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The boundary properties of functions representable as limit-periodic continued fractions of the form A 1(z)/(B 1(z) + A 2(z)/(B 2(z) +...)) are studied; here the sequence of polynomials {A n } =1 ∞ has periodic limits with zeros lying on a finite set E, and the sequence of polynomials {B n } =1 ∞ has periodic limits with zeros lying outside E. It is shown that the transfinite diameter of the boundary of the convergence domain of such a continued fraction in the external field associated with the fraction coincides with the upper limit of the averaged generalized Hankel determinants of the function defined by the fraction. As a consequence of this result combined with the generalized Polya theorem, it is shown that the functions defined by the continued fractions under consideration do not have a single-valued meromorphic continuation to any neighborhood of any nonisolated point of the boundary of the convergence set.

For integers $m \geq 2$, we study divergent continued fractions whose numerators and denominators in each of the $m$ arithmetic progressions modulo $m$ converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern-Stolz theorem. We give a theorem on a general class of Poincar{\'e} type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity. We also generalize a curious $q$-continued fraction of Ramanujan's with three limits to a continued fraction with $k$ distinct limit points, $k\geq 2$. The $k$ limits are evaluated in terms of ratios of certain unusual $q$ series. Finally, we show how to use Daniel Bernoulli's continued fraction in an elementary way to create analytic continued fractions with $m$ limit points, for any positive integer $m \geq 2$., Comment: 29 pages. Updated/new content

The boundary properties of functions represented by limit periodic continued fractions of a fairly general form are investigated. Such functions are shown to have no single-valued meromorphic extension to any neighbourhood of any non-isolated boundary point of the set of convergence of the continued fraction. The boundary of the set of meromorphy has the property of symmetry in an external field determined by the parameters of the continued fraction. Bibliography: 26 titles.

We consider the limit periodic continued fractions of Stieltjes type $$\frac{1}{1-}\frac{g_{1}z}{1-}\frac{g_{2}(1-g_{1})z}{1-}\frac{g_{3}(1-g_{2})z}{1-\cdots,},\quad z\in\mathbb{C},g_{i}\in(0,1),\lim\limits_{i\to\infty}g_{i}=1/2,$$ appearing as Schur–Wall g-fraction representations of certain analytic self maps of the unit disc |w