On the convergence of limit periodic continued fractions \(K(a_n/1)\) where \(a_n\to -\frac 14\). IV (Q5956456)

Convergence of the continued fraction \(K(a_n/b_n)\) where \({a_n}\) and \({b_n}\) are given sequences of complex numbers is studied here. It is known that \(K(a_n/1)\) converges if \(a_n\) approaches \(-1/4\) from the inside of the parabola which passes through \(-1/4\) and has focus at 0 and axis along the ray arg \(z = 2 \alpha\), where \(|\alpha |< \pi/2 \) [cf. \textit{W. J. Thron}, Math. Z. 69, 173-182 (1958; Zbl 0081.05704)]. In the present paper the author generalises the foregoing criterion of convergence.