On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

Abstract:

This paper studies ordinary and general convergence of the Rogers-Ramanujan continued fraction. Let the continued fraction expansion of any irrational number $t \in (0,1)$ be denoted by $[0,e_{1}(t),e_{2}(t),\cdots ]$ and let the $i$-th convergent of this continued fraction expansion be denoted by $c_{i}(t)/d_{i}(t)$. Let \[ S=\{t \in (0,1): e_{i+1}(t) \geq \phi ^{d_{i}(t)} \text { infinitely often}\}, \] where $\phi = (\sqrt {5}+1)/2$. Let $Y_{S} =\{\exp (2 \pi i t): t \in S \}$. It is shown that if $y \in Y_{S}$, then the Rogers-Ramanujan continued fraction $R(y)$ diverges at $y$. $S$ is an uncountable set of measure zero. It is also shown that there is an uncountable set of points $G \subset Y_{S}$ such that if $y \in G$, then $R(y)$ does not converge generally. It is further shown that $R(y)$ does not converge generally for $|y| > 1$. However we show that $R(y)$ does converge generally if $y$ is a primitive $5m$-th root of unity, for some $m \in \mathbb {N}$. Combining this result with a theorem of I. Schur then gives that the continued fraction converges generally at all roots of unity.