On the Van Vleck Theorem for Limit-Periodic Continued Fractions of General Form

V. I. Buslaev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and its applications, Tome 298 (2017), pp. 75-100

Résumé

The boundary properties of functions representable as limit-periodic continued fractions of the form $A_1(z)/(B_1(z)+A_2(z)/(B_2(z)+\dots ))$ are studied; here the sequence of polynomials $\{A_n\}_{n=1}^\infty $ has periodic limits with zeros lying on a finite set $E$, and the sequence of polynomials $\{B_n\}_{n=1}^\infty $ 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.

Détail

Keywords: continued fractions, Hankel determinants, transfinite diameter, meromorphic continuation.