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.
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