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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.


References [Enhancements On Off] (What's this?)

  • 1. Andrews, G. E.; Berndt, Bruce C.; Jacobsen, Lisa; Lamphere, Robert L. The continued fractions found in the unorganized portions of Ramanujan's notebooks. Mem. Amer. Math. Soc. 99 (1992), no. 477, vi+71pp MR 93f:11008
  • 2. Berndt, Bruce C.; Chan, Heng Huat; Zhang, Liang-Cheng Explicit evaluations of the Rogers-Ramanujan continued fraction. J. Reine Angew. Math. 480 (1996), 141-159. MR 98c:11007
  • 3. Berndt, Bruce C.; Chan, Heng Huat Some values for the Rogers-Ramanujan continued fraction. Canad. J. Math. 47 (1995), no. 5, 897-914. MR 97a:33043
  • 4. Borwein, Peter; Erdélyi, Tamás, Polynomials and polynomial inequalities, Springer, New York, 1995. x+48 0 pp. MR 97e:41001
  • 5. Hardy, G. H. (Godfrey Harold). Lectures by Godfrey H. Hardy on the mathematical work of Ramanujan; fall term 1936 / Notes by Marshall Hall. The Institute for Advanced Study. Ann Arbor, Mich., Edwards Bros., Inc., 1937.
  • 6. Huang, Sen-Shan. Ramanujan's evaluations of Rogers-Ramanujan type continued fractions at primitive roots of unity. Acta Arith. 80 (1997), no. 1, 49-60. MR 98h:11012
  • 7. Jacobsen, Lisa General convergence of continued fractions. Trans. Amer. Math. Soc. 294 (1986), no. 2, 477-485. MR 87j:40004
  • 8. Lorentzen, Lisa; Waadeland, Haakon Continued fractions with applications. Studies in Computational Mathematics, 3. North-Holland Publishing Co., Amsterdam, 1992, 35-36, 67-68. MR 93g:30007
  • 9. Milovanovic, G. V; Mitrinovic, D. S; Rassias, Th. M; Topics in polynomials: extremal problems, inequalities, zeros, World Sci. Publishing, River Edge, NJ, 1994. xiv+821 pp. MR 95m:30009
  • 10. Ramanujan, S. Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957. MR 20:6340
  • 11. Ramanujan, S. Collected Papers, Chelsea, New York, 1962.
  • 12. Ramanathan, K. G. On Ramanujan's continued fraction. Acta Arith. 43 (1984), no. 3, 209-226 MR 85d:11012
  • 13. Rockett, Andrew M.; Szüsz, Peter, Continued fractions. World Scientific Publishing Co., Inc., River Edge, NJ, 1992, pp 140-141. MR 93m:11060
  • 14. L.J.Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318-343.
  • 15. Schur, Issai, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüchen, in Gesammelte Abhandlungen. Band II, Springer-Verlag, Berlin-New York, 1973, 117-136. (Originally in Sitzungsberichte der Preussischen Akadamie der Wissenschaften, 1917, Physikalisch-Mathematische Klasse, 302-321) MR 57:2858b
  • 16. Yi,J. Evaluation of the Rogers-Ramanujan continued fraction R(q) by modular equations, Acta Arith. 98 (2001), no. 3, 103-127. MR 2002i:11021

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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

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