Remarks on a theorem of Perron (Q890201)

Let be given a linear differential equation of order two \[ y''={q_0}(z)y+{q_1}(z)y,\;{q_0},{q_1}\in \mathbb{C}(z).\tag{1} \] The \(n\)-th derivative of \(y\) can be presented as \[ {y^{(n)}}={q_{0,n}}(z)y+{q_{1,n}}(z)y,{q_{i,n}}\in \mathbb{C}(z),\;i=0,1. \] Let \(S=\left\{ {t_1},{t_2},\dots,{t_r} \right\}\) be the set of singular points of (1) in \({\mathbb C}\). With \(S\) one can associate the open sets \({U_1},{U_2},\dots ,{U_r}\) defined by \[ {U_i}:=\left\{ z\in \mathbb{C} \mid |z-{t_i}| < |z-{t_j}|,\;j=1,2,\dots,r,\;j\neq i \right\}. \] The authors say that (1) is generic if \(\forall {t_i}\in S\) there is a basis \({y_1},{y_2}\) of the space of solutions of (1) such that near \({t_i}\) the solution \({y_2}\) extends holomorphically to \({t_i}\) and the solution \({y_1}\) does not extend. For the equation (1) the following theorem is proved in the paper: Theorem. For a generic Fuchsian differential equation (1), the fraction \(\frac{q_{0,n}(z)}{q_{1,n}(z)}\) converges uniformly in compact subsets of \(U_i\) to \(-\frac{y'_2}{y_2}\) , where \(y_2\) is the unique (up to multiplication by a constant) holomorphic solution of (1) in \(U_i\). Then, the authors show that \(\frac{q_{0,n}(z)}{q_{1,n}(z)}\) can be written as the continued fraction. So they get a generalization of a theorem of Perron for the hypergeometric equation (see [\textit{O. Perron}, Die Lehre von den Kettenbrüchen. Band II. Analytisch-funktionentheoretische Kettenbrüche. 3. verbesserte und erweiterte Auflage. Stuttgart: B. G. Teubner Verlagsgesellschaft (1957; Zbl 0077.06602)]).