appropriate Lie Group, here the special linear group, we can establish several properties of, and relations on, hypergeometric functions. Introduced in the late sixties by Miller Jr, among others, this approach seemed to be promising. It is based on the Lie group structure and the Lie Algebra which is the derivative at zero of the elements of the Lie group. The exponential function, using infinite series, permits to go from the Lie Algebra elements to the Lie group elements. Using a basis based on hypergeometric functions and commutators based on differential operators, several relations on hypergeometric functions can be derived. The following table (Wasson and Gilmore [65] ) gives below the correspondence between the Lie group to be considered for the chosen special function.

Miller Jr [66] and Miller Jr [67] have presented the arguments concerning the two functions and. However, they are too lengthy to be reproduced here. But the main difficulties seem to be the selection of the Lie group to start with, and then the choice of these bases themselves, which can be quite complicated.

This highly mathematical approach is in the domain of theoretical mathematical physics, with few applications in Statistics. But the concept of symmetry frequently used here can be related to several symmetry problems in Statistics. Wijsman’s monograph [64] is helpful in understanding several related abstract mathematical notions. Consequently, we have reservations that this approach can be used, in either Classical or Bayesian Statistics, although it is very elegant and seems helpful in establishing new relations for special functions. Lie group theory and Lie algebra have found some real applications, however, in Statistics on manifolds and on Image processing (see, for example, Fletcher, Lu and Joshi [68] ).

Carlson [10] introduced several hypergeometric functions of his own, which are different from the classical ones, e.g. and functions, which are obtained by averaging and, using a Dirichlet measure. The motivation is that expressions of and in the several complex variables domain are free of branch points, and can be better studied. Prof Carlson passed away quite recently.

Several notions developed here can be linked to the classical ones. For example, the so-called Euler measure is just the Lebesgue measure using the gamma density,

,

and the average derived

is our relation (10) above.

According to Carlson [10] , is supposed to play several roles, those of, those of the elliptic integral and those of Appell’s, while the couple replaces.

We have, in particular:

.

Several classical special functions can be shown to be particular cases of and hence, are Dirichlet averages of elementary functions. Even the Schwarz-Christoffel mapping in complex analysis can be shown to be an -function too.

There is a parallel theory of hypergeometric functions based on q-hypergeometric series. Here, the ratios of successive terms are a rational function of. We then have:

for any real or complex, , and the corresponding q-basic hypergeometric series is:

.

Several results here are similar to the ones we have seen, but some are quite different. We will not discuss this approach further and refer the reader to Srivastava and Karllson [29] . It should be mentioned that Ramanujan has established several interesting results in this domain.

Hypergeometric distribution in unidimensional statistics:

a) There are X “good” elements in a population of N. The probability of having x “good” when choosing at random n elements is (in finite sampling without replacement):

,. (38)

The moment generating function of this distribution is

.

This fact gives this discrete distribution its name. It must be mentioned that it is the conditional distribution, on which Fisher’s exact test on proportions is based.

b) A generalization of this distribution leads to the Kemp family, which is based on a generalization of the above probability, i.e.

, ,

for arbitrary positive values of a and b. Several types of distributions are obtained and reported in Johnson and Kotz ( [71] , chapter 6). Derived distributions include the Non-Central Hypergeometric distribution, and the related positive and negative hypergeometric distributions.

The discrete multivariate hypergeometric distribution is a straightforward extension of the univariate case: Instead of one good subset we have distinct good ones and the k-th subset is the bad ones. For a sample of size m, and, , ,

with we have:

, ,. (39)

We refer to chapter 39 of Johnson, Kotz and Balakrishnan [72] for other properties of this discrete multivariate distribution.

a) Gauss hypergeometric function has found applications mostly in distribution theory (e.g. Pham-Gia and Turkkan [73] and [74] ).

A nice property of hypergeometric functions, especially and, is that they could provide, by mere multiplication with the central density, the expression of the non-central density. For example, the density of the central F, , defined here

as the ratio of two independent chi-square, is .

The related non-central variable G, with non-centrality parameter, , will have as density

.

This fact is particularly useful when we study the power of a test, which uses the non-central distribution of a statistic. If we define, we have similar re-

sults relating to. A similar result holds for the non- central beta distribution. Pham-Gia and Turkkan [36] provides the comparison between ratios of random variables and ratios of random matrices, and hypergeometric functions of various types are used in both cases.

G and H-functions are used in the expressions of the densities of several positive random variables and in the distributions of determinants of random matrices, as shown by Pham-Gia [75] . The following variables with their densities limited to their positive part, have their density expressed as a G or H-function: the half- standard normal, the half-Cauchy, the half-Student t (Springer [26] , pp. 202-207). The Cumulative Distribution Function of a H-function density variable is also expressible as a H-function, and so are its Laplace transform and characteristic function.

When considering a random Beta matrix variate, its determinant has its density expressed as a G-function since it is a product of independent univariate betas, and so do products and ratios of independent random matrices and several test statistics in multivariate analysis (e.g. Pham-Gia and Choulakian [76] , Rathie [27] ). The three types of

G-functions mostly encountered here are:. But, as Mathai and Saxena ( [23] , sect 5.6) have remarked, often we have here the cases where the parameters differ by integers and computations of residues have to be adjusted accordingly.

b) Relations between hypergeometric functions and the normal distribution: What are the relations between these two most important notions in Statistics?

