A linear Fuchsian equation with variable indices
Résumé
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.