The Convergence Behavior of $q$-Continued Fractions on the Unit Circle

Abstract

In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of $q$-continued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each $q$-continued fraction, $G(q)$, in this class, that there is an uncountable set of points, $Y_G$, on the unit circle such that if $y\in Y_G$ then $G(y)$ does not converge to a finite value. We discuss the implications of our theorems for the convergence of other $q$-continued fractions, for example the Göllnitz-Gordon continued fraction, on the unit circle.

11 pages. arXiv admin note: text overlap with arXiv:1812.10873

Cited by in corpus (1)

• A Theorem on Divergence in the General Sense for Continued Fractions