Gauss's \(_ 2F_ 1(1)\) cannot be generalized to \(_ 2F_ 1(x)\)
Using ideas of J. Wimp and R. McIntosh, it is proved that Gauss's explicit evaluation of \(_ 2F_ 1(a,b;c;1)\) cannot be generalized to \(_ 2F_ 1(a,b;c;x)\), for arbitrary \(a\), \(b\), \(c\) and \(x\). A short proof of Wimp's theorem that asserts that \(_ 3F_ 2(a,b,c;d,e;1)\) cannot be expressed in closed form is also given.
scientific article; zbMATH DE number 89597