Open Access

Convergence properties of the classical and generalized Rogers-Ramanujan continued fraction

Research in Number Theory20151:15

DOI: 10.1007/s40993-015-0016-4

Received: 12 May 2015

Accepted: 21 July 2015

Published: 2 November 2015

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

Keywords

Convergence Divergence Rogers-ramanujan continued fractions Roots of unity