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arXiv · 1812.11221

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

Douglas Bowman·James Mc Laughlin
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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.

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Douglas Bowman, James Mc Laughlin. 2018-12-28. The Convergence Behavior of $q$-Continued Fractions on the Unit Circle. https://doi.org/10.1007/s11139-006-0072-4

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