Fractional models of anomalous relaxation based on the Kilbas and Saigo function

We revisit the Kilbas and Saigo functions of the Mittag-Leffler type of a real variable , with two independent real order-parameters. These functions, subjected to the requirement to be completely monotone for , can provide suitable models for the responses and for the corresponding spectral distributions in anomalous (non-Debye) relaxation processes, found e.g. in dielectrics. Our analysis includes as particular cases the classical models referred to as Cole-Cole (the one-parameter Mittag-Leffler function) and to as Kohlrausch (the stretched exponential function). After some remarks on the Kilbas and Saigo functions, we discuss a class of fractional differential equations of order with a characteristic coefficient varying in time according to a power law of exponent , whose solutions will be presented in terms of these functions. We show 2D plots of the solutions and, for a few of them, the corresponding spectral distributions, keeping fixed one of the two order-parameters. The numerical results confirm the complete monotonicity of the solutions via the non-negativity of the spectral distributions, provided that the parameters satisfy the additional condition , assumed by us.