We study functions g$\alpha${}(x) which are one-sided, heavy-tailed L\'evy stable probability distributions of index $\alpha${}, 0<$\alpha${}<1, of fundamental importance in random systems, for anomalous diffusion and fractional kinetics. We furnish exact and explicit expressions for g$\alpha${}(x), 0$\le${}x<$\infty${}, for all $\alpha${}=l/k4, all expressed in terms of known functions. This will allow a ``fine-tuning'' of $\alpha${} in order to adapt g$\alpha${}(x) to a given experimental situation.

We study generalized Cattaneo (telegrapher's) equations involving memory effects introduced by smearing the time derivatives. Consistency conditions where the smearing functions obey restrict freedom in their choice but the proposed scheme goes beyond the approach based on using fractional derivatives. We find conditions under which solutions of the equations considered so far can be recognized as probability distributions, i.e., are normalizable and nonnegative on their domains. Nonnegativity of solutions is demonstrated by methods of positive definite and completely monotonic functions with the Bernstein theorem being the cornerstone of the ongoing proofs. Analysis of exactly solvable examples and relevant mean-squared displacements enables us to classify diffusion processes described by such got solutions and to identify them with either ordinary or anomalous diffusion which character may change over time. To complete the present research we compare our results with those obtained using the continuous-time random-walk and the continuous-time persistent random-walk approaches.

We study functions related to the experimentally observed Havriliak–Negami dielectric relaxation pattern proportional in the frequency domain to [ 1 + ( i ω τ 0 ) α ] − β with τ 0 > 0 being some characteristic time. For α = l / k < 1 ( l and k being positive and relatively prime integers) and β > 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 β = ( 2 − q ) / ( q − 1 ) and τ 0 = ( q − 1 ) 1 / α ( 1 < q < 2 ) we show that for 0 < α < 1 the response functions f α , β ( t / τ 0 ) go to the one-sided Lévy stable distributions when q tends to one. Moreover, applying the self-similarity property of the probability densities g α , β ( u ) , we introduce two-variable densities and show that they satisfy the integral form of the evolution equation.

We investigate functions g($\alpha${},$\beta${};x) which are heavy-tailed L\'evy stable probability distributions of index 0<$\alpha${}$\leqslant${}2 and appropriate asymmetry parameter $\beta${}. They are of central importance in physics of amorphous and disordered systems, econophysics, geology, hydrology, internet traffic, dynamics of human relations, etc. We present an ensemble of exact and explicit solutions of g($\alpha${},$\beta${};x) for all admissible rational values of $\alpha${} and $\beta${}. We reproduce all the previously known cases and furnish distributions for values $\alpha${} and $\beta${} which were previously inaccessible. We point out many instances of experimental and statistical data that could be described by our results.

Using the Bernstein theorem we prove the complete monotonicity of the three parameter Mittag?Leffler function E??,? (?w) for w ? 0 and suitably constrained parameters ?, ? and ?.