  In this paper we introduce a novel Mittag--Leffler-type function and study
its properties in relation to some integro-differential operators involving
Hadamard fractional derivatives or Hyper-Bessel-type operators. We discuss then
the utility of these results to solve some integro-differential equations
involving these operators by means of operational methods. We show the
advantage of our approach through some examples. Among these, an application to
a modified Lamb--Bateman integral equation is presented.
