On the analytical development of incomplete Riemann-Liouville fractional calculus

Abstract

The theoretical development of fractional calculus includes the formulation of different definitions, the extension of properties from standard calculus, and the application of fractional operators to special functions. In two recent papers, incomplete versions of classical fractional operators were formulated in connection with special functions. Here, we develop the theory of incomplete fractional calculus more deeply, investigating further properties of these operators and answering some fundamental questions about how they work. By considering appropriate function spaces, we discover that incomplete fractional calculus may be used to analyse a wider class of functions than classical fractional calculus can consider. By using complex analytic continuation, we formulate definitions for incomplete Riemann-Liouville fractional derivatives, hence extending the incomplete integrals to a fully-fledged model of fractional calculus. Further properties proved here include a rule for incomplete differintegrals of products, and composition properties of incomplete differintegrals with classical calculus operations.