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Hypergeometric functions have been increasingly present in several disciplines including Statistics, but there is much confusion on their proper uses, as well as on their existence and domain of definition. In this article, we try to clarify several points and give a general overview of the topic, going from the univariate case to the matrix case, in one and then in several arguments. We also survey some results in fields close to Statistics, where hypergeometric functions are actively used, studied and developed.

Hypergeometric functions in one or several variables, introduced first in Mathematics, have been used in Physics and Applied Mathematics for some time. But their presence in Statistics is quite recent, within various topics, particularly in operations on random variables and on non-null distributions. In Multivariate analysis, as reported by Bose [

There is, however, some confusion regarding the different forms under which the hypergeometric function appears. In particular, the equalities between the infinite series, the Euler integral representation, the Laplace representation and the Mellin-Barnes representation can be confusing. Since they are only valid under certain conditions, one form can converge while the others do not, or take different values. We will discuss the necessary conditions for their equivalences, for

In this article, we are mostly concerned with the presence of hypergeometric functions in Statistics and to this end, have adopted two measures: Section 7 is completely devoted to Statistics, and in the last part of the article we will survey hypergeometric functions in various domains, and discuss their potential relations with, and applications in, Statistics. Throughout the text, whenever possible, we will also express similar opinions, which are strictly ours, and are necessarily subjective.

We try to be informative without being too technical. Naturally, we can only give a general landscape on the hypergeometric functions’ presence in neighboring fields. We will not go into details when coming into a specific domain, since this would require advanced knowledge in that domain itself. But relevant references are given so that the reader can deepen his/her knowledge on a certain topic if she/he so wishes. We have also given up the effort of trying to present a unified set of notations/symbols throughout the paper because these notions vary so much from one field to the other. We believe that a good grasp of the whole picture will allow readers to have an appreciation of the diversity and richness of hypergeometric functions. Then, they can make possible connections between these ideas and their own statistical domain, to derive other results and conclusions.

There are, at present, three survey articles on hypergeometric functions in the literature: one is from the Encyclopedia of Statistical sciences [

Leon Ehrenpreis [

In the same spirit, Askey [

1) The versatility of hypergeometric functions is due to the fact that they can be expressed as an infinite series, or as very different forms of integrals. The three basic forms, Euler, Laplace and Mellin-Barnes, can then be studied and extended, using mathematical analysis tools.

2) Some common approaches used by researchers are: averaging (through different processes) and progressive definitions (e.g. from

3) In Statistics, understandably, Hypergeometric functions are not developed, but used, mostly in distribution theory. However, James [

In section 2 we will consider the univariate scalar case and progressive generalizations of the hypergeometric functions, from three parameters to n parameters and to H and G-functions. Since integral representations play a key role, we have presented them clearly at every step. In Section 3, we generalize to several scalar variables, again giving the three integral representations. In Section 4, we consider one or several matrix variates and the three current approaches to introduce them. In Section 5, computational issues will be discussed. Section 6 gives some other approaches used to derive the hypergeometric functions, different from the classical one. In Section 7, the presence of hypergeometric functions in Statistics, will be presented, with no pretention of being exhaustive. Finally, in section 8 we present the hypergeometric function in neighboring domains, with potential connections to Statistics. Since there are so many such domains, we do not pretend to be exhaustive, or objective here either, and can only give basic ideas of interest, or results of importance. Deeper results would, naturally, require specialized advanced technical knowledge from the reader in that domain.

NOTE: In this survey we will limit our consideration to the real case, for scalar, vector and matrix variables, since the complex case is seldom encountered in Statistics, and its inclusion would considerably lengthen the article. Classical treatises on this topic are Erdelyi et al. [

We also realize that to cover such an immense topic as hypergeometric functions, within a limited number of pages, our survey is very ambitious and necessarily incomplete in many respects. Several properties of Gauss hypergeometric function related to continued fractions, linear and quadratic transformations, etc., could not be treated due to lack of space. We hence ask for your comprehension and understanding.

To put more clarity into our presentation we have worked out the following plan, which also reflects our point of view on surveying the whole topic: integral representations within progressive generalizations. Naturally, our view is only one among so many others, that could differ sharply from ours.

