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On the summability of a class of formal power series

The formal power series solutions for some classes of moment diﬀerential equations, induced by polynomial moment diﬀerential operators, are characterized in terms of their summability properties, and in terms of estimates for recursive expressions involving their coeﬃcients. Of special interest are the particularization of these results to classes of fractional and of ordinary diﬀerential equations. The Stokes’ phenomenon can be described in some of these situations. The main results are extended into the framework of q -Gevrey asymptotics and q -diﬀerence equations.