  In this paper, we investigate some properties related to a multi-index
special function $\mathcal{W}^{\left(\bar{\alpha},\bar{\nu}\right)}$ that arose
from an eigenvalue problem for a multi-order fractional hyper-Bessel operator,
involving Caputo fractional derivatives. We show that for particular values of
the parameters involved in this special function
$\mathcal{W}^{\left(\bar{\alpha},\bar{\nu}\right)}$, this leads to the
hyper-Bessel function of Delerue. The Laplace transform of the
$\mathcal{W}^{\left(\bar{\alpha},\bar{\nu}\right)}$ is discussed obtaining, in
particular cases, the well-known functional relation between hyper-Bessel
function and multi-index Mittag-Leffler function, or, quite simply, between
classical Wright and Mittag-Leffler functions. Moreover, it is shown that the
multi-index special function satisfies the recurrence relation involving
fractional derivatives. In a particular case, we derive, to the best of our
knowledge, a new differential recurrence relation for the Mittag-Leffler
function. We also provide derivatives of the 3-parameters function
$\mathcal{W}_{\alpha,\beta,\nu}$ with respect to parameters, leading to
infinite power series with coefficients being quotients of digamma and gamma
functions.
