One loop integration with hypergeometric series by using recursion relations

Abstract: General one-loop integrals with arbitrary mass and kinematical parameters in $d$-dimensional space-time are studied. By using Bernstein theorem, a recursion relation is obtained which connects $(n+1)$-point to $n$-point functions. In solving this recursion relation, we have shown that one-loop integrals are expressed by a newly defined hypergeometric function, which is a special case of Aomoto-Gelfand hypergeometric functions. We have also obtained coefficients of power series expansion around 4-dimensional space-time for two-, three- and four-point functions. The numerical results are compared with LoopTools for the case of two- and three-point functions as examples.