  We show how a rescaling of fractional operators with bounded kernels may help
circumvent their documented deficiencies, for example, the inconsistency at
zero or the lack of inverse integral operator. On the other hand, we build a
novel class of linear operators with memory effects to extend the L-fractional
and the ordinary derivatives, using probability tools. A Mittag-Leffler-type
function is introduced to solve linear problems, and nonlinear equations are
solved with power series, illustrating the methods for the SIR model. The
inverse operator is constructed, and a fundamental theorem of calculus and an
existence-and-uniqueness result for differential equations are proved. A
conjecture on deconvolution is raised, that would permit completing the
proposed theory.
