8 research outputs found
On the Integral of Fractional Poisson Processes
In this paper we consider the Riemann--Liouville fractional integral
, where , , is a
fractional Poisson process of order , and . We give
the explicit bivariate distribution , for , , the mean and the
variance . We study the
process for which we are able to produce explicit
results for the conditional and absolute variances and means. Much more
involved results on are presented in the last section
where also distributional properties of the integrated Poisson process
(including the representation as random sums) is derived. The integral of
powers of the Poisson process is examined and its connections with generalised
harmonic numbers is discussed
Hilfer-Prabhakar Derivatives and Some Applications
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. Further, we show some applications of these generalized
Hilfer-Prabhakar derivatives in classical equations of mathematical physics,
like the heat and the free electron laser equations, and in
difference-differential equations governing the dynamics of generalized renewal
stochastic processes