Convergence properties of the classical and generalized Rogers-Ramanujan continued fraction
© The Author(s) 2015
Received: 12 May 2015
Accepted: 21 July 2015
Published: 2 November 2015
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