We present a generalization of Hilfer derivatives in which Riemann-Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties. Furthermore, we show some applications of these generalized Hilfer-Prabhakar derivatives in classical equations of mathematical physics such as the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes.

In this paper we introduce the space-fractional Poisson process whose state probabilities pα is the fractional difference operator found in the study of time series analysis. We explicitly obtain the distributions p α k (t), the probability generating functions G α (u, t), which are also expressed as distributions of the minimum of i.i.d. uniform random variables. The comparison with the time-fractional Poisson process is investigated and finally, we arrive at the more general space-time fractional Poisson process of which we give the explicit distribution.

The Mittag-Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function.

We consider a fractional version of the classical nonlinear birth process of which the Yule Furry model is a particular case. Fractionality is obtained by replacing the first order time derivative in the differencedifferential equations which govern the probability law of the process with the Dzherbashyan Caputo fractional derivative. We derive the probability distribution of the number N(v)(t) of individuals at an arbitrary time t. We also present an interesting representation for the number of individuals at time t, in the form of the subordination relation N(v)(t) = N(T(2v) (t)), where N(t) is the classical generalized birth process and T(2v) (t) is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in Section 3 and various forms of its distribution are given and discussed

In this paper we introduce a novel Mittag-Leffler-type function and study its properties in relation to some integro-differential operators involving Hadamard fractional derivatives or Hyper-Bessel-type operators. We discuss then the utility of these results to solve some integro-differential equations involving these operators by means of operational methods. We show the advantage of our approach through some examples. Among these, an application to a modified Lamb-Bateman integral equation is presented.