Specializations of partial differential equations for Feynman integrals

Starting from the Mellin-Barnes integral representation of a Feynman integral depending on a set of kinematic variables zi , we derive a system of partial differential equations w.r.t. new variables xj , which parameterize the differentiable constraints zi = yi(xj ). In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeo-metric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding systems of partial differential equations for master integrals with arbitrary powers of propagators. As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers.(c) 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).