The functions of the Wright-type in fractional calculus

Here we provide a survey of the high transcendental functions related to the Wright special function. Like the functions of the Mittag-Leffler type, the functions of the Wright type are known to play fundamental roles in various applications of the fractional calculus. This is mainly due to the fact that they are interrelated with the Mittag-Leffler functions through a Laplace transformation. We start providing the definitions in the complex plane for the general Wright function and for two special cases that we call auxiliary functions. Then we devote particular attention to the auxiliary functions in the real field, because they admit a probabilisticinterpretation related to the fundamental solutions of certain evolution equations of fractional order. These equations are fundamental to understand phenomena of anomalous diffusion or intermediate between diffusion and wave propagation. At the end we add some historical and bibliographical notes.