Buslaev V. I.
Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1139–1144
We present a criterion of rationality for a function determined by its expansion in a series in orthogonal polynomials. This criterion can be regarded as an analog of the well-known Kronecker criterion of rationality for functions given by power series.
Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 315–326
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.