On univariate fractional calculus with general bivariate analytic kernels

Abstract

Several fractional integral and derivative operators have been defined recently with a bivariate structure, acting on functions of a single variable but with kernels defined using double power series. We propose a general structure to contain all such operators, and establish some important mathematical facts, such as a series formula, a Leibniz rule, a fundamental theorem of calculus, and Laplace and Fourier transform relations, which are applicable to all operators within our general structure.