Fractional non-homogeneous Poisson and Polya-Aeppli processes of order $k$ and beyond

We introduce two non-homogeneous processes: a fractional non-homogeneous Poisson process of order $k$ and and a fractional non-homogeneous Polya-Aeppli process of order $k$. We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property.