_{1}

^{*}

A brief history of the Centre for Mathematical and Statistical Sciences, Kerala, India, is given and an overview of Mathai’s research and education programs in the following topics is outlined: Fractional Calculus; Functions of Matrix Argument — M-transforms, M-convolutions; Krätzel integrals; Pathway Models; Geometrical Probabilities; Astrophysics — reaction rate theory, solar neutrinos; Special Functions — G and H-functions; Multivariate Analysis; Algorithms for Non-linear Least Squares; Characterizations — characterizations of densities, information measure, axiomatic definitions, pseudo analytic functions of matrix argument and characterization of the normal probability law; Mathai’s Entropy — entropy optimization; Analysis of Variance; Population Problems and Social Sciences; Quadratic and Bilinear Forms; Linear Algebra; Probability and Statistics .

Fractional integrals, fractional derivatives and fractional differential equations were available only for real scalar variables. The most popular fractional integrals in the literature are Riemann-Liouville fractional integrals given by the following:

D a x − α f = 1 Γ ( α ) ∫ a x ( x − t ) α − 1 f ( t ) d t , ℜ ( α ) > 0, (1.1)

where ℜ ( ⋅ ) denotes the real part of ( ⋅ ) .

D x b − α f = 1 Γ ( α ) ∫ x b ( t − x ) α − 1 f ( t ) d t , ℜ ( α ) > 0. (1.2)

Here − α in the exponent of D indicates an integral. The D with positive exponent D a x α f , D x b α f is used to denote the corresponding fractional derivatives. Here (1.1) is called Riemann-Liouville left-sided or first kind fractional integral of order α and (2.2) is called Riemann-Liouville fractional integral of order α of the second kind or right-sided. If a = − ∞ and b = ∞ then (1.1) and (1.2) are called Weyl fractional integrals of order α and of the first kind and second kind, respectively, or the left-sided and right-sided ones. Mathai was trying to find an interpretation or connection of fractional integrals in terms of statistical densities and random variables. In Mathai (2009), an interpretation is given for Weyl fractional integrals as densities of sum (first kind) and difference (second kind) of independently distributed real positive random variables having special types of densities. Fractional integrals were also given interpretations as fractions of total integrals coming from gamma and type-1 beta random variables. Also Weyl fractional integrals were extended to real matrix-variate cases there.

Then while working on Mellin convolutions of products and ratios, Mathai found that a fusion of fractional calculus and statistical distribution theory was possible which also opened up ways of extending fractional calculus to real scalar functions of matrix argument, when the argument matrix is real or in the complex domain. Let us consider real scalar variables first. The Mellin convolution of a product of two functions f 1 ( x 1 ) and f 2 ( x 2 ) says the following: Consider the integral

g 2 ( u 2 ) = ∫ v 1 v f 1 ( u v ) f 2 ( v ) d v . (1.3)

Then the Mellin transform of g 2 ( u 2 ) , with Mellin parameter s, is the product of the Mellin transforms of f 1 and f 2 . That is

M g 2 ( s ) = M f 1 ( s ) M f 2 ( s ) , (1.4)

where

M f 1 ( s ) = ∫ 0 ∞ x 1 s − 1 f 1 ( x 1 ) d x 1 and ∫ 0 ∞ x 2 s − 1 f 2 ( x 2 ) d x 2 = M f 2 ( s ) ,

whenever they exist. If g 2 ( u 2 ) is written as

g 2 ( u 2 ) = ∫ v 1 v f 1 ( v ) f 2 ( u v ) d v (1.5)

then also the formula in (1.4) holds. Thus, the Mellin convolution of a product has the two integral forms in (1.3) and (1.5). But, Mellin convolution of a ratio will have four different representations. Two of these are the following:

M g 1 ( u 1 ) = M f 1 ( s ) M f 2 ( 2 − s ) (1.6)

where

g 1 ( u 1 ) = ∫ v v f 1 ( u v ) f 2 ( v ) d v (1.7)

and

M g 1 ( s ) = M f 1 ( 2 − s ) M f 2 ( s ) (1.8)

where

g 1 ( u 1 ) = ∫ v v u 1 2 f 1 ( v u 1 ) f 2 ( v ) d v . (1.9)

Let x 1 and x 2 be real scalar positive random variables, independently distributed, with densities f 1 ( x 1 ) and f 2 ( x 2 ) , respectively. Let u 2 = x 1 x 2 and u 1 = x 2 x 1 , v = x 2 . Then the Jacobians are 1 v and − v u 1 2 respectively or

d x 1 ∧ d x 2 = 1 v d u 2 ∧ d v

and

d x 1 ∧ d x 2 = − v u 1 2 d u 1 ∧ d v .

