On properties of hypergeometric functions of three variables, \(F_ M\) and \(F_ G\) (Q1824075)

The triple hypergeometric series \(F_ M(x,y,z)\) and \(F_ G(x,y,z)\) are investigated near certain points on the boundary of the region of convergence, e.g. \(x+z=1\) with \(x>0\), under certain constraints upon the parameters. Four asymptotic expressions are derived, from which it is seen that the boundary points considered are logarithmic singularities. \(F_ G\) and \(F_ M\) are here expressed in terms of Srivastava's \(F^{(3)}\), Gauß' \({}_ 2F_ 1\), the logarithmic derivative of the Gamma functions, and elementary functions. The four expressions are too complicated to reproduce here. The derivations are based upon similar results for simpler hypergeometric functions.