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This book focuses on Erdelyi-Kober fractional calculus from a
statistical perspective inspired by solar neutrino physics. Results
of diffusion entropy analysis and standard deviation analysis of
data from the Super-Kamiokande solar neutrino experiment lead to
the development of anomalous diffusion and reaction in terms of
fractional calculus. The new statistical perspective of
Erdelyi-Kober fractional operators outlined in this book will have
fundamental applications in the theory of anomalous reaction and
diffusion processes dealt with in physics. A major mathematical
objective of this book is specifically to examine a new definition
for fractional integrals in terms of the distributions of products
and ratios of statistically independently distributed positive
scalar random variables or in terms of Mellin convolutions of
products and ratios in the case of real scalar variables. The idea
will be generalized to cover multivariable cases as well as matrix
variable cases. In the matrix variable case, M-convolutions of
products and ratios will be used to extend the ideas. We then give
a definition for the case of real-valued scalar functions of
several matrices.
Chapter 1 introduces elementary classical special functions.
Gamma, beta, psi, zeta functions, hypergeometric functions and the
associated special functions, generalizations to Meijer's G and
Fox's H-functions are examined here. Discussion is confined to
basic properties and selected applications. Introduction to
statistical distribution theory is provided. Some recent extensions
of Dirichlet integrals and Dirichlet densities are discussed. A
glimpse into multivariable special functions such as Appell's
functions and Lauricella functions is part of Chapter 1. Special
functions as solutions of differential equations are examined.
Chapter 2 is devoted to fractional calculus. Fractional integrals
and fractional derivatives are discussed. Their applications to
reaction-diffusion problems in physics, input-output analysis, and
Mittag-Leffler stochastic processes are developed. Chapter 3 deals
with q-hyper-geometric or basic hypergeometric functions. Chapter 4
covers basic hypergeometric functions and Ramanujan's work on
elliptic and theta functions. Chapter 5 examines the topic of
special functions and Lie groups. Chapters 6 to 9 are devoted to
applications of special functions. Applications to stochastic
processes, geometric infinite divisibility of random variables,
Mittag-Leffler processes, alpha-Laplace processes, density
estimation, order statistics and astrophysics problems, are dealt
with in Chapters 6 to 9. Chapter 10 is devoted to wavelet analysis.
An introduction to wavelet analysis is given. Chapter 11 deals with
the Jacobians of matrix transformations. Various types of matrix
transformations and the associated Jacobians are provided. Chapter
12 is devoted to the discussion of functions of matrix argument in
the real case. Functions of matrix argument and the pathway models
along with their applications are discussed.
Chapter 1 introduces elementary classical special functions. Gamma,
beta, psi, zeta functions, hypergeometric functions and the
associated special functions, generalizations to Meijer's G and
Fox's H-functions are examined here. Discussion is confined to
basic properties and selected applications. Introduction to
statistical distribution theory is provided. Some recent extensions
of Dirichlet integrals and Dirichlet densities are discussed. A
glimpse into multivariable special functions such as Appell's
functions and Lauricella functions is part of Chapter 1. Special
functions as solutions of differential equations are examined.
Chapter 2 is devoted to fractional calculus. Fractional integrals
and fractional derivatives are discussed. Their applications to
reaction-diffusion problems in physics, input-output analysis, and
Mittag-Leffler stochastic processes are developed. Chapter 3 deals
with q-hyper-geometric or basic hypergeometric functions. Chapter 4
covers basic hypergeometric functions and Ramanujan's work on
elliptic and theta functions. Chapter 5 examines the topic of
special functions and Lie groups. Chapters 6 to 9 are devoted to
applications of special functions. Applications to stochastic
processes, geometric infinite divisibility of random variables,
Mittag-Leffler processes, alpha-Laplace processes, density
estimation, order statistics and astrophysics problems, are dealt
with in Chapters 6 to 9. Chapter 10 is devoted to wavelet analysis.
An introduction to wavelet analysis is given. Chapter 11 deals with
the Jacobians of matrix transformations. Various types of matrix
transformations and the associated Jacobians are provided. Chapter
12 is devoted to the discussion offunctions of matrix argument in
the real case. Functions of matrix argument and the pathway models
along with their applications are discussed.
This is a modified version of Module 10 of the Centre for
Mathematical and Statistical Sciences (CMSS). CMSS modules are
notes prepared on various topics with many examples from real-life
situations and exercises so that the subject matter becomes
interesting to students. These modules are used for undergraduate
level courses and graduate level training in various topics at
CMSS. Aside from Module 8, these modules were developed by Dr. A.
M. Mathai, Director of CMSS and Emeritus Professor of Mathematics
and Statistics, McGill University, Canada. Module 8 is based on the
lecture notes of Professor W. J. Anderson of McGill University,
developed for his undergraduate course (Mathematics 447). Professor
Dr. Hans J. Haubold has been a research collaborator of Dr. A.M.
Mathais since 1984, mainly in the areas of astrophysics, special
functions and statistical distribution theory. He is also a
lifetime member of CMSS and a Professor at CMSS. A large number of
papers have been published jointly in these areas since 1984. The
following monographs and books have been brought out in conjunction
with this joint research: Modern Problems in Nuclear and Neutrino
Astrophysics (A.M. Mathai and H.J. Haubold, 1988, Akademie-Verlag,
Berlin); Special Functions for Applied Scientists (A.M.Mathai and
H.J. Haubold, 2008, Springer, New York); and The H-Function: Theory
and Applications (A.M.Mathai, R.K. Saxena and H.J. Haubold, 2010,
Springer, New York). These CMSS modules are printed at CMSS Press
and published by CMSS. Copies are made available to students free
of charge, and to researchers and others at production cost. For
the preparation of the initial drafts of all these modules,
financial assistance was made available from the Department of
Science and Technology, the Government of India (DST), New Delhi
under project number SR/S4/MS:287/05. Hence, the authors would like
to express their thanks and gratitude to DST, the Government of
India, for its financial assistance.
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