The joint density of x 1 and x 2 is f 1 ( x 1 ) f 2 ( x 2 ) due to statistical independence and then the marginal densities of u 2 and u 1 , denoted by

and

In

Similarly

This means

and

In other words,

Let

and zero elsewhere. In statistical problems, the parameters are real but the integrals hold for complex parameters and hence the conditions are given for complex parameters. Let

where

is an Erdélyi-Kober fractional integral of order

Motivated by this observation, Mathai has given a new definition for fractional integrals of the first and second kinds of order

and

But (1.15) gives a Weyl fractional integral of the second kind of order

Let

Let

Then

is a statistical density, when f ia a statistical density. This is an Erdélyi-Kober fractional integral operator of the first kind of order

Mathai (2009) introduced fractional integrals in the real matrix-variate case but they could not be given any physical interpretations. In Mathai (2013) there are interpretations in terms of statistical distribution problem and M-convolutions introduced in Mathai (1997) (

All the matrices appearing here are

and it is

Lemma 1.1. Let

where

Lemma 1.2. Let

Lemma 1.3. Let X be a

Lemma 1.4. Let

With the help of (1.22) one can evaluate a matrix-variate gamma integral and write the result as

where the real matrix-variate gamma integral is

with

Since the total integral is 1, the identity follows from (1.25),

This identity will be used to establish fractional derivatives in a class of matrix-variate functions. The real matrix-variate type-1 beta density is defined as

There is a corresponding type-2 beta density, which is of the form

With the preliminaries in Section 1.6 one can define fractional integrals in the real matrix-variate case. Let

Denoting the densities of

and

Let

This

The density of

where the first kind Erdélyi-Kober fractional integral in the matrix-variate case is given in(1.33). The above notations as well as a unified notation for fractional integrals and fractional derivatives were introduced by Mathai (2013, 2014, 2015).

The above results in the real matrix-variate case are extended to complex matrix-variate cases, see Mathai (2013), to many matrix-variate cases, see Mathai (2014) and also the corresponding fractional derivatives in the matrix-variate case are worked out in Mathai (2015). The matrix differential operator introduced in Mathai (2015) is not a universal one, even though it works on some wide classes of functions. The matrix differential operator is introduced through the following symbolic representation. Let D be a differential operator defined for real matrix-variate case. Then

This is the αth order fractional derivative in Riemann-Liouville sense. Consider

is the αth order fractional derivative in the Caputo sense. In the Caputo case,

Let x be a real scalar positive variable. Consider the integrals

and

Structures such as the ones in (2.1) and (2.2) appear in many different areas. This (2.2) for

where

or

or

Then

This is the form in (2.1). Now, consider Mellin convolution of a product when

This is the form in (2.2). Hence (2.1) and (2.2) can be treated as Mellin convolutions of ratio and product when

Note that if

The integrand in (2.2) for

and

Then the unconditional density of y,

Now, compare (2.2) and (2.11). They are of one and the same forms. Hence (2.2), multiplied by an appropriate constant, can be considered as an unconditional density in a Bayesian structure.

For

In a physical system the stable solution may be exponential or power function or Gaussian. This is the idealized situation. But in reality the solution may be somewhere nearby the ideal or the stable situation. In order to capture the ideal situation as well as the neighboring unstable situations, a model with a switching mechanism was introduced by Mathai (2005). A form of this was proposed in the 1970’s by Mathai in connection with population studies. This was a scalar variable case. Then the ideas were extended to matrix-variate cases and brought out in 2005. For the real scalar positive variable situation, the model is the following:

If (3.1) is to be used as a statistical density then

When

Note that

When

where

where

where

for some

for some

The original paper Mathai (2005) deals with rectangular matrix-variate case. Let

where

and when

If a location parameter matrix is to be introduced then replace X by

Note that the structure

Also

Note that (3.8) for

While exploring a reliability problem, Mathai (2003) came across a multivariate family of densities, which could be taken as a generalization of type-1 Dirichlet family of densities. Then Mathai and his co-workers introduced several generalizations of type-1 and type-2 Dirichlet densities, see for example Thomas and Mathai (2009). For the different generalizations of type-1 and type-2 Dirichlet family, a number of characterization results are established showing that these models could also be generated by products of statistically independently distributed real scalar random variables. This is exactly the same structure available in the likelihood ratio criteria in the null cases of testing hypotheses on the parameters of one or more Gaussian populations as well as in the determinant

