On Hankel determinants of functions given by their expansions in $P$-fractions

Abstract

We obtain explicit formulas that express the Hankel determinants of functions given by their expansions in continued $P$-fractions in terms of the parameters of the fraction. As a corollary, we obtain a lower bound for the capacity of the set of singular points of these functions, an analog of the van Vleck theorem for $P$-fractions with limit-periodic coefficients, another proof of the Gonchar theorem on the Leighton conjecture, and an upper bound for the radius of the disk of meromorphy of a function given by a $С$-fraction.