  We consider the Cauchy problem $(\mathbb D_{(k)} u)(t)=\lambda u(t)$,
$u(0)=1$, where $\mathbb D_{(k)}$ is the general convolutional derivative
introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory {\bf
71} (2011), 583--600), $\lambda >0$. The solution is a generalization of the
function $t\mapsto E_\alpha (\lambda t^\alpha)$ where $0<\alpha <1$, $E_\alpha$
is the Mittag-Leffler function. The asymptotics of this solution, as $t\to
\infty$, is studied.
