Gravitational lensing by a generalised-NFW halo via the Fox $H$-function and its application to the super-NFW

We present an analytical framework for a family of axisymmetric gravitational lenses, in which we express the lensing properties in terms of the Fox $H$-function. We apply this framework to a generalised-NFW (gNFW) profile, where we provide the power series representation of the Fox $H$-functions involved, and explore their performance and accuracy. From these power series we show that the corresponding Fox $H$-functions reduce to simple expressions in terms of the Gauss hypergeometric function. We apply these results to the particular case of the super-NFW (sNFW) profile, obtaining simpler expressions, this time in terms of complete elliptic functions (which are easier to work with). When the number of images formed is maximum, the sum of their signed magnifications denoted as $I$, is constant for several lenses. We study its behaviour for the sNFW, NFW and Hernquist lenses, and show that for a fixed $\kappa_0$ (characteristic convergence), in general, $I$ is not constant ($I_{\text{min}}\leq I \leq I_{\text{max}}$), as it exhibits a strong dependence on the source position inside the radial caustic. The boundaries depend on $\kappa_0$ (and so does the average $\langle I \rangle$). Our numerical experiments suggest that for these lenses $I_{\text{min}}\to 1$ as $\kappa_0$ increases, and $I\to 1$ as $\kappa_0\to \infty$. Additionally, $I$ is constant only for a specific $\kappa_0$, which is different for each model.