On the Van Vleck theorem for limit-periodic continued fractions of general form

Abstract

The boundary properties of functions representable as limit-periodic continued fractions of the form A₁(z)/(B₁(z) + A₂(z)/(B₂(z) +...)) are studied; here the sequence of polynomials {A_n}_n=1^∞ has periodic limits with zeros lying on a finite set E, and the sequence of polynomials {B_n}_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 Pólya 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.