Gauss Relations in Feynman Integrals

Embedding Feynman integrals in Grassmannians, we can write Feynman integrals as some finite linear combinations of generalized hypergeometric functions. In this paper we present a general method to obtain Gauss relations among those generalized hypergeometric functions. The hypergeometric expressions of Feynman integral are continued from a connected component to another by the inverse Gauss relations, then continued to the whole domain of definition by the Gauss-Kummer relations. The Laurent series of the Feynman integral around the space-time dimension $D=4$ is obtained by the Gauss adjacent relations where the coefficient of the term with power of $D-4$ is given as the linear combinations of hypergeometric functions with integer parameters. As examples, we illustrate how to obtain the expressions for the Feynman integrals of the 1-loop self-energy and a 2-loop massless triangle diagram in the domains of definition.