On the divergence of the Rogers-Ramanujan continued fraction on the unit circle期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Douglas Bowman James Mc Laughlin

Institution:	Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115 ; Department of Mathematics, Trinity College, 300 Summit Street, Hartford, Connecticut 06106-3100

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 -th convergent of this continued fraction expansion be denoted by $c_{i}(t)/d_{i}(t)$ . Let $\begin{displaymath}S=\{t \in (0,1): e_{i+1}(t) \geq \phi^{d_{i}(t)} \text{ infinitely often}\}, \end{displaymath}$ 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 diverges at . 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 does not converge generally. It is further shown that does not converge generally for $\vert y\vert > 1$ . However we show that does converge generally if is a primitive -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.

Keywords:	Continued fractions Rogers-Ramanujan

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