This book focuses on Erdélyi-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 Erdélyi-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.