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 pm linearly independent 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.
Kaneko et al. (Wed,) studied this question.