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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On the divergence of the Rogers-Ramanujan continued fraction on the unit circle
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by Douglas Bowman and James Mc Laughlin PDF
Trans. Amer. Math. Soc. 356 (2004), 3325-3347 Request permission


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.
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Additional Information
  • Douglas Bowman
  • Affiliation: Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115
  • Email:
  • James Mc Laughlin
  • Affiliation: Department of Mathematics, Trinity College, 300 Summit Street, Hartford, Connecticut 06106-3100
  • Email:
  • 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.
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 3325-3347
  • MSC (2000): Primary 11A55; Secondary 40A15
  • DOI:
  • MathSciNet review: 2052952