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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