The Havriliak-Negami relaxation and its relatives: the response, relaxation and probability density functions

We study functions related to the experimentally observed Havriliak-Negami dielectric relaxation pattern proportional in the frequency domain to [1 + (i omega tau(0))(alpha)](-)(beta) with tau(0) > 0 being some characteristic time. For alpha = l/ k < 1 (l and k being positive and relatively prime integers) and beta > 0 we furnish exact and explicit expressions for response and relaxation functions in the time domain and suitable probability densities in their domain dual in the sense of the inverse Laplace transform. All these functions are expressed as finite sums of generalized hypergeometric functions, convenient to handle analytically and numerically. Introducing a reparameterization beta = (2 - q)/(q - 1) and tau(0) = ( q - 1)(1/alpha) (1 < q < 2) we show that for 0 < alpha < 1 the response functions f(alpha,beta)(t/tau(0)) go to the one-sided Levy stable distributions when q tends to one. Moreover, applying the self-similarity property of the probability densities g(alpha,beta)(u), we introduce two-variable densities and show that they satisfy the integral form of the evolution equation.