  A family of formal power series, such that its coefficients satisfy a
recursion formula, is characterized in terms of the summability, in the sense
of J. P. Ramis, of its elements along certain well chosen directions. We
describe a set of ordinary differential equations which have, as a formal
solution, one of such formal power series, and also describe the actual
solutions which own such formal power series as Gevrey asymptotic expansion.
  The main results are extended into the framework of $q-$Gevrey series and
$q-$difference equations, and also to the case of moment-differential
equations, whose formal solutions have their coefficients' growth governed by a
so-called strongly regular sequence, appearing as a generalization of the
Gevrey-like sequences.