In (3.1) if one puts

In the pathway idea itself there is an open area which is not yet explored. The scalar version of the pathway model in (3.1) to (3.3) can be looked upon as the behavior of a hypergeometric series

From the point of view of a hypergeometric series, the process (3.11) to (3.12) is the process of a binomial series going to an exponential series. But a Bessel series

Therefore, a generalized form, covering the path towards the exponential form

A multivariate function usually means a function of many scalar variables. This is different from a matrix-variate function or a function of matrix argument. Functions of matrix argument are real-valued scalar functions

If one restricts A and

only candidate here. Hence, if X is

or

Therefore

the Laplace transform of

then

is the Laplace transform in the real symmetric positive definite matrix-variate case, where

Under this approach, a hypergeometric function of matrix argument, denoted by

where

The second approach is through zonal polynomials, developed by James (1961)and Constantine (1963). Here also functional commutativity is implicitly assumed, though not stated explicitly. Under this definition, a hypergeometric series is defined as follows:

where

Here

Lemma 4.1.

where

with

Lemma 4.2.

Starting from 1970, Mathai developed functions of matrix argument through M-transforms and M-convolutions. Under M-transform definition, a hypergeometric function

where

For example

Since the left side in (4.12) is a function of only one parameter

The work until 1999 is summarized in Mathai (1999a). The work started as an off-shoot of the work in multivariate statistical analysis. Mathai noted that the moment structure for many types of random geometric configurations was that of product of independently distributed type-1 beta, type-2 beta or gamma random variables. Such structures were already handled by Mathai and his co-workers in connection with problems in multivariate statistical analysis. Earlier contributions of Mathai in this area are available from Chapter 4 of the book Mathai (1999a). Then Ruben, a colleague of Mathai at McGill University, one day gave a copy of his paper showing a conjecture in geometrical probabilities, called Miles’ conjecture about a re-scaled, relocated random volume, generated by uniformly distributed random points in n-space, as asymptotically normal when

As an application of geometrical probabilities, Mathai and his co-authors looked into a geography problem of city designs of rectangular grid cities, as in North America, versus circular cities as in Europe, with reference to travel distance, and the associated expense and loss of time, from suburbs to city core, see Mathai (1998), Mathai and Moschopoulos (1999a).

After publishing the books Mathai and Saxena (1973, 1978) physicists were using results in special functions in their physics problems. HJH came to Montreal, Canada in 1982 with open problems on reaction-rate theory, solar models, solar neutrinos, and gravitational instability. The idea was to get exact analytical results and analytical models where computations and computer models were available. Mathai figured out that the problems connected with reaction-rate theory and solar neutrinos could be tackled once the following integral was evaluated explicitly (Critchfield, 1972; Fowler, 1984):

The corresponding general integral is

In 1982 Mathai could not find any mathematical technique of handling (6.2) or its particular case (6.1). He noted that (6.2) could be written as a product of two integrable functions and thereby as statistical densities by multiplying with appropriate normalizing constants. Then the structure in (6.2) could be converted into the form

and the right side of (6.3) is the density of a product of two real scalar positive independently distributed random variables with densities

where

Now, it is only a matter of evaluating the density

where

which also shows that

Therefore

Hence

Comparing (6.4) with (6.5) the required integral is given by the following:

The right side of (6.6) is a H-function.

