-6304885x185.png" />. Let. Then with the help of the above lemmas one can show that, ignoring sign,
Denoting the densities of and as and, one can compute these by using the densities and through transformation of variables and they will be the following:
Let be a type-1 beta density of the type in (1.27) with parameters. Note that in (1.27) the parameters are. Then will be of the following form:
This in (1.32) for is an Erdélyi-Kober fractional integral of the second kind of order and parameter and hence Mathai called the integral as Erdélyi-Kober fractional integral of order and parameter of the second kind in the real matrix-variate case. In a similar fashion, Erdélyi-Kober fractional integral of order of the first kind with parameter, available from (1.31) by taking as a real matrix-variate type-1 beta with parameters is the following:
The density of, again denoted by, is given by
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 and represent αth order fractional derivative and fractional integral respectively. 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, operates on f first and then the fractional integral is taken, whereas in the Riemann-Liouville sense, the th order fractional integral is taken first and then operates on this. A universal differential operator D in the real as well as complex matrix-variate case is still an open problem for further research.
2. Krätzel Integral: Thermonuclear Functions
Let x be a real scalar positive variable. Consider the integrals
Structures such as the ones in (2.1) and (2.2) appear in many different areas. This (2.2) for is the basic Krätzel integral, see Krätzel (1979). For and general is the generalized Krätzel integral. An integral transform of the form
where is arbitrary so that exists, is known as Krätzel transform. Mathai has investigated various aspects of (2.1) and (2.2) in detail and he has also introduced a statistical density in terms of Krätzel integral. The structures in (2.2) and (2.1) can be generated as Melin convolutions of product and ratio. Consider the real scalar variables and and the corresponding functions and. Then it is seen from (1.10) that the Mellin convolution of a product is given by
or, and the Mellin convolution of a ratio, from (1.7), as
or. Let and be generalized gamma functions of the form
Then of (2.5) reduces to the form
This is the form in (2.1). Now, consider Mellin convolution of a product when and are generalized gamma functions in (2.6). Then reduces to the form
This is the form in (2.2). Hence (2.1) and (2.2) can be treated as Mellin convolutions of ratio and product when and are generalized gamma functions.
Note that if and are multiplied by the corresponding normalizing constants and then and become statistical densities. Let and be independently distributed real scalar positive random variables. Let. Then the densities of and are given by (2.4) and (2.5) multiplied by the appropriate constants and reduce to the forms in (2.8) and (2.7), multiplied by appropriate constants. In other words, (2.1) and (2.2), multiplied by appropriate constants, can be looked upon as the density of a ratio and product respectively.
The integrand in (2.2) for and normalized is the inverse Gaussian density available in stochastic processes. The integral in (2.2) for is the basic reaction-rate probability integral, which will be considered later. Mathai (2012) has introduced a Krätzel density associated with (2.1) and (2.2) and it is shown that one has general Bayesian structures in (2.1) and (2.2). For example, let us consider a conditional density of y, given x, in the form
and can act as the normalizing constant. In other words, the conditional density is a generalized gamma density. Let the marginal density of x be given by, and can act as a normalizing constant, a generalized gamma density. Then the joint density of y and x is given by
Then the unconditional density of y, , is available by integrating out x from this joint density. That is,
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 (2.1) and (2.2) one can extend the integrals to the real and complex matrix-variate cases. Mathai has also looked into this problem of Krätzel integrals in the matrix-variate cases. There will be difficulty with the Jacobians if one considers general parameters and in the matrix-variate case. The type of difficulties that can arise is described in Mathai (1997) by considering the transformation when. In the real matrix-variate case the scalar quantity is replaced by the determinant and exponent is replaced by for or if a is also replaced by a positive definite constant matrix. Mathai has also extended Baysian structures, densities of product and ratio, inverse Gaussian density, Krätzel integral and Krätzel density, to matrix-variate cases. When the matrix is in the complex domain is replaced by = absolute value of the determinant of, where is a matrix in the complex domain.
3. Pathway Model: Entropy, Probability, Dynamics
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 is the normalizing constant there. Otherwise is a constant, may be and then (3.1) will be a mathematical model. For we can write and then (3.1) becomes
When then and go to
Note that in (3.1) is in the family of generalized type-1 beta family of functions, whereas is in the family of generalized type-2 beta family of functions and belongs to the generalized gamma family of functions. Thus, when the pathway parameter q, goes from to 1 we have one family of functions, when q is from 1 to we have another family of functions and when we have a third family of functions. Thus, all the three cases are contained in (3.1), which is the pathway model for the real positive scalar variable case. Replace x by, , to extend the families over the real line.