We have already mentioned the half-standard normal density expressed as a G-function. And an interesting relation exists on moments. Let. We then have the raw moments:

and absolute moments, where is Kummer confluent hypergeometric function. Central moments have, however, simple expressions:, and .

Here, again, we can see that is associated with the non-centrality factor.

We have already mentioned the works of James [13] and Constantine and Muirhead [14] on zonal polynomials. Farrell [77] , Pillai [78] and Olkin and Rubin [79] also made significant contributions. For functions of matrix arguments there are several results where these functions are associated with fractional calculus, under various forms (see Mathai and Haubold [62] ), most of them still at the very theoretical level, however. They will probably make an impact on statistics in the years ahead. An application of interest is given by Gross and Richards [80] .

Handbook of the beta distribution (see Gupta and Nadarajah [81] ) has a selection of articles containing various hypergeometric functions in one or two scalar variables. In particular, Pham-Gia, in that reference, and Pham-Gia and Turkkan [82] has hypergeometric functions applications in Bayes inference. Exton ( [28] , chapter 7) and Mathai and Saxena [23] should be consulted for a large list of applications of multivariate hypergeometric functions in Statistics. Hypergeometric functions of matrix arguments are encountered in non-central matrix distributions and in power calculation for hypothesis testing involving vector and matrix variates. Mathai [44] and Mathai and Pederzoli [46] offered several theoretical results on this topic. Applications of functions with matrix arguments in engineering include Chiani et al. [83] , Gross and Richards [80] and Tulino and Verdu [84] .

Hypergeometric integrals are the main concern of these fields, in which some important results can be presented under.

a) Varchenko [88] considers the integral

,

and later, the more general form:

,

where, the polytope, is now a variable also, are linear functions and are complex numbers. They are also called hypergeometric integrals and generalize the beta function. An interesting example is a configuration of 3 consecutive points on a line. Put as the consecutive distances separating them, and

, , ,.

Then we have:

meaning that the determinant of integrals of hypergeometric forms of a configuration, over all bounded components of the complement of that configuration, can be simply computed. This formula can be extended to a configuration of hyperplanes.

There are several important results on hypergeometric functions in Conformal Field theory, on representation theory of Lie Algebra, in quantum groups, etc. However, they do not fit into this survey and the reader is invited to consult Varchenko [88] . For example, the integral

can be interpreted as average of interactions of the last m points with the first n points, and can be shown to be associated with a representation of Kac-Moody algebra.

b) An interesting point of view can be taken for hypergeometric integrals, using the fact that definite integrals are considered as pairings of homology and cohomology groups according to de Rham Theory.

Let T be an m-dimension complex manifold, or equivalently, as a 2m-dimension real smooth manifold. Let be a smooth map from the p-dim simplex to T. A finite sum, , is called a p-chain, and a p-cycle if, where is the boundary operator.

The homology group is vector space of p-cycles modulo (Image of chains by operator). Let’s consider the de Rham cohomology group which is the quotient space:

.

We know that there is an isomorphism:. By Stokes Theorem

.

We define, which leads to a bilinear form: .

The GKZ or A-hypergeometric integral is

, with,

where, and similarly for.

We can see that the hypergeometric integral is a pairing between homology groups and cohomology groups, with its value being a function of x. A simple illustration using, is the winding number in complex analysis, which is a pairing between and. It satisfies the GKZ-hypergeometric system as presented above. This theoretical result is of importance although it does not permit to calculate the value of the integral.

For complex variables we have twisted homology and cohomology, as explained in Aomoto and Kita [86] .

Hypergeometric functions have been used to find solutions of algebraic equations of fifth order and higher. The reason is that its expression as an infinite series can be conveniently used for the search for a solution. For example, with the equation: we can use Lagrange inversion formula that states that one solution to is given by the power series

.

Here, we have:

.

Setting

we have the solution of the equation as the hypergeometric function

.

There is a classification list by H.A. Schwarz, of hypergeometric functions which are at the same time algebraic (Beukers [87] ). This list has been recently extended by Beukers and Heckman [89] . Perelomov [90] gives hypergeometric solutions to more general algebraic equations.

It is not surprising that hypergeometric functions are used in Economics and related fields, where advanced mathematics are often used for modeling and computation. We refer the reader to Abadir [4] for an extensive survey on their presence there. In Finance, the well-known Black-Scholes model now has its generalization to hypergeometric functions (Albanese et al. [91] ).

Hypergeometric functions are frequently seen in theoretical physics and Appell’s function is associated with several results related to the Shrodinger equation (see Exton [28] ). It should be mentioned that the Theory of Random Matrices, developed independently in theoretical physics, has strong connections with matrix variate distributions in Mutivariate statistics. The various distributions associated with the eigenvalues of the Wishart matrix distribution were the connecting link between the two disciplines, and Wishart’s [92] pioneering work on the distribution of the covariance matrix has been often cited in Physics. But the laws of Wigner, Tracy-Widom and Marcenko-Pastur developed there, have now found applications in Statistics (Johnstone [93] ). On the other hand, G and H-functions have numerous applications in Astrophysics, as can be seen in Chapter 5 of Mathai and Haubold [43] .