PLAN OF THE PRESENTATION

1. Introduction

2. Hypergeometric series and functions in one scalar variable

2.1. The Laplace , Fourier and Mellin Transforms

2.2. Sums versus integrals

2.3. Integral representations

2.3.1. Euler integral on a finite segment of the real line

2.3.2. Laplace representation on the positive half-line

2.3.3. Mellin-Barnes representation by contour integral in the complex plane

2.3.4. Contiguous relations

2.4. Generalization to several parameters

2.4.1. Generalized hypergeometric functions

2.4.2. Analytic continuation

2.4.3. Euler integral representation

2.4.4. Laplace representation on the positive axis

2.4.5. Mellin-Barnes representation

2.5. Generalization to G and H- functions

3. Hypergeometric series and functions in several independent scalar variables

3.1. Appell, Lauricella and others sums

3.2. Integral representations and further generalization

3.2.1. Integral representation of Euler type

3.2.2. Integral representation of Laplace type on

3.2.3. Representation of Mellin-Barnes type

3.3. Differential Equations and systems

3.4. Generalized G and H -functions in several independent scalar variables

4. Hypergeometric functions in matrix arguments: three proposed approaches

4.1. Functions in one matrix variate

4.1.1. Laplace transform approach

4.1.2. Zonal Polynomials approach

4.1.3. Matrix-transforms approach

4.2. Hypergeometric function in two matrix variates

5. Computational Issues

5.1. Computation of the hypergeometric function

5.2. Old and new relations between hypergeometric functions managed by computer

6. Hypergeometric functions derived via other approaches

6.1. Fractional Calculus

6.2. Lie Group approach

6.3. Carlson’s approach

6.3.1. Definitions of functions

6.3.2. Results of interest

6.3.3. Single integral representation and Elliptic integrals

6.4. Basic q-hypergeometric functions

7. Presence of Hypergeometric Functions in Statistics

7.1. Discrete case

7.2. Continuous case

7.3. Matrix case

7.4. Other Applications

8. Hypergeometric Functions in Neighboring Domains

8.1. Algebraic topology, Algebraic K-Theory, Algebraic Geometry

8.1.1. Integral representations

8.1.2. Single Integral representation

8.1.3. A-Hypergeometric functions

8.2. Hypergeometric integrals in Conformal Field theory, Homology and Cohomology

8.3. Algebraic functions and roots of equations

8.4. Economics, Quantitative Economics and Econometrics

8.5. Random matrices in Theoretical Physics

9. Conclusion

10. References

End

These three transforms play key roles in this article:

a) For a function

where r is a complex variable. Conversely, if

evaluated over any line

Two functions with same Laplace transform are identical. If

is the moment generating function of X.

b) The Fourier transform of

and its inverse is

c) The Mellin transform of

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

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

The corresponding series is then,

which becomes, after rearranging and change of scale:

The hypergeometric series

where the Pochhammer symbol is

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 [

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 [

There are three integral representations of

Let

and

For

Outside the unit disc

But the terminology can become confusing.

Example 1:

a) Using MAPLE, with

the last value being, however, taken (arbitrarily) from

b) For

These integral representations (5) and (5’) are very convenient because even when a, b and c differ by integers, thi(e)s(e) integral(s) still converge(s), and equal(s) the series within the convergence domain of the latter. This is to be compared with the Mellin- Barnes representation in 2.3.3 where the poles must be simple, which does not happen when a, b and c differ by integers.

This representation is useful when dealing with Laplace transform methods and moment generating functions, which is frequent in Statistics. However,

where

or double integral representation:

This hypergeometric function is an important function in its own right (see Slater [

MATHEMATICA gives this transform a quite complex sum of three hypergeometric functions, as follows:

NOTE: Some results on this transform, and its inverse, are given on p.212 and 291 of Tables of Integral Transforms [

Complex analysis developed in the 19th century brought powerful tools such as the calculus of residues, and Mellin-Barnes formula gives a third representation, based on contour integration. The value of the integral is computed, not as a complex integral, but as the sum of the residues at poles of

Computing the residues at the simple poles of

Mathai and Saxena ( [

Example 2: For

Mellin-Barnes integral formula has its origins in the work of Pincherle in 1888 (see Mainardi and Pagnini [

Let

Although

, (8)

with Pochhammer’s notation:

For particular values of p and q we have the following series:

Series are very useful in the resolution of differential or algebraic equations, but to study the solution’s analytic properties we rather use its integral form.

As we have seen, conditional on the values of a, b and c in

For the general case, Olsson [

a) Laplace integral representation:

This relation is not to be mistaken as the Laplace transform below.

b) Laplace transforms:

Considering

where L is a curve in the complex plane, properly indented to separate the two kinds of poles.

(The above expressions become Laplace and inverse Laplace transforms of when and respectively. They would permit us to “circulate” between, , and, under some conditions on the values of p and q.)