When are statistical densities, then (3.1) to (3.3) give a distributional pathway. Mathai has also established a parallel pathway in terms of entropy optimization and in terms of differential equations. These give entropic and differential pathways as well. For example, consider the optimization of Mathai’s entropy, namely
where is a density function of x, and x can be real scalar or vector or matrix variable. A density means that for all x and. If one takes the limit when then (3.4), for real scalar x, reduces to
where is Shannon’s entropy or measure of “uncertainty’’ or the complement of “information’’. In (3.5), is taken as zero when. Consider the optimization of (3.4) subject to the conditions (a): and (b):. For, condition (b) becomes since the total probability is 1. For, (a) means that the first moment is fixed. This can correspond to the physical law of conservation of energy when dealing with energy distribution. If one uses calculus of variation to optimize (3.4) then the Euler equation is
where and are Lagrangian multipliers. Note that (3.6) gives the structure
for some and, which means
for some and. For and one has the model in (3.1) with replaced by. Thus, for one has an entropic pathway. Similarly one can consider the corresponding differential equations to obtain a differential pathway.
The original paper Mathai (2005) deals with rectangular matrix-variate case. Let be and of rank m be a matrix of distinct real scalar variables’s. Let A be and B be constant positive definite matrices. Consider the function
where are scalars, I is a identity matrix and is a constant. If (3.8) is to be taken as a density then is the normalizing constant there. For, goes to
and when, and go to
If a location parameter matrix is to be introduced then replace X by where M is a constant matrix.
Note that the structure is the structure of the volume content of a parallelotope in Euclidean n-space. Look at the m rows of X. These are vectors. These can be taken as m points in n-dimensional Euclidean space. These m vectors, , are linearly independent when the rank of X is m. These taken in a given order can form a convex hull and a m-parallelotope. The volume of this m-parallelotope is the determinant. Hence is the volume content of a generalized m-parallelotope.
Also is a generalized quadratic form. For and it is a quadratic form in the vector variable. Thus the theory of quadratic form and generalized quadratic form can be extended to a wider class represented by the pathway model (3.8). The current theory of quadratic form and bilinear form in random variables is confined to samples coming from a Gaussian population, see Mathai and Provost (1992), Mathai, Provost and Hayakawa (1995) (Figure 10). The results on quadratic and bilinear forms can now be extended to the wider class of pathway models. One problem in this direction is discussed in Mathai (2007). The matrix-variate pathway model in Mathai (2005) is extended to complex domain in Mathai and Provost (2005, 2006). Some works in the scalar complex variable case, associated with normal or Gaussian population, are available in the literature with applications in sonar, radar, communication and engineering problems. Some applications of hermitian forms, corresponding to the quadratic forms in Mathai and Provost (1992), in light scattering and quantum mechanics are also available in the literature.
Note that (3.8) for is a matrix-variate type-1 beta density or is a type-1 beta matrix. This is the exact form of the matrix appearing in the generalized analysis of variance and design of experiments areas, in the likelihood ratio test involving one or more multivariate normal or Gaussian populations etc, a summary of the contributions of Mathai and his co-workers is available from Mathai and Saxena (1973). The theory available there is based on Gaussian populations. Now, generalized analysis of variance can be examined in a wider pathway family so that the limiting form corresponding to (3.10), will be the Gaussian case.
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 or in the model (3.8) for. Thus, it is already shown that these three areas are connected.
In (3.1) if one puts then one gets Tsallis statistics in non-extensive statistical mechanics. Also, (3.2) for as well as for some general is superstatistics (Beck, 2004; Cohen, 2004). (3.2) and its limiting form (3.3) are covered in superstatistics but (3.1) is not covered because superstatistics considerations deal with a conditional density of generalized gamma form as well as the marginal density a generalized gamma form then the unconditional density, which is superstatistics in statistical terms from a Bayesian point of view, can only produce a type-2 beta form, namely (3.2) form and not (3.1) form. Thus, superstatistics is also a special case of the pathway model in the real scalar positive variable case.
