Abstract
The aim of this paper is to study the convergence and divergence of the Rogers-Ramanujan and the generalized Rogers-Ramanujan continued fractions on the unit circle. We provide an example of an uncountable set of measure zero on which the Rogers-Ramanujan continued fraction R(x) diverges and which enlarges a set previously found by Bowman and Mc Laughlin. We further study the generalized Rogers-Ramanujan continued fractions R a (x) for roots of unity a and give explicit convergence and divergence conditions. As such, we extend some work of Huang towards a question originally investigated by Ramanujan and some work of Schur on the convergence of R(x) at roots of unity. In the end, we state several conjectures and possible directions for generalizing Schur’s result to all Rogers-Ramanujan continued fractions R a (x).
2010 Mathematics Subject Classification 11A55, 11P84
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Acknowledgements
This paper was written during the Cologne Young Researchers in Number Theory Program 2015, organized by Larry Rolen. The authors would like to thank him for his tireless support and numerous ideas regarding the project and the University of Cologne for hospitality. The program was funded by the University of Cologne postdoc grant DFG Grant D-72133-G-403-151001011, under the Institutional Strategy of the University of Cologne within the German Excellence Initiative.
The authors would also like to thank Kathrin Bringmann, Michael Griffin and Armin Straub for the fruitful discussions and helpful suggestions. In particular, Armin Straub suggested that the polynomials T m,k in Lemma 9 might factor into cyclotomic polynomials and Michael Griffin proposed the precise formula for T m,k . The authors had further inspiring discussions with Karl Mahlburg, whom they would like to thank for his comments. Last, but not least, the authors thank the anonymous referees for the useful observations and suggested corrections.
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Ciolan, EA., Neiss, R.A. Convergence properties of the classical and generalized Rogers-Ramanujan continued fraction. Res. number theory 1, 15 (2015). https://doi.org/10.1007/s40993-015-0016-4
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DOI: https://doi.org/10.1007/s40993-015-0016-4