A linear Fuchsian equation with variable indices (Q1874484)

The authors are interested in formal solutions to Fuchsian partial differential equation of the form \[ Lu:= (t\partial_t- c)(t\partial_t- x- d) u- t^2 P_2(x,\partial_x)u= f, \] where \(P_2\) is a second-order operator with analytic coefficients. The following two problems are under discussion: \(\bullet\) Let \(f\equiv 0\). What is the influence of \(c\) and \(d\) on the existence of power series representations with respect to \(t\) for solutons of \(Lu\equiv 0\), that is, \(u= \sum_{j\geq 0} u_j(x, t)t^{a+ j}\)? \(\bullet\) How does a special structure of \(f\) (e.g., \(f(t,x)= t^k x^m\), \(f(t,x)= t^k(x^m- p(t)))\), of \(p\) especially, influence the existence of power series representations with respect to \(t\) for solutions of \(Lu\equiv f\)? The authors give several answers to these questions. They \(\bullet\) characterize the function space to which \(u_j\) belong to, these are spaces of analytic functions in new basic variables, containing \(x^{-k}\), \(t^x\) and \(\ln t\); \(\bullet\) explain the degree of freedom, that is, the number of functions one can choose arbitrarily; \(\bullet\) show how a special behaviour of \(p= p(t)\) is reflected to a solution of \(Lu\equiv f\) (polynomial behaviour, logarithmic behaviour).