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 (binomial series) going to (exponential series). That is,
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 can also be sent to an exponential series. For example, consider the Bessel series
Therefore, a generalized form, covering the path towards the exponential form
, is also a Bessel form. This path of a Bessel formgoing to an exponential form can produce a large variety of results. This area has open problems for further research.
4. Special Functions of Matrix Argument
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 where X is a square or rectangular matrix. For example, for a matrix X, = determinant of X, = trace of X are real-valued scalar functions when X is real. Even for a square matrix X, the square root cannot be uniquelydetermined unless further conditions are imposed on X. If one uses the definition, then the square root of A, one can have manycandidates for B. For example, for a simple matrix like a identity matrix, are square roots:
If one restricts A and to be positive definite matrices then is the
only candidate here. Hence, if X is real positive definite or Hermitian positive definite then can be uniquely defined. Therefore, functions of matrix argument are developed mainly when the argument matrix is either real positive definite or Hermitian positive definite. There are three approaches available in the literature for functions of matrix argument, that is, real-valued scalar functions of matrix argument X. For convenience, all the matrices appearing in this section are positive definite denoted by, real or Hermitian, unless stated otherwise. One definition is through Laplace and inverse Laplace transforms. This development is due to Herz (1955) and others. Here the basic assumption of functional commutativity is used, that is, even if. For example, determinant and trace will satisfy this property. When X is real symmetric then there exists an orthonormal matrix Q such that where are the eigenvalues of X. Then
or, which is a function of real variables’s, when and real, has become a function of D which is of p real variables, under this assumption of functional commutativity. If and then
the Laplace transform of because (4.3) is not consistent with the definition of multivariate Laplace transform. In (4.2) the non-diagonal terms appear twice. In the multivariate Laplace transform, the variables and the corresponding parameters must appear only once each. If one considers a modified parameter matrix for all i and j, and
is the Laplace transform in the real symmetric positive definite matrix-variate case, where is the parameter matrix and stands for the wedge product of the differentials’s or
Under this approach, a hypergeometric function of matrix argument, denoted by
where and are scalar parameters and X is a real positive definite matrix, is defined by a Laplace and inverse Laplace pair. Under this definition, explicit forms are available only for and. Details of the definition and properties may be seen from Herz (1955) and from Mathai (1997).
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 are zonal polynomials of order k, and
Here is the Pochhammer symbol and is the generalized Pochhammer symbol. All terms of the series in (4.7) are explicitly available but since zonal polynomials are complicated to compute, only the first few terms up to are computed. Details of zonal polynomials may be found, for example from the book Mathai, Provost and Hayakawa (1995). The definition through (4.5) and its inverse Laplace form and the definition through (4.7) are not very powerful in extending results in the univariate case to the corresponding matrix-variate case. When (4.7) is used to extend univariate results to matrix-variate cases the following two basic results will be essential. These will be stated here as lemmas without proofs.
with defined in (4.8).
Starting from 1970, Mathai developed functions of matrix argument through M-transforms and M-convolutions. Under M-transform definition, a hypergeometric function with matrix argument is defined as that class of functions satisfying functional commutativity and the integral equation
Since the left side in (4.12) is a function of only one parameter, one cannot normally recover a function of p scalar variables. It is conjectured that when is analytic in the cone of positive definite matrices, one has uniquely recovered from the right side of (4.12). This is not established yet and also an explicit form of an inverse or through the right-side of (4.12) is an open problem. But (4.12) is the most convenient form to extend univariate results on hypergeometric functions to the corresponding class of matrix-variate cases. In general, when one goes from a univariate case, such as a univariate function, to a multivariate case, there is nothing called a unique multivariate analogue. Whatever be the properties of the univariate function that one wishes to preserve in the multivariate analogue, one may be able to come up with different functions as multivariate analogues of a univariate function. Hence, a class of multivariate analogues is more appropriate than a single multivariate analogue. Properties of M-transforms and properties of hypergeometric family coming from (4.12) are available in the book Mathai (1997). When Mathai introduced M-convolutions and M-transforms, details in Mathai (1997), no physical meaning could be found. Now, a physical interpretation is available for M-convolutions as densities of products and ratios of matrix random variables, as illustrated in Sections 1.7.
