Kipriyanov's Fractional Calculus Prehistory and Legacy

This paper is partly a historical survey of various approaches and methods in the fractional calculus, partly a description of the Kipriyanov extraordinary theory in comparison with the classical one. The significance and outstanding methods in constructing the independent Kipriyanov fractional calculus theory are convexly stressed, also we represent modern results involving the Kipriyanov operator and corresponding generalization under the semigroup theory point of view.