On fractional calculus with analytic kernels with respect to functions

Many different types of fractional calculus have been proposed, which can be organised into some general classes operators. For a unified mathematical theory, results should proved in the most possible setting. Two important fractional-calculus operators are integrals and derivatives with respect to functions (dating back 1970s) those analytic kernels (introduced 2019). To cover both these settings single study, we consider functions, never studied detail before. Here establish basic properties operators, including series formulae, composition relations, function spaces, Laplace transforms. The tools convergent series, from kernels, operational calculus, essential ingredients analysis class that covers both.