Meromorphic continuation of functions given by limit k-periodic continued fractions (Q1100580)

Let \(K(a_ n(z)/b_ n(z))\) converge (pointwise) to a meromorphic function f(z) in some domain \({\mathcal D}\). From previous results follows that its modified approximants \[ S_ n(w_ n(z),z)=\frac{a_ 1(z)}{b_ 1(z)+}\frac{a_ 2(z)}{b_ 2(z)+...+}\frac{a_ n(z)}{b_ n(z)+w_ n(z)} \] may converge in a larger domain \({\mathcal D}^*\supseteq {\mathcal D}\) to a meromorphic continuation of f(z) to \({\mathcal D}^*\). This requires that the modifying factors \(w_ n(z)\) are very well chosen. In this paper the choice \[ w_ n(z)=\frac{\tilde a_{n+1}(z)}{\tilde b_{n+1}(z)+}\frac{\tilde a_{n+2}(z)}{\tilde b_{n+2}(z)+...} \] and its meromorphic continuation is used, where \(K(\tilde a_ n(z)/\tilde b_ n(z))\) is purely k-periodic so that its tail values \(w_ n(z)\) are known. The main result gives sufficient conditions on \(K(a_ n(z)/b_ n(z))\) to obtain meromorphic continuation by this method.