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• Methods of generalized fractional calculus merged with the stochastic processes approach opens possibility to link memory functions with completely Bernstein functions determined from the stochastic processes underlying the relaxation phenomena. • We show how our approach works for Havriliak-Negami and Jurlewicz-Weron-Stanislavsky models. The non-Debye, i.e., non-exponential, behavior characterizes a large plethora of dielectric relaxation phenomena. Attempts to find their theoretical explanation are dominated either by considerations rooted in the stochastic processes methodology or by the so-called fractional dynamics based on equations involving fractional derivatives which mimic the non-local time evolution and as such may be interpreted as describing memory effects. Using the recent results coming from the stochastic approach we link memory functions with the Laplace (characteristic) exponents of infinitely divisible probability distributions and show how to relate the latter with experimentally measurable spectral functions characterizing relaxation in the frequency domain. This enables us to incorporate phenomenological knowledge into the evolution laws. To illustrate our approach we consider the standard Havriliak-Negami and Jurlewicz-Weron-Stanislavsky models for which we derive well-defined evolution equations. Merging stochastic and fractional dynamics approaches sheds also new light on the analysis of relaxation phenomena which description needs going beyond using the single evolution pattern. We determine sufficient conditions under which such description is consistent with general requirements of our approach. [ABSTRACT FROM AUTHOR]

• Generalized fractional calculus. • Prabhakar fractional calculus. • Cole–Cole relaxation. We investigate a possibility to describe the non-Debye relaxation processes using the Volterra-type equations with kernels given by the Prabhakar functions with the upper parameter ν being negative. Proposed integro-differential equations mimic the fading memory effects and are explicitly solved using the umbral calculus and the Laplace transform methods. Both approaches lead to the same results valid for admissible domain of the parameters α, μ and ν characterizing the Prabhakar function. For the special case α ∈ (0, 1], μ = 0 and ν = − 1 we recover the Cole-Cole model, in general having a residual polarization. We also show that our scheme gives results equivalent to those obtained using the stochastic approach to relaxation phenomena merged with integral equations involving kernels given by the Prabhakar functions with the positive upper parameter. [ABSTRACT FROM AUTHOR]