On a hypergeometric function of four variables with a new aspect of \(SL\)- symmetry (Q1200287)

The hypergeometric function under consideration is the author's \(K_{16}\). (It is also denoted by \(H_ C^{(4)}\).) A system of four partial differential equations for this function is obtained, and their singular manifolds are given. The system has 21 essentially different singular points in \(\mathbb{C}^ 4\). Some properties of \(K_{16}\) are given: an Euler transformation and a Pochhammer integral representation. Next, it is shown that the system of PDE's has further solutions in terms of \(K_{16}\) and two more quadruple hypergeometric functions. After these preparations the author can finally present, for each of the 21 singularities, a list of six linearly independent integrals of the system of PDE's. The expressions are too long to reproduce here.