Positive definite branched continued fractions of special form

Abstract

Research of the class of branched continued fractions of special form, whose denominators do not equal to zero, is proposed and the connection of such fraction with a certain quadratic form is established. It furnishes new opportunities for the investigation of convergence of branching continued fractions of special form.

Some properties of branched continued fractions of special form

“…Applying lemma 1 and taking into account (3) and (4), we have In [6] the notion of the nth denominator B n (z) of the approximant f n (z), n ≥ 1, of BCF (1) is given. By arguments similar to the proof of the [3,Theorem 4.8], we can show that following theorem holds.…”

“…Several works are devoted to the establishment of different properties of branched continued fractions (BCF) of special form. For example, [1] is dedicated to the investigation of BCF with real positive and complex elements, [4] -to 1-periodic BCF of special form, [2] -to functional BCF with nonequivalent variables and BCF of special form with complex variables, [6] -to positive definite BCF of special form.…”

“…In this paper, using a representation of the approximants of BCF of special form (defined in [6]) through composition one-and two-dimensional fractional-linear maps, we have established that under certain conditions the values of the approximants of the positive definite BCF of special form…”