  A definition of summability is put forward in the framework of general
Carleman ultraholomorphic classes in sectors, so generalizing $k-$summability
theory as developed by J.-P. Ramis. Departing from a strongly regular sequence
of positive numbers, we construct an associated analytic proximate order and
corresponding kernels, which allow us to consider suitable Laplace and
Borel-type transforms, both formal and analytic, whose behavior closely
resembles that of the classical ones in the Gevrey case. An application to the
study of the summability properties of the formal solutions to some
moment-partial differential equations is included.
