  In this paper, we propose a new concept of derivative with respect to an
arbitrary kernel-function. Several properties related to this new operator,
like inversion rules, integration by parts, etc. are studied. In particular, we
introduce the notion of conjugate kernels, which will be useful to guaranty
that the proposed derivative operator admits a right inverse. The proposed
concept includes as special cases Riemann-Liouville fractional derivatives,
Hadamard fractional derivatives, and many other fractional operators. Moreover,
using our concept, new fractional operators involving certain special functions
are introduced, and some of their properties are studied. Finally, an existence
result for a boundary value problem involving the introduced derivative
operator is proved.
