Functions defined by continued fractions meromorphic continuation
Let f(z) be the pointwise limit of the continued fraction \(K(a_ n(z)/b_ n(z))\) on \(E=\{z\in {\mathcal D}^*\); \(K(a_ n(z)/b_ n(z))\) converges\(\}\), where \(a_ n,b_ n: {\mathcal D}^*\to {\mathbb{C}}\) are analytic functions, \(a_ n(z)\neq 0\), on a domain \({\mathcal D}^*\) on a complex Riemann surface. It is then well known that the sequence of modified approximants \[ S_ n(w_ n,z)=\frac{a_ 1(z)}{b_ 1(z)}\begin{matrix} \quad \\ +\end{matrix} \frac{a_ 2(z)}{b_ 2(z)}\begin{matrix} \quad \\ +...+\end{matrix} \frac{a_ n(z)}{b_ n(z)+w_ n(z)},\quad n=1,2,3,... \] converges pointwise on E to the same limit under some mild conditions on the modifying factors \(w_ n(z)\). If in particular \(w_ n(z)\) is analytic on a domain \({\mathcal D}\subseteq E\) from some n on, and the convergence of \(S_ n(w_ n,z)\) is locally uniform on \({\mathcal D}\), then clearly f(z) is analytic on \({\mathcal D}\). In some cases it is possible to extend \(w_ n(z)\) analytically beyond \({\mathcal D}\), and even beyond E, in such a way that \(\{S_ n(w_ n,z)\}\) converges locally uniformly on this larger domain \(\subseteq {\mathcal D}^*\). Then lim \(S_ n(w_ n,z)\) is an analytic continuation of f(z). In this paper sufficient conditions for \(w_ n(z)\) to give such an extension are given. The method is a generalization of a result by Thron and Waadeland.
scientific article; zbMATH DE number 3944202