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