On the convergence of continued fractions at Runckel's points and the Ramanujan conjecture
Abstract
We consider the limit periodic continued fractions of Stieltjes
1
1−
g
1
z
1−
g
2
(1−
g
1
)z
1−
g
3
(1−
g
2
)z
1−...,
,z∈C,
g
i
∈(0,1),
lim
i→∞
g
i
=1/2,(1)
appearing as Shur--Wall
g
-fraction representations of certain analytic self maps of the unit disc
|w|<1
,
w∈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,+∞)
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.