5. Geometric Probabilities: Probability Density Function
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. The proof was very roundabout. Mathai noted that it could be proved easily with the help of the asymptotic expansions of gamma functions. This paper was published in Mathai (1982). Then Mathai formulated and proved parallel conjectures regarding type-1 beta distributed, type-2 beta distributed points and gamma distributed random points and published a series of papers. Then Mathai noted that many European researchers were working on distances between random points, and random areas when the random points are in particular shapes such as triangles, parallelograms, squares, rhombuses etc. As generalizations of all these classes of problems, Mathai generalized Buffon’s clean-tile problem, the starting point of geometrical probabilities. He considered placing a ball at random in a pyramid with polygonal base, defining “at random’’ in terms of kinematic measure, Mathai (1999c). When mixing geometry with probability or measure theory, or in the area of stochastic geometry, the basic axioms of probability are not sufficient, as pointed out by Bertrand’s or Russell’s paradoxes. We need an additional axiom of invariance under Euclidean motion. Another contribution of Mathai in this area is Mathai (1999b) where he has shown that the usual complicated procedures coming from integral geometry and differential geometry are not necessary for handling certain types of random volumes but only the simple properties of functions of matrix argument and Jacobians of matrix transformation are sufficient. The procedure is illustrated in the distribution of volume content of parallelotope generated by random points in Euclidean n-space. The work on geometrical probabilities is currently progressing in the areas of random sets, image processing etc. The book, Mathai (1999a), only deals with distributional aspects of random geometric configurations.
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).
6. Astrophysics: Solar Neutrinos
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 and respectively with. Take
where are normalizing constants. When the density of u, denoted by, is given by
Now, it is only a matter of evaluating the density by using some other method. Note that where and are independently distributed means
where denotes the expected value of. Then for (6.4)
which also shows that since at is 1. Evaluations of and are not necessary for our procedure to hold.
Hence is available from the inverse Mellin transform. That is,
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, and. In this case, the H-function in (6.6) reduces to a G-function and explicit computable series forms are also given by Mathai and his co-workers. Problems considered were resonant reactions, non-resonant reactions, depleted case, and high energy tail cut off. A summary of the work until 1988 is available in the research monograph Mathai and Haubold (1988). After publishing papers in physics by using statistical techniques it was realized that the density of a product of independently distributed real positive random variables was nothing but the Mellin convolution of a product. Hence, one could have applied Mellin and inverse Mellin transform techniques there. The work in this area of reaction-rate also resulted in two encyclopedia articles, see Haubold and Mathai (1997, 1998).
Analytic Solar Models
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 be the radius of the Sun. Let so that. The model for the matter density distribution is taken as
where c is the central core density when. The parameters and are selected to agree with observational data. Then by using (6.7), expressions for mass, pressure, temperature and luminosity are computed by using physical laws. Then by using known observations, or comparing with known data on mass, pressure etc the best values for and are estimated so that close agreement is there with observational values of mass, pressure etc. Some of the results until 1988 are given in the monograph Mathai and Haubold (1988).
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 to. Then the differential equation got simplified. Then he changed the dependent variable and found that the differential equation became a particular case of G-function differential equation. This resulted in the first paper of Mathai in integer order differential equations and it was published in Mathai (1989). Then the results were applied to gravitational instability problem (Haubold & Mathai, 1988).
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 and an exponential type output so that the residual part. When and are independently and identically exponentially distributed then u has a Laplace distribution. One model HJH and Mathai tried was Laplace type random variables over time so that the graph will look like blips at equal or random points on a horizontal line. If the time-lag is shortened then the blips will start joining together. If exponential models of different intensities, that is, in the input-output model , if is different for different blips then the pattern can be brought to the pattern seen in nature or the pattern seen from the data.
7. Special Functions of Mathematical Physics
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 and so that the integral in (7.1) can be evaluated? One can construct some general mixing families of and.
