General fractional integrals and derivatives of arbitrary order

Abstract

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented, and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives of arbitrary order possess integrable singularities at the point zero, the kernels of the general fractional integrals can—depending on their order—be both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved.