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In this paper, some properties related to the-parametric Mittag-Leffler (M-L) functions are investigated. More precisely, three families of such functions with different kinds of indices are examined. Appropriate estimates and asymptotic formulae are obtained and which are further used in proving the convergence of series of functions of these families in the whole complex plane, as well as in its compact subsets. Using these auxiliary results, different relations referring to the discussed families are obtained; generalizing the ones already known for particular cases of the parameters. Those are considered different representations of the-parametric M-L functions pertaining to the families discussed, and with which various equalities are established relating different sums that involve matrix coefficients. Some interesting properties are reached for particular choices of the matrices. Finally, bounds of the-parametric M-L function are given under a perturbed matrix argument.

On some relations in the class of multi-index Mittag-Leffler functions