On the Divergence of the Rogers-Ramanujan Continued Fraction on the Unit Circle

Abstract

Let the continued fraction expansion of any irrational number $t\in (0,1)$ be denoted by $[0,a_1(t),a_2(t),...]$ and let the i-th convergent of this continued fraction expansion be denoted by $c_i(t)/d_i(t)$. Let \[ S=\{t \in (0,1): a_{i+1}(t) \geq ϕ^{d_{i}(t)} \text{infinitely often}\}, \] where $\varphi =(\sqrt{5}+1)/2$. Let $Y_S=\{\exp (2\pi it):t\in S\}$. 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 and that R(y) does converge generally if y is a primitive 5m-th root of unity, some $m\in \mathbb{N}$.

arXiv comment27 pages. Some interesting new examples

Cited by in corpus (4)

• The Convergence and Divergence of $q$-Continued Fractions outside the Unit Circle
• A Theorem on Divergence in the General Sense for Continued Fractions
• The Convergence Behavior of $q$-Continued Fractions on the Unit Circle
• On the Divergence in the General Sense of $q$-Continued Fraction on the Unit Circle