Computing ${}_{q+1}{F}_q$ via rational conformal maps

It is well-known that certain Möbius (linear fractional) transformations provide analytic continuation of the Gauss hypergeometric function. These transformations work by exchanging the roles of the three regular singular points and $0$, $1$, and $\infty$ of the Gauss hypergeometric differential equation. Not one of these transformations, however, brings the points $e^{\pm{\rm i}π/3}$ closer to the origin, and convergence in a neighbourhood of these is also troublesome. In this work, we produce a Maclaurin series in optimal complexity after rational conformal mapping that induces pointwise convergence to any ${}_{q+1}F_q$ throughout the cut complex plane $\mathbb{C}\setminus[1,\infty)$. We show that for the optimal parameter values this series converges faster than an approximate mapped Chebyshev series obtained by the ultraspherical spectral method.