Gauss Relations in Feynman Integrals

Embedding Feynman integrals in Grassmannians, we express Feynman integrals as linear combinations of generalized hypergeometric functions. Here we present general methods to obtain Gauss relations among those generalized hypergeometric functions. The hypergeometric expressions of Feynman integral are analytically continued from some connected component to another by the Gauss inverse relations, then continued to the whole domain of definition by the Gauss-Kummer relations. The Laurant series of the Feynman integral around the time-space dimension D=4 is obtained by the Gauss adjacent relations where the coefficients of powers of D-4 are given as some finite linear combinations of hypergeometric functions with integer parameters. As an example, we illustrate how to use the method to obtain the analytic expression of the Feynman integral of one-loop self energy in its whole domain of definition.