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

Abstract

We consider the limit periodic continued fractions of Stieltjes $$\frac{1}{1-}\frac{g_{1}z}{1-}\frac{g_2(1-g_1)z}{1-}\frac{g_3(1-g_2)z}{1-...,},z\in \mathbb{C},g_i\in (0,1),\underset{i\to \infty }{\lim }{g}_i=1/2,(1)$$ 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.

arXiv comment8 pages

Cited by in corpus (1)

• On convergence of generalized continued fractions and Ramanujan's conjecture