  The L-fractional derivative is defined as a certain normalization of the
well-known Caputo derivative, so alternative properties hold: smoothness and
finite slope at the origin for the solution, velocity units for the vector
field, and a differential form associated to the system. We develop a theory of
this fractional derivative as follows. We prove a fundamental theorem of
calculus. We deal with linear systems of autonomous homogeneous parts, which
correspond to Caputo linear equations of non-autonomous homogeneous parts. The
associated L-fractional integral operator, which is closely related to the beta
function and the beta probability distribution, and the estimates for its norm
in the Banach space of continuous functions play a key role in the development.
The explicit solution is built by means of Picard's iterations from a
Mittag-Leffler-type function that mimics the standard exponential function. In
the second part of the paper, we address autonomous linear equations of
sequential type. We start with sequential order two and then move to arbitrary
order by dealing with a power series. The classical theory of linear ordinary
differential equations with constant coefficients is generalized, and we
establish an analog of the method of undetermined coefficients. The last part
of the paper is concerned with sequential linear equations of analytic
coefficients and order two.