Conversely, it can be shown that if

where

Since the poles are in infinite numbers, we can see that

NOTE: We have most common functions in mathematics represented by

convert (hypergeom

gives as answer:

In an effort to generalize

where

The Meijer function

From (3) and (14) we can see that G and H-functions are Inverse Mellin Transforms of

The G-function converges when L is taken as one of the two paths

1) The three paths of integration are similar to those of

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 [

3) The H-function can be brought to the G-function for computation, when all

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

The Laplace transforms pair of G:

and its inverse

(Taking

Also, the relation

lytic continuation 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 [

This function should not be confused with Carlson’s

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.

When we go from one variable to two variables there are different ways to sum the variables, reflected in different expressions for the coefficients given to

a) Each of these functions can be expressed as an infinite series in x alone, with coefficients containing Gauss function

and, similarly for other functions.

Also,

b) Other hypergeometric functions, 34 in total, have been defined by Jacob Horn. The main ones are

c) Functions

They have a particular role in the representation of Appell functions. For example, we have

Lauricella functions are extensions of Appell functions to n variables, where

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

These integrals represent hypergeometric functions in n variables. For example,

and similarly for other functions, which can serve to extend the function outside the domains of convergence of the series. The n-tuple

In particular, for

But deeper results are obtained using A-hypergeometric functions (see section 8.1.2). Also,

Convenient forms for these integrals have been suggested by Carlson, using his own hypergeometric functions (see sect. 6.3.3).

Lauricella functions are expressed in terms of n-fold integrals of

Again, for

and also a single integral representation, using Humbert function:

Integrals are taken along the infinite imaginary axis, suitably indented. For example, for

Analytic continuation for Appell and Lauricella series: They can be continued analytically outside their convergence domain using their Euler integral representation or recurrence relations that exist between themselves. Exton ( [

The presence of so many forms of hypergeometric functions in n variables is embarrassing when we do not know the relations between them, which was the situation in the first half of the 20th century. But this situation started to change by the mid-eighties (see sect. 8.1.3).

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

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

and, for partial differential systems, there is one for each Lauricella function

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 ( [

b) The differential equation satisfied by

where

pendent, when the difference between any two of the values:

Differential equations for one-matrix hypergeometric functions can be considered. A short introduction to this topic is given by Muirhead ( [

As for one variable, we use the Mellin-Barnes approach to define this function. Buschman [

where

But, as pointed out by Nguyen Thanh Hai and Yakubovich [

In multivariate Analysis variables encountered can be matrices, which will be arguments of hypergeometric functions.

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 [

Domain of integration: Let

Since it is usually very difficult to carry out direct integration over a complex region

We have also the region

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

The multigamma function: Let

The Matrix Laplace Transform: Let

We assume that the integral converges in the half-plane

Gupta and Nagar [

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

This approach was pioneered by Bochner, developed by Herz [

Here, m is the dimension of the matrices and in (25). Also, for the multivariate Laplace transform, the elements off-diagonal of Z are taken as

This approach was introduced by James, and developed by James and Constantine, using results on group decomposition by Lo Keng Hua (see Gross and Richards [

the expression

metric homogeneous polynomial of degree k in elements of X. Here,

When

The decomposition into a direct sum of subrings is assured by ring theory (Gross and Richards [

(Muirhead [

Alternately, we can obtain

For

Other methods, not necessarily simpler, have been suggested (Kates [

and

with

A hypergeometric functions of one matrix X then have the familiar form:

and we have

Like the scalar variable case (see (9)), using zonal polynomials, we have the Euler-type representation:

Similarly, again using zonal polynomials, the Laplace and inverse Laplace representations of

This zonal polynomials approach is favored when we aim at deriving theoretical results, using and obtaining expressions similar to the scalar case. Since higher order zonal polynomials are difficult to obtain we have here a topic still under development. It is worth mentioning that numerical computations have been carried out successfully for low values of p and q only (see sect.5). Several breakthroughs are due to James [

Mathai [

if its M-transform, i.e.

Similarly, the Lauricella function

where

Mathai [

Hypergeometric function in two matrix variates is present in a basic result of multivariate analysis (Muirhead ( [

Here,

It is straightforward to extend the number of matrices to

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 [

G-functions are used lately to carry out difficult definite integrals computations (Adamchik [

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 [

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 [

The theory of Grobner basis has great influence on computations lately, in several domains of mathematics and algebraic statistics. Saito, Sturmfels and Nakayama [

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. [

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 [

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 [

We have so far relied on infinite series and integrals to deal with hypergeometric functions in one scalar variable. Can it be done otherwise? Yes, and it can be derived from at least three other directions which differ drastically from the approaches starting with hypergeometric series (4) or (8). However, only the third one, the Carlson’s approach, could be of immediate use in Statistics, in our opinion, the other two seem to be very advanced exercises to derive known or new results.