Another family of problems that was looked into were the null and non-null distributions of the likelihood ratio criterion or -criterion for testing hypotheses on the parameters of one or more multinormal populations. Consider the vector having the density
where is a constant vector, known as the mean-value vector here. For if ‘s are independently distributed with the same density in (7.2) then we say that we have a simple random sample of size N from the p-variate normal or Gaussian population (7.2). Suppose that we want to test a hypothesis Ho:V = is diagonal. This is called the test for independence in the Gaussian case. Then the -criterion can be shown to have the structure:
where and are independently distributed matrix-variate gamma variables of (1.25) with the same B. Then the structure in (7.3) is distributed as a product of independently distributed type-1 beta random variables,. Then the density of u can be written as a G-function of the type. The density of will go in terms of a H-function. The H-function is more or less the most generalized special function in real scalar variable case and it is defined by the following Mellin-Barnes integral and the following standard notation is used:
where are real and positive numbers,’s and’s are complex numbers. L is a contour separating the poles of to one side and those of to the other side. Existence of the contours and convergence conditions are available from the books Mathai (1993a), Mathai and Saxena (1973, 1978), Mathai and Haubold (2008), Mathai, Saxena and Haubold (2010). When then the H-function reduces to a G-function denoted as
Explicit computable series forms of and for the general, were given by Mathai in a series of papers. The first three forms correspond to product of independently distributed gamma variables, type-1 beta variables and type-2 beta variables respectively. The details of the computable representations are available in the book Mathai (1993a). This is achieved by developing an operator which can handle poles of all orders. This operator may be seen from Mathai & Rathie (1972) and its use from Mathai (1993a). This is a modification of a procedure developed in Mathai and Rathie (1971) to handle generalized partial fractions. Let
where for and the coefficients’s are to be evaluated. The technique developed in Mathai and Rathie (1971) enables one to compute’s explicitly.
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 Integral
A type-1 Dirichlet integral is over a simplex , namely
But one can construct a multivariate integral over a hypercube giving rise to the same where the integrand is different from type-1 Dirichlet format of (7.9). Mathai (2018) constructed such a function which he called it the pseudo Dirichlet function. Consider the following integral:
The method of proving this result is to expand by using a binomial expansion, integrate out variables one by one and then use the properties of Gauss’ hypergeometric function of argument 1 to obtain the result in (7.10). Mathai also extended the integral (7.10) to the real matrix-variate case. In the real matrix-variate case the corresponding integral gives a constant multiple of the form where
where is the real matrix-variate gamma given by the following:
The integral is the following:
where A is a symmetric product of matrices given by
with denoting the positive definite square root of the real positive
definite matrix. The structure in (7.10) gives the same gamma product in (7.9) with replaced by k.
8. Multivariate Statistical Analysis and Statistical Distribution Theory: Fractional Reaction and Diffusion
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 -criteria for testing hypotheses on one or more multivariate Gaussian populations and exponential populations were given by Mathai and his co-workers. The -criterion is explained in (7.3). Null distributions mean the distributions when the null hypotheses are assumed to hold and non-null distributions mean without the restrictions imposed by the hypotheses. In the non-null situations some of the cases are still open problems. In the null cases, u, a one-to-one function of the -criterion, has usually the following representations:
where are independently distributed real scalar type-1 beta random variables, are the same type of type-2 beta random variables and are the same type of gamma random variables. The density of u in (8.1) can be written in terms of a, that of (8.2) as a and that of (8.3) as a. Computational aspects of these forms are already discussed in Section 7 above. In geometrical probabilities also the squares of the volume content of a p-parallelotope can be written as (8.1) when the random points are type-1 beta distributed, as (8.2) when the random points are type-2 beta distributed and as (8.3) when the random points are gamma distributed. There also densities can be evaluated in terms of the three types of G-functions, as explained above.
Also, Mathai and his co-workers have established a connection between -criterion in testing of statistical hypotheses, connected with multivariate normal populations, and certain generalizations of type-1 Dirichlet models. Various generalizations of type-1 and type-2 Dirichlet models were introduced by Mathai and his co-workers starting with Mathai (2003). In this area also G and H-functions appear. The forms and, coming from products of scalar variables of type-1 and type-2 beta, appear in this area of generalized Dirichlet models.
Exact 11-digit accurate percentage points connected with the null distributions of the -criteria were developed by Mathai and Katiyar starting with the Biometrika paper Mathai and Katiyar (1979). As a byproduct, an algorithm for non-linear least squares was also developed by them, see Mathai and Katiyar (1993a). Mathai has contributions in integer programming and optimization also, see Kounias and Mathai (1988).
Mittag-Leffler Function and Mittag-Leffler Density
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.