For the reaction-rate probability integral,

Another attempt was to replace the current computer model for the Sun with analytic models. The idea was to assume a basic model for the matter density distribution in the Sun or in main sequence stars which could be treated as a sphere in hydrostatic equilibrium. Let r be an arbitrary distance from the center of the Sun and let

where c is the central core density when

Another area that was looked into was the gravitational instability problem concerning the evolution of large scale structure in the Universe. The problem was formulated in the form of differential equations. Mathai tried to change the operator

Another area looked into was the solar neutrino problem (Davis Jr., 2003; Sakurai, 2014). HJH and Mathai tried to come up with appropriate models to model the solar neutrino data. Mathai had noted that the graph of the time series data looked similar to the pattern that he had seen when working on modeling of the chemical called Melatonin in human body. Usually what is observed is the residual part of what is produced minus what is consumed or converted or lost. Hence the basic model should be an input-output type model. The necessary theory is available in Mathai (1993c). The simplest input-output model is an exponential type input

Mathai and his co-workers are credited with popularizing special functions, especially G and H-functions, in statistics and physics. Major part of the special function work was done with co-worker Saxena. They thought that they were the first one to use G and H-functions in statistical literature. But D. G. Kabe pointed out that he had expressed a statistical density in terms of a G-function in 1958. This may be the first paper in statistics where G or H-function was used. Most probably the use of G and H-function in physics an engineering areas started after the publication of the books Mathai and Saxena (1973, 1978). The first work on the fusion of statistical distribution theory and special functions started by creating statistical densities by using generalized special functions. In this connection the most general such density is based on a product of two H-functions, which appeared in Mathai and Saxena (1969). Another area that was looked into was Bayesian structures. The unconditional density in Bayesian analysis is of the form

What are the general families of functions for

Another family of problems that was looked into were the null and non-null distributions of the likelihood ratio criterion or

where _{o}:V = is diagonal. This is called the test for independence in the Gaussian case. Then the

where

where

where

Explicit computable series forms of

where

The G and H-functions are also established by Mathai for the real matrix-variate cases through M-transforms, along with extensions of all special functions of scalar variables to the matrix-variate cases. Also Mathai extended multivariate functions such as Apple functions, Lauricella functions, Kampé de Fériet functions etc to many matrix-variate cases. Some details may be seen from Mathai (1993a, 1997).

By making use of the explicit series forms, MAPLE and MATHEMATICA have produced computer programs for numerical computations of G-functions and MATHEMATICA has a computer program for the evaluation of H-function also. Solutions of fractional differential equations usually end up in terms of Mittag-Leffler function, its generalization as Wright’s function and its generalization as H-function. In connection with fractional differential equations for reaction, diffusion, reaction-diffusion problems HJH, Mathai and Saxena have given solutions for a large number of situations, which may be seen from the joint works of Haubold, Mathai and Saxena (2011), see also Mathai and Haubold (2018c). In all these solutions, either Mittag-Leffler function or Wright’s function or H-function appears. Also many other physicists, mathematicians and engineers have tried other fractional partial differential equations where also the solutions are available in terms of H-functions.

A Pseudo Dirichlet IntegralA type-1 Dirichlet integral is over a simplex

But one can construct a multivariate integral over a hypercube giving rise to the same

The method of proving this result is to expand

where

The integral is the following:

where A is a symmetric product of matrices given by

with

definite matrix

In the area of multivariate analysis, almost all exact null distributions in the most general cases and a large number of non-null distributions of

where

Also, Mathai and his co-workers have established a connection between

Exact 11-digit accurate percentage points connected with the null distributions of the

HJH, Mathai and Saxena have solved fractional differential equations, starting from 2000, where the solutions invariably come in terms of Mittag-Leffler function, Wright’s function or H-function. Exponential type solutions of integer order differential equations automatically change to Mittag-Leffler functions when we go from integer order to fractional order differential equations. There is also a Mittag-Leffler stochastic process based on a Mittag-Leffler density, which is a non-Gaussian stochastic process. Work in this area is summarized in Haubold, Mathai and Saxena (2011). Mathai has also introduced a generalized Mittag-Leffler density and has shown that it is attracted to heavy-tailed models such as Lévy and Linnik densities, rather than to Gaussian models.

In this area, two basic books are Mathai and Rathie (1975) and Mathai and Pederzoli (1977). Characterization is the unique determination through some given properties. Characterization of a density means to show that certain property or properties uniquely determine that density. Unique determination of a concept means to give an axiomatic definition to that concept. That is, to show that the proposed axioms will uniquely determine the concept. The techniques used in this area, to go from the given properties to the density or from the given axioms to the concept such as “uncertainty’’ or its complement “information’’ etc, are functional equations, differential equations, Laplace, Mellin, and Fourier transforms. For example, look at the distribution of error. The error

where

Assume that the total variance of

Check the consequence of these three assumptions.

