Some Circular Regions of Convergence for Branched Continued Fractions of a Special Form

Abstract

517.524For branched continued fractions of a special form, we obtain circular regions of convergence. These regions are related to the multidimensional generalizations of some well-known theorems ) on twin convergence regions for continued fractions. In the case where a branched continued fraction is transformed into a continued fraction ( N = 1 ), the obtained circular regions can of convergence be wider (under certain conditions imposed on the parameters) than some known twin convergence regions for continued fractions.An important place in the theory of continued fractions is occupied by twin convergence regions. Twin convergence regions are called pairs of regions E 1 , E 2 of the complex plane such that the conditions a 2 k−1 ∈E 1 and a 2 k ∈E 2 , k ≥ 1 , ensure the convergence of the fraction k=1 ∞ D a k 1 . The first twin convergence regions were obtained by W. Leighton and H. S. Wall [11]: the fraction k=1 ∞ D a k 1 converges if a 2 k−1 ≤ 1 4 and a 2 k ≥ 25 4 , k ≥ 1 . Setting a k = c k 2 , k ≥ 1 , W. J. Thron [7] established the convergence of the continued fraction k=1 ∞ D c k 2 1 in the case where c 2 k−1 ≤ ρ and c 2 k ± i ≥ ρ , 0 < ρ < 1, k ≥ 1 . In [7], it was shown that, for ρ = 1 , the fraction k=1 ∞ D c k 2 1 converges if c 2 k−1 ≤ 1, c 2 k ± i ≥ 1 , k ≥ 1 , and c 2 k > ε , where ε is any positive number. L. J. Lange [9] proved the convergence of the fraction k=1 ∞ D c k 2 1 for c 2 k−1 ± ia ≤ ρ and c 2 k ± i(a + 1) ≥ ρ , k ≥ 1 , where a ∈C , and a and ρ satisfy the inequality a < ρ < a + 1 . J. Mc Laughlin and N. J. Wyshinski [10] showed that the fraction k=1 ∞ D a k 1 converges if a 2 k−1 ≤ c and a 2 k ≥ 1 + 3c + 2 c 2c + 1 , k ≥ 1 , where c > 0 . The twin convergence regions for continued fractions were also studied, e.g., in the work by L. Lorentzen [12]. The branched continued fractions are multidimensional generalizations of continued fractions. For the branched continued fractions of various structures, the twin convergence regions were studied by T. M. Antonova [1], E. A. Boltarovich [5], V. R. Gladun [6]. The twin convergence regions for two-dimensional continued fractions were described in the works by T. M. Antonova and O. M. Sus' [2] and Kh. Yo. Kuchmins'ka [8].