Analytic continuations of the Horn $H_1$ and $H_5$ functions

The analytic continuations (ACs) of the double variable Horn $H_1$ and $H_5$ functions have been derived for the first time using the automated symbolic $\mathit{\text{Mathematica}}$ package $\mathtt{\text{Olsson.wl}}$. The use of Pfaff-Euler transformations have been emphasised to derive AC to cover regions which are otherwise not possible. The corresponding region of convergence (ROC) is obtained using its companion package $\mathtt{\text{ROC2.wl}}$. A $\mathit{\text{Mathematica}}$ package $\mathtt{\text{HornH1H5.wl}}$, containing all the derived ACs and the associated ROCs, along with a demonstration file of the same is made publicly available in https://github.com/souvik5151/Horn_H1_H5 .