Properties of the multi-index special function $\mathcal{W}^{\left(\bar{\alpha},\bar{\nu}\right)}(z)$

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.