On the convergence of continued fractions at Runckel’s points

Abstract

We consider the limit periodic continued fractions of Stieltjes type

$$\frac{1}{1-}\frac{g_{1}z}{1-}\frac{g_{2}(1-g_{1})z}{1-}\frac{g_{3}(1-g_{2})z}{1-\cdots,},\quad z\in\mathbb{C},g_{i}\in(0,1),\lim\limits_{i\to\infty}g_{i}=1/2,$$

appearing as Schur–Wall g-fraction representations of certain analytic self maps of the unit disc |w|<1, w∈ℂ. We make precise the convergence behavior and prove the general convergence [2, p. 564] of these continued fractions at Runckel’s points [6] of the singular line (1,+∞). It is shown that in some cases the convergence holds in the classical sense. As a result we provide an interesting example of convergence relevant to one result found in the Ramanujan’s notebook [1, pp. 38–39].