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


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Аннотация

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 {An}n=1 has periodic limits with zeros lying on a finite set E, and the sequence of polynomials {Bn}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.

Авторлар туралы

V. Buslaev

Steklov Mathematical Institute of Russian Academy of Sciences

Хат алмасуға жауапты Автор.
Email: buslaev@mi.ras.ru
Ресей, ul. Gubkina 8, Moscow, 119991

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