On continued fractions \(K(a_ n/1)\) where all \(a_ n\) are lying on a Cartesian oval (Q1209567)

Let \(V\) be a closed, circular disk in the complex plane with center \(\Gamma\) and radius \(\rho\). The oval theorem by Jacobsen and Thron, modified by Lorentzen and Ruscheweyh, says that \(K(a_ n/1)\) converges to a value in \(V\) if all its elements \(a_ n\) are contained in a certain closed domain \(E(\Gamma,\rho)\). The boundary \(\partial E\) of \(E(\Gamma,\rho)\) is a cartesian oval. Haakon Waadeland considers here the special case where \(\Gamma>0, \Gamma<\rho< \Gamma+1/2\) and all \(a_ n\in \partial E\), and he describes the range of this family of continued fractions; i.e. the set of values such continued fractions take. He proves that this range \(\mathcal L\) is \(V\) itself, minus a certain hole which he describes. He also proves that \(V\) itself, minus the origin, is the range of the family of continued fractions with all \(a_ n\in E(\Gamma,\rho)\), and that \[ \mathcal L=\frac{\partial E}{1+V}\setminus\{0\}, \] i.e. \(\mathcal L\) is the same as the range when \(a_ 1\in\partial E\) and all the other elements \(a_ n\in E(\Gamma,\rho)\). The range of families of continued fractions is important for numerical applications. It is strongly related to the concept of best value regions for continued fractions.