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

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

2004-12-15

arxiv.org/abs/math/0412298v1

Mathematics

Complex Variables

8 pages

Scientific paper

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.

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