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We study functions related to the experimentally observed Havriliak-Negamidielectric relaxation pattern in the frequency domain$\sim[1+(i\omega\tau_{0})^{\alpha}]^{-\beta}$ with $\tau_{0}$ being somecharacteristic time. For $\alpha = l/k< 1$ ($l$ and $k$ positive integers) and$\beta > 0$ we furnish exact and explicit expressions for response andrelaxation functions in the time domain and suitable probability densities intheir "dual" domain. All these functions are expressed as finite sums ofgeneralized hypergeometric functions, convenient to handle analytically andnumerically. 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-sidedL\'{e}vy stable distributions when $q$ tends to one. Moreover, applying theself-similarity property of the probability densities $g_{\alpha, \beta}(u)$,we introduce two-variable densities and show that they satisfy the integralform of the evolution equation.

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