On the convergence of continued fractions at Runckel's points and the Ramanujan conjecture

Abstract

We consider the limit periodic continued fractions of Stieltjes $$ \frac{1}{1-} \frac{g_1 z}{1-} \frac{g_2(1-g_1) z}{1-} \frac{g_3(1-g_2)z}{1-...,}, z\in \mathbb C, g_i\in(0,1), \lim\limits_{i\to \infty} g_i=1/2, \quad (1) $$ appearing as Shur--Wall $g$-fraction representations of certain analytic self maps of the unit disc $|w|< 1$, $w \in \mathbb C$. We precise the convergence behavior and prove the general convergence [2, p. 564 ] of (1) at the Runckel's points of the singular line $(1,+\infty)$ It is shown that in some cases the convergence holds in the classical sense. As a result a counterexample to the Ramanujan conjecture [1, p. 38-39] stating the divergence of a certain class of limit periodic continued fractions is constructed.