  We define a hypergeometric series in $m$ variables with $p+(p-1)m$
parameters, which reduces to the generalized hypergeometric series $_pF_{p-1}$
when $m=1$, and to Lauricella's hypergeometric series $F_C$ in $m$ variables
when $p=2$. We give a system of hypergeometric differential equations
annihilating the series. Under some non-integral conditions on parameters, we
give an Euler type integral representation of the series, and linearly
independent $p^m$ solutions to this system around a point near to the origin.
We show that this system is of rank $p^m$, and determine its singular locus.
