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Transactions of the American Mathematical Society

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


Authors: Douglas Bowman and James Mc Laughlin
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 3325-3347
MSC (2000): Primary 11A55; Secondary 40A15
DOI: https://doi.org/10.1090/S0002-9947-03-03390-7
Published electronically: December 15, 2003
MathSciNet review: 2052952
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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

\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 $R(y)$ 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 $\vert y\vert > 1$. 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.


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Additional Information

Douglas Bowman
Affiliation: Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115
Email: bowman@math.niu.edu

James Mc Laughlin
Affiliation: Department of Mathematics, Trinity College, 300 Summit Street, Hartford, Connecticut 06106-3100
Email: james.mclaughlin@trincoll.edu

DOI: https://doi.org/10.1090/S0002-9947-03-03390-7
Keywords: Continued fractions, Rogers-Ramanujan
Received by editor(s): January 17, 2003
Received by editor(s) in revised form: April 15, 2003
Published electronically: December 15, 2003
Additional Notes: The second author’s research supported in part by a Trjitzinsky Fellowship.
Article copyright: © Copyright 2003 American Mathematical Society