9. Characterization Problems: Gauss and Beyond
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 may be the error in measurement in an experiment, the error between observed and predicted values etc. If the factors contributing to the error are known then the experimenter will try to control these factors. Very often the error is contributed by infinitely many unknown factors each factor contributing infinitesimal quantities towards. Put some conditions on this. Let
where are assumed to be independently distributed. Assume that each or -a with equal probabilities. That is,
Assume that the total variance of is finite or
Check the consequence of these three assumptions., where a is fixed and finite. For large n one may take. The moment generating function of is
which is the moment generating function of a normal density with mean value zero and variance or the density is
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 is a density of the real scalar variable x. There is a corresponding discrete analogue, which is given by
where. A modified form of (9.2) and (9.3) is Tsallis entropy given by
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. Note that a direct optimization of (9.4), under the assumption that the first moment in is fixed, does not yield (9.5) directly. One has to go through an escort density
and then assume that the first moment is fixed in the escort density, to get the form in (9.5). Mathai’s entropy
when optimized under the condition of first moment in being fixed leads to Tsallis statistics directly. Also the optimization of (9.6) under two moment-type conditions leads to the pathway model, discussed in Section 3, where (9.5) will be a particular case.
10. Biological Modeling: Formation of Pattern
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, a second point starts and moves at the same speed. When the second point comes to, a third point starts, and so on. The rule governing the movement is or
where k is a constant, giving Archimedes’ spiral. When or, one obtains sunflower, coconut tree crown, certain cactus heads and so on. Such a movement can be generated by a viscous fluid flowing up through a capillary with valves so that when a certain pressure is built up in one chamber the liquid moves up to the next chamber. The continuous flow is made pulse-like at the end. The upward motion can be effected by an evaporation process in the leaves, and there is no need for a heart-like mechanism in trees, pumping the fluid up. Mathai and Davis (1973) showed that the arrangments of leaves on a coconut tree crown is ideal from many mathematical points of view.
11. Design of Experiments and Analysis of Variance
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 for all i and j, is called the incidence matrix and the sum of the elements in each row is equal to 1. Thus is singular and hence one cannot write it as where A and b are known and the vector is unknown and is to be estimated. Mathai noted that one could profitably use the conditions in the design and make a nonsingular matrix. One condition in the design is that where are the elements in. Let C be a matrix where all elements in the i-th row of C are the median of the i-th row elements in A, namely the median of for. Then evidently (null). Then
where and is the median of the i-th row elements in A. Then a norm of B is. But since the mean deviation from the median is the least, is the minimum under the circumstances. Therefore, not only that is nonsingular but the series is the fastest converging series for the problem at hand. Then
A good approximation for is available as. This is found to be sufficient for all practical purposes of testing of statistical hypotheses on the components of.
12. Population Problems and Social Sciences
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 be two discrete populations. Consider the representation of P and Q as points on a hypersphere of radius 1,. Then the points are and. Consider where is the angle between these vectors or points on the hypersphere. Note that the lengths of the vectors are
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.
13. Quadratic and Bilinear Forms
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, if and only if A is idempotent and of rank r, where X the vector having the standard normal distribution, that is, and is a chisquare random variable with r degrees of freedom. Is there a corresponding concept when dealing with bilinear forms? When the samples come from a bivariate Gaussian or normal population it is not difficult to work out the density of the sample correlation coefficient. But what about the density of the sample covariance, without the scaling factors of the standard deviations? Both these questions were answered by Mathai (1993c) where Mathai introduced a concept called Laplacianness of bilinear forms (Mathai, 1993b) and also worked out the density of the covariance structures observing that covariance structure is a bilinear form. The necessary and sufficient conditions for a bilinear form to be noncentral generalized Laplacian are given in Corollary 2.5.2 of Mathai, Provost and Hayakawa (1995). For a noncentral generalized Laplacian the moment generating function is of the form
where is the non-centrality parameter.
14. Reliability Analysis: Extension to Pathway Model and Matrix-Variate Case
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).
15. Mellin Convolutions of Products and Ratios and M-Convolutions
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 real symmetric positive definite matrices. One can take symmetric products as, , etc. When the original densities are assumed to be functionally symmetric then all such symmetric forms will produce the same densities whereas each symmetric product produces an integral representation which will be all different. Thus, one gets a large number of different integral representations for the same density or M-convolution of a product.
One can also obtain further representations by taking for example, , and consider the original symmetric product of three matrices as
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 be real scalar positive random variables, independently distributed and let the product.
where will be gamma products when’s have densities belonging to the pathway family of functions, namely type-1 beta, type-2 beta and gamma. Then the inverse of is a G-function. Then this G-function has several different types of integral representations. A series of papers are written by Mathai by using these ideas connecting statistical distributions, fractional calculus, Mellin convolutions and M-convolutions, see for example Mathai (2017, 2018, 2019).