Hence

That is

Therefore, as

which is the moment generating function of a normal density with mean value zero and variance

This is the derivation of the Gaussian or normal density given by Gauss, and hence it is also called the Gaussian density or error curve. Mathai and Pederzoli (1977) contains such characterizations of the normal probability law by using structural properties, regression properties etc. One fundamental idea was introduced in this area by Gordon and Mathai (1972). They tried to come up with pseudo-analytic functions of matrix argument involving rectangular matrices and by using this, characterization theorems were established for a multivariate normal density.

In Mathai and Rathie (1975), axiomatic definitions of information theory measures and basic statistical concepts are given. This is the first book giving axiomatic definitions of information measures. The techniques used are mainly from functional equations by using the proposed axioms create a functional equation and obtain its unique solution by imposing more conditions, if necessary, thus coming up with a unique definition or characterization of the concept. One such measure there is the one introduced as Havrda-Charvát measure, which for the continuous case is the following:

where

where

for the continuous case, with a corresponding discrete analogue. Optimization of (9.4) under the condition that the total energy is preserved or the first moment is fixed, leads to Tsallis statistics of non-extensive statistical mechanics. Tsallis statistics is of the following form:

which is also a power law in the sense

and then assume that the first moment is fixed in the escort density

when optimized under the condition of first moment in

The most significant contribution in this area is the proposal of a theory of growth and form in nature and the explanation of the emergence of beautiful patterns in sunflower, along with explanation for the appearance of Fibonacci sequence and golden ratio there. The mathematical reconstruction of the sunflower head, with all the features that are seen in nature, is still the cover design of the journal of Mathematical Biosciences. The paper of Mathai and Davis appeared in that journal in 1974 and in 1976 the journal adopted the mathematically reconstructed sunflower head of Mathai and Davis (1974) as the journal’s cover design with acknowledgement to the authors. When this paper was sent for publication to this journal, the editor wrote back saying: “enthusiastically accepted for publication’’ because this was the first time all natural features were explained in full. As per Davis and Mathai, the programming of the sunflower head is like a point moving along an Archimedes’ spiral at a constant speed so that when the point makes an angle

where k is a constant, giving Archimedes’ spiral. When

The first paper of Mathai (Mathai, 1965), was on an approximate analysis of variance. It was on the analysis of a two-way classification with multiple observations per cell. Here the orthogonality is lost, and when estimating the main effects, one ends up in a singular system of linear equations of the form

where

where

A good approximation for

A problem that was looked into was how to come up with a measure of “distance’’ or “closeness’’ or “affinity’’ between two sociological groups or how to say that one community is close to another community with respect to a given characteristic. Mathai introduced the concepts of “directed divergence’’, “affinity’’ etc from information theory to social statistics. Let

and hence

This is a measure of angular dispersion and it is usually called “affinity’’ between P and Q or Matusita’s measure of affinity between two discrete distributions. George and Mathai computed “affinity” between communities with reference to the characteristic of production of children and found that the politicians’ statements did not match with the realities. Thus, some politicians’ claims of certain communities producing more children, was repudiated in a scientific way in George and Mathai (1974). They also studied the most important variable responsible for population growth, namely the interval between two live births in woman of child-bearing age group and proposed a model, George and Mathai (1975). They also gave a formula for estimating an event from information supplied by different agencies, replacing the popular Deming formula in this regard.

Major contributions in these areas are summarized in the books Mathai and Provost (1992), and Mathai, Provost and Hayakawa (1995). There is a very important concept in quadratic forms in Gaussian random variables called chisquaredness of quadratic forms. That is,

where

In the area of reliability analysis, the basic concepts are survival function, hazard function, cumulative hazard, system reliability, reliability in the presence of other variables such as covariates etc. In a series of papers, Mathai and Princy in 2016 introduced the pathway model into the area so that the desired shapes for hazard function and the desired reliability for systems with components in series and parallel architecture could be obtained by selecting appropriate models from the pathway family of functions. Then, these ideas were extended to situations where the input variable or the variable under consideration is a rectangular matrix. As a byproduct, Maxwell-Boltzmann distribution, Raleigh distribution, Dirichlet averages etc were extended to matrix-variate cases, see for example Mathai and Princy (2017a, 2017b).