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

Lavoie et al. [

The generalized hypergeometric function

and the more general relation is:

Using fractional calculus, Kiryakova [

a)

b)

c)

Kiryakova [

Using Poisson type representation we obtain the cosine function.

The generalized m-tuple fractional derivative is then:

where

We have, as expected,

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

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 [

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 [

Group theory has had important influence on Statistics. As stated by Giri [

It can be proved that, starting from the structure of an appropriate Lie Group, here the special linear group

Miller Jr [

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 [

Carlson [

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 [

We have, in particular:

Several classical special functions can be shown to be particular cases of

Using a general averaging process with a Dirichlet distribution on a simplex

with

For any measurable function

Here,

Hence, the averages w.r.t. power functions,

1)

2)

Similarly, the average w.r.t. to the exponential is

3)

1) There are several relations between these functions, and with the classical hypergeometric functions. In fact,

2) Relations between

a)

b)

c) Several other relations relating

1) Representation by a single integral:

where

This single integral gives the holomorphic continuation of

2) Connections between Appell function

A particular case of the hypergeometric integral considered in section 8.2, in a theoretical context, is the elliptic integral

that can be now shown to be equal to

where

Furthermore, setting:

we now have

Carlson’s various hypergeometric functions are found to be quite useful by Askey [

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

for any

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 [

As stated earlier, in Statistics, Hypergeometric functions are generally not developed, but used, and mostly in distribution theory.

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):

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 ( [

The discrete multivariate hypergeometric distribution is a straightforward extension of the univariate case: Instead of one good subset we have

with

We refer to chapter 39 of Johnson, Kotz and Balakrishnan [

a) Gauss hypergeometric function

A nice property of hypergeometric functions, especially

as the ratio of two independent chi-square

The related non-central variable G, with non-centrality parameter

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

sults relating

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 [

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 [

G-functions mostly encountered here are:

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

and absolute moments

Here, again, we can see that

We have already mentioned the works of James [

Handbook of the beta distribution (see Gupta and Nadarajah [

Hypergeometric functions, being special mathematical functions, are traditionally associated with classical mathematical analysis, recurrence formulas and other special functions. An interesting account of their history is given by Stephen Wolfram [

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

We define first the Hypergeometric series of type

defined by the lattice formed by the set

We can see that Gauss

a)

with

b)

with

Similarly, Aomoto and Kita [

This topic is related to the preceding one, and has attracted attention for a long time, since integrating in one variable is supposedly much simpler than doing it in several ones. There are at least three known cases, and we start with the Dirichlet distribution,

1) Let

Also,

2) Carlson

3) For hypergeometric functions, Picard’s Theorem (Equation (19)) on

4) Here, we can see that Picard’s integral is a particular case of Equation (41) above, when

In the late eighties, Gelfand, Kapranov and Zelevinsky considered all the vector generalizations of Gauss hypergeometric functions, and the related differential equations, and fit them into the system of A-hypergeometric functions.

A GKZ (Gelfand, Kapranov, Zelevinsky) hypergeometric system is recently renamed A-hypergeometric system. It starts with an A-Matrix,

For hypergeometric functions we assume that the last row of A is constant, i.e.

(hence this integral satisfies the GKZ hypergeometric system).

A solution for the above system can be investigated under the form of a multiple series of the following form, which include most series in section 3.

We can verify that Gauss hypergeometric function

It should be mentioned that there are several applications in Combinatorics of A-hypergeometric functions, for example in arranging a number of hyperplanes in a multi-dimension complex space.

a) Varchenko [

and later, the more general form:

where

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 [

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

The homology group

We know that there is an isomorphism:

We define

The GKZ or A-hypergeometric integral is

where

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

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

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:

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 [

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 [

Hypergeometric functions are frequently seen in theoretical physics and Appell’s

The hypergeometric function and its generalizations have a place of choice in mathematics and its allied fields. We have given an overview of the roles this function plays across various domains and disciplines. In particular in Statistics, and Applied Statistics, its influence can be important in the years ahead and the statistician should be aware of its development in neighboring disciplines. We conclude this review by mentioning a reference bearing a special title [

The first author wishes to thank colleagues at the Université de Moncton (Eric, Claude) and the Hochiminhcity University of Science (Trong, Bao, Nhat, Phong I and II, Dong, Hoa, Thin and Doan) for helping him understand and present various abstract mathematics concepts inherent to the latter part of this paper

Pham-Gia, T. and Thanh, D.N. (2016) Hypergeometric Functions: From One Scalar Variable to Several Matrix Arguments, in Statistics and Beyond. Open Journal of Statistics, 6, 951-994. http://dx.doi.org/10.4236/ojs.2016.65078