Abstract
This paper shows that meaningful interpretations for Mellin convolutions of products and ratios involving two, three or more functions, can be given through statistical distribution theory of products and ratios involving two, three or more real scalar random variables or general multivariate situations. This paper shows that the approach through statistical distributions can also establish connection to fractional integrals, reaction-rate probability integrals in nuclear reaction-rate theory, Krätzel integrals and Krätzel transform in applied analysis, continuous mixtures, Bayesian analysis etc. This paper shows that the theory of Mellin convolutions, currently available for two functions, can be extended to many functions through statistical distributions. As illustrative examples, products and ratios of generalized gamma variables, which lead to Krätzel integrals, reaction-rate probability integrals, inverse Gaussian density etc, and type-1 beta variables, which lead to various types of fractional integrals and fractional calculus in general, are considered.
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References
R. Attenburger, C. Haitz and J. Timmer, Analysis of phase-resolved partial discharge patterns of voids based on a stochastic process approach. J. of Physics D, Applied Phys. 35 (2002), 1149–1163.
C. Beck and E.G.D. Cohen, Superstatistics. Physica A 322, Nos 2-3 (2003), 267–275.
R. Gorenflo, A.A. Kilbas, F. Mainardi and S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, New York (2014).
R. Gorenflo, Y. Luchko, F. Mainardi, Analytic properties and applications of the Wright function. Fract. Calc. Appl. Anal. 2 (1999), 383–414.
V. Kiryakova, Multiple (multi-index) Mittag-Leffler functions and relations to generalized fractional calculus. J. Comput. and Appl. Math. 118 (2000), 241–259.
K. Kobayashi, Plane wave diffraction by a strip: Exact and asymptotic solutions. J. of Phys. Soc. of Japan 60 (1990), 1891–1905.
K. Kobayashi, Generalized gamma function occurring in wave scattering problem. J. of Phys. Soc. of Japan 60, No 5 (1991), 1501–1512.
E. Krätzel, Integral transformations of Bessel type. In: Generalized Functions and Operational Calculus, Proc. Conf. Varna, 1975, Bulgarian Academy of Sciences, Sofia (1979), 148–165.
F. Mainardi, Y. Luchko, and G. Pagnini, The fundamental solution of the space-time fractional diffusion equations. Fract. Calc. Appl. Anal. 4 (2001), 153–192.
A.M. Mathai, A Handbook of Generalized Special Functions for Statistical and Physical Sciences. Oxford University Press, Oxford (1993).
A.M. Mathai, A pathway to matrix-variate gamma and normal densities. Linear Algebra and its Applications 396 (2005), 317–328.
A.M. Mathai, Fractional integrals in the matrix-variate cases and connection to statistical distributions. Integr. Transf. Spec. Funct. 20, No 12 (2009), 871–882.
A.M. Mathai, Generalized Krätzel integral and associated statistical densities. Internat. J. of Math. Anal. 6, No 51 (2012), 2501–2510.
A.M. Mathai, Fractional integral operators in the complex matrix-variate case. Linear Algebra and its Applications 439 (2013), 2901–2913.
A.M. Mathai, Fractional integral operators involving many matrix variables. Linear Algebra and its Applications 446 (2014), 196–215.
A.M. Mathai, Fractional differential operator in the complex matrix-variate case. Linear Algebra and its Applications 478 (2015), 200–217.
A.M. Mathai, On products and ratios of three or more generalized gamma variables. J. of Indian Soc. for Probability and Statistics 17, No 1 (2016), 79–94.
A.M. Mathai and H.J. Haubold, Modern Problems in Nuclear and Neutrino Astrophysics. Akademie-Verlag, Berlin (1988).
A.M. Mathai and H.J. Haubold, A pathway from Bayesian statistical analysis to superstatistics. Applied Mathematics and Computation 218 (2011), 799–804.
A.M. Mathai and H.J. Haubold, Pathway model, superstatistics, Tsallis statistics and a generalized measure of entropy. Physica A 375 (2007), 110–122.
A.M. Mathai and H.J. Haubold, Matrix-variate statistical distributions and fractional calculus. Fract. Calc. Appl. Anal. 14, No 1 (2011), 138–155; DOI: 10.2478/s13540-011-0010-z; https://www.degruyter.com/view/j/fca.2011.14.issue-1/issue-files/fca.2011.14.issue-1.xml.
A.M. Mathai and H. J. Haubold, Fractional operators in the matrix-variate case. Fract. Calc. Appl. Anal. 16, No 2 (2013), 469–478; DOI: 10.2478/s13540-013-0029-4; https://www.degruyter.com/view/j/fca.2013.16.issue-2/issue-files/fca.2013.16.issue-2.xml.
A.M. Mathai, H.J. Haubold and C. Tsallis, Pathway model and nonextensive statistical mechanics. Sun and Geosphere 10, No 2 (2015), 157–162; arXiv:1010.4597 (2011).
A.M. Mathai and T. Princy, Multimodel stress-strength under pathway model. South East Asian J. of Math. and Math. Sci. 12, No 2 (2016), 30–43.
A.M. Mathai and T. Princy, Multivariate and matrix-variate Maxwell-Boltzmann and Raleigh densities. Physica A 468 (2017), 668–676.
A.M. Mathai and T. Princy, Analogues of reliability analysis for matrix-variate case. Linear Algebra and its Applications 532 (2017), 287–311.
A.M. Mathai and S.B Provost, Some complex matrix-variate statistical distributions on rectangular matrices. Linear Algebra and its Applications 410 (2005), 198–216.
A.M. Mathai, R.K. Saxena and H.J. Haubold, The H-Function: Theory and Applications. Springer, New York (2010).
I. Podlubny, Fractional Differential Equations. Ser. Math. in Science and Eng. Vol. 198, Academic Press, San Diego (1999).
C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics. J. of Statistical Phys. 52 (1988), 479–487.
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Mathai, A.M. Mellin Convolutions, Statistical Distributions and Fractional Calculus. FCAA 21, 376–398 (2018). https://doi.org/10.1515/fca-2018-0022
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DOI: https://doi.org/10.1515/fca-2018-0022