A system of hypergeometric differential equations in m variables of rank pᵐ

We define a hypergeometric series in m variables with $p+(p-1)m$ parameters, which reduces to the generalized hypergeometric series ₚFₚ₋₁ when $m=1$, and to Lauricella's hypergeometric series FC 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 pᵐ solutions to this system around a point near to the origin. We show that this system is of rank pᵐ, and determine its singular locus.