Numerical evaluation of Appellʹs F1 hypergeometric function Original Research

In this work we present a numerical scheme to compute the two-variable hypergeometric function F1(α,β,β′,γ;x,y) of Appell for complex parameters α,β,β′ and γ, and real values of the variables x and y. We implement a set of analytic continuations that allow us to obtain the F1 function outside the region of convergence of the series definition. These continuations can be written in terms of the Hornʹs G2 function, Appellʹs F2 function related, and the F1 hypergeometric itself. The computation of the function inside the region of convergence is achieved by two complementary methods. The first one involves a single-index series expansion of the F1 function, while the second one makes use of a numerical integration of a third order ordinary differential equation that represents the system of partial differential equations of the F1 function. We briefly sketch the program and show some examples of the numerical computation.