16. Above Topics Itemized: New Concepts and Procedures
The following are the new concepts, new ideas and new procedures introduced by Mathai (Figure 16):
- 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 from its M-transform is still a conjecture;
Figure 16. H.-J. Treder.
- 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.
 Critchfield, C. L. (1972). Analytic Forms of the Thermonuclear Function. In F. Reines (Ed.), Cosmology, Fusion & Other Matters: George Gamow Memorial Volume (pp. 186-191). Colorado: Colorado Associated University Press.
 Fowler, A. W. (1984). Experimental and Theoretical Nuclear Astrophysics: The Quest for the Origin of the Elements. Reviews of Modern Physics, 56, 149-179.
 George, A., & Mathai, A. M. (1974). Applications of the Concepts of Affinity and Distance to Population Problems. Journal of Biosocial Sciences (London), 6, 347-356.
 Gordon, F. S., & Mathai, A. M. (1972). Characterization of Multivariate Normal Distribution using Regression Properties. Annals of Mathematical Statistics, 43, 53-65.
 Haubold, H. J., & Mathai, A. M. (1997). Sun. In J. H. Shirley, & R. W. Fairbridge (Eds.), Encyclopaedia of Planetary Sciences (pp. 786-794). London: Chapman and Hall.
 Haubold, H. J., & Mathai, A. M. (Eds.) (2010). Proceedings of the Third UN/ESA/NASA Workshop on the International Heliophysical Year 2007 and Basic Space Science. Berlin and Heidelberg: Springer.
 Haubold, H. J., Mathai, A. M., & Saxena, R. K. (2011). Mittag-Leffler Functions and Their Applications. Journal of Applied Mathematics, 2011, Article ID: 298628.
 Mathai, A. M. (1989). On a System of Differential Equations Connected with the Gravitational Instability in a Multi-Component Medium in Newtonian Cosmology. Studies in Applied Mathematics, 80, 75-93.
 Mathai, A. M. (1998). Pollution by Vehicular Travels from the Suburbs to the City Core. Environmetrics, 9, 617-628.
 Mathai, A. M. (2003). Order Statistics from a Logistic Distribution and Applications to Survival & Reliability Analysis. IEEE Transactions on Reliability, 52, 200-206.
 Mathai, A. M. (2009). Fractional Integrals in the Matrix-Variate Cases and Connection to Statistical Distributions. Integral Transforms and Special Functions, 20, 871-882.
 Mathai, A. M. (2019). Extended Fractional Integrals, Products and Ratios of Matrices and Statistical Distributions. 67th Annual Conference of Bharat Ganita Parishad, Lucknow, India, November 2019.
 Mathai, A. M., & Davis, T. A. (1973). A Mathematical Explanation of the Emergence of Foliar Spirals in Coconut and Other Palm Trees. Proceedings of the National Science Academy (India), 39, 194-202.
 Mathai, A. M., & Haubold, H. J. (2018d). Erdélyi-Kober Fractional Calculus: From a Statistical Perspective, Inspired by Solar Neutrino Physics. In Springer Briefs in Mathematical Physics Volume 31. Singapore: Springer.
 Mathai, A. M., & Katiyar, R. S. (1993). A New Algorithm for Nonlinear Least Squares. Researches in Mathematical Statistics (Russian), 207, 143-157. (Translation by the American Mathematical Society)
 Mathai, A. M., & Moschopoulos, P. G. (1999). Pollution by Vehicular Travels from Satellite Townships to the City. Environmetrics, 10, 791-802.
 Mathai, A. M., & Provost, S. B. (2005). Some Complex Matrix-Variate Statistical Distributions on Rectangular Matrices. Linear Algebra and its Applications, 410, 198-216.
 Pyenson, L., Mathai, A. M., & Haubold, H. J. (2019). United Nations Education Program in Space Science and Technology 1988-2018. Creative Education, 10, 2219-2231.
 Treder, H.-J. (1974). Gravitation und weitreichende schwache Wechselwirkungen bei Neutrino-Feldern (Gedanken zu einer Theorie der solaren Neutrinos). Astronomische Nachrichten, 295, 169-184.