A linear Fuchsian equation with variable indices

We construct three types of solutions for a Fuchsian equation with variable indices: (1) branched solutions involving logarithms of the time variable t; (2) solutions involving t^x, where x is a space variable; and, (3) for a model case, exact solutions involving hypergeometric functions. These three solutions have completely different singularities. The constructions are given in a form suitable for application to more general equations. As an illustration, we resolve in particular an apparent discrepancy between two recent results on this problem.