  In this paper we show several connections between special functions arising
from generalized COM-Poisson-type statistical distributions and
integro-differential equations with varying coefficients involving
Hadamard-type operators. New analytical results are obtained, showing the
particular role of Hadamard-type derivatives in connection with a recently
introduced generalization of the Le Roy function. We are also able to prove a
general connection between fractional hyper-Bessel-type equations involving
Hadamard operators and Le Roy functions.
