Scalar 1-loop Feynman integrals as meromorphic functions in space-time dimension d

The long-standing problem of representing the general massive one-loop Feynman integral as a meromorphic function of the space-time dimension $d$ has been solved for the basis of scalar one- to four-point functions with indices one. In 2003 the solution of difference equations in the space-time dimension allowed to determine the necessary classes of special functions: self-energies need ordinary logarithms and Gauss hypergeometric functions $_2F_1$, vertices need additionally Kamp\'{e} de F\'{e}riet-Appell functions $F_1$, and box integrals also Lauricella-Saran functions $F_S$. In this study, alternative recursive Mellin-Barnes representations are used for the representation of $n$-point functions in terms of $(n-1)$-point functions. The approach enabled the first derivation of explicit solutions for the Feynman integrals at arbitrary kinematics. In this article, we scetch our new representations for the general massive vertex and box Feynman integrals and derive a numerical approach for the necessary Appell functions $F_1$ and Saran functions $F_S$ at arbitrary kinematical arguments