Mellin convolutions of products and ratios involving two functions are available in the literature. Mathai illustrated how these concepts are connected to statistical distribution theory and fractional calculus. In fact, a general definition for fractional integrals is given by Mathai using Mellin convolutions of products and ratios involving two functions. Corresponding M-convolutions involving two functions of matrix argument is Mathai’s contribution. He has also given physical interpretations for M-convolutions as densities of symmetric products and symmetric ratios of matrices. Mathai also extended Mellin convolutions and M-convolutions to three or more functions, see Mathai (2018). When three functions are involved, one can obtain several integral representations for the same Mellin convolution and M-convolution. For example, consider the symmetric product of three

One can also obtain further representations by taking for example,

symmetric products of two matrices each, which will produce several more integral representations. In the real scalar case the Mellin convolutions can be evaluated in terms of generalized special functions, thus producing integral representations for these special functions. For example, let

where

The following are the new concepts, new ideas and new procedures introduced by Mathai (

- Developed “dispersion theory’’ in 1967;

- Developed a generalized partial fraction technique, with Rathie, in 1971;

- Developed an operator to evaluate residues when poles of all types of orders occur (1971);

- Introduced the phrases “statistical sciences’’ in 1971 thereby the phrase “mathematical sciences’’ came into existence;

- Proposed a theory of growth and forms in nature (1974), the theory still standed, mathematically reconstructed a sunflower head;

- Introduced the concepts of “affinity’’, “distance’’ etc. in social sciences and created a procedure to compare sociological groups (1974);

- Introduced functions of matrix argument through M-transforms and M-convolutions;

- Introduced a non-linear least square algorithm (1993);

- Solved Miles’ conjecture in geometrical probabilities, created and solved parallel conjectures (1982);

- Introduced Jacobians of matrix transformations in solving problems of random volumes, replacing the complicated integral and differential geometry procedures (1982);

- Now meaningful physical interpretations are given for M-convolutions; Unique recovery of

- Extended Jacobians of matrix transformations from the real case to complex matrix-variate cases in a large number of situations;

- Introduced the concept of Laplacianness of bilinear forms and established the density of covariance structures (1993);

- Introduced pathway model and pathway idea (2005);

- Extended fractional calculus to real matrix-variate cases (2007);

- Established a connection between fractional calculus and statistical distribution theory (2007);

- Introduced Mathai’s (2007) entropy;

- Geometrical interpretation and a general definition for fractional integrals were given (2013-2015);

- Extended fractional calculus to complex matrix-variate case and complex domain in general (2013);

- Extended fractional calculus to many matrix-variate cases (real and complex) (2014);

- Developed a fractional differential operator in the matrix-variate case (2015);

- Extended reliability analysis concepts to rectangular matrix-variate case (2017).

HJH expresses his deep appreciation for a lifelong support from and cooperation with Prof. Dr. A. M. Mathai, Department of Mathematics and Statistics, McGill University, Montreal, Canada, and Director of the Centre for Mathematical and Statistical Sciences, Peechi, Kerala, India. HJH also takes the opportunity to place on record his gratefulness for the encouragment of research by AkM Prof. Dr. Dr. e.h. mult. H.-J. Treder, Director of the Einstein Laboratory for Theoretical Physics, Caputh, Germany (see Schulz-Fieguth, 2018). Treder was the director of the Central Institute for Astrophysics of the Academy of Sciences (Berlin, GDR). In 1965 he was the principal organizer of the widely respected Einstein Symposium at the 50th anniversary of the invention of general relativity theory. For the Berlin international celebrations of Einstein’s 100th birthday, 1979, he managed to secure the summer house of Einstein in Caputh, Brandenburg, as the Einstein Laboratory of Theoretical Physics in consultation with the administrators of the estate of Otto Nathan and Einstein. In 1981 he hosted the Michelson Colloquium at Potsdam to celebrate and recall the first Michelson experiment performed in 1881 at the Astrophysical Observatory in Potsdam. Treder was able to secure space and time for intense research work in his professional environment ranging from the solar neutrino problem (Treder, 1974) to fractional calculus (Treder, 1989). He supported actively the United Nations efforts to make available education and research in science to nations worldwide.

The authors declare no conflicts of interest regarding the publication of this paper.

Haubold, H. J. (2020). A. M. Mathai Centre for Mathematical and Statistical Sciences: A Brief History of the Centre and Prof. Dr. A. M. Mathai’s Research and Education Programs at the Occasion of His 85th Anniversary. Creative Education, 11, 356-405. https://doi.org/10.4236/ce.2020.113028