Intersection numbers and twisted period relations for the generalized hypergeometric function ${}_{m+1} F_m$

Abstract: We study the generalized hypergeometric function ${}{m+1} F_m$ and the differential equation ${}{m+1}E_m$ satisfied by it. We use the twisted (co)homology groups associated with an integral representation of Euler type. We evaluate the intersection numbers of some twisted cocycles which are defined as $m$-th exterior products of logarithmic $1$-forms. We also give twisted cycles corresponding to the series solutions to ${}{m+1}E_m$, and evaluate the intersection numbers of them. These intersection numbers of the twisted (co)cycles lead twisted period relations which give relations for two fundamental systems of solutions to ${}{m+1}E_m$.