Convergence of some branched continued fractions with independent variables

Abstract

In this paper, we investigate a convergence of associated multidimensional fractions and multidimensional J -fractions with independent variables that are closely related to each other; the coefficients of its partial numerators are positive constants or are non-zero complex constants from parabolic regions. We have established the uniform convergence of the sequences of odd and even approximants of the above mentioned fractions to holomorphic functions on compact subsets of certain domains of C N . And also, we have proved that a condition of convergence for the considered branched continued fractions in certain subsets of C N is the divergence of the series composed of its coefficients. Moreover, we have established that the convergence is uniform to a holomorphic function on all compact subsets of domains of C N , which are interior of the above mentioned subsets.

Branched Continued Fraction Expansions of Horn’s Hypergeometric Function H3 Ratios

“…Central to the theory of branched continued fractions is their problem of convergence. Various methods are used to prove the convergence of branched continued fractions, in particular, methods using the theorem on the continuation of convergence from an already known small domain to a larger [6,30], the value set technique for branched continued fraction [31], the even part of another branched continued fraction [32], the difference formula between its two approximants [33][34][35][36][37], and induction by dimension of a branched continued fraction [36,38,39].…”

“…From (30) and (31) it follows that Q (n) i(k) (z) = 0 for all i(k) ∈ I, 1 ≤ k ≤ n, n ≥ 1, and for all z ∈ D L . Therefore, from (26) for each m > n ≥ 1 and for each z ∈ D L we get…”

Truncation error bounds for branched continued fraction whose partial denominators are equal to unity

“…Let n be an arbitrary integer number, and let n ≥ 2. Using inequalities (11)-(13), (18), for each multiindex i(k) ∈ I, 1 ≤ k ≤ n − 1, and for each index 1 ≤ j ≤ i k we estimate the following value (14) and (15) respectively, for an arbitrary multiindex i(k) ∈ I, 1 ≤ k ≤ n − 1, and an arbitrary index 1 ≤ j ≤ i k we have…”

“…The BCF (1) is said to converge if its sequence of approximants converges, and lim n→∞ f n is called its value. Many methods for proving the convergence of continued fractions and their generalization BCF are methods for proving the existence of limits of sequences of their approximants and, therefore, they do not give truncation error bounds (see, e.g., [3,7,9,15,17]). However, these estimates are important for applying them to the approximation of functions of one or several complex variables.…”

Multidimensional regular C-fraction with independent variables corresponding to formal multiple power series

“…Constructions of the fractions with increase in numbers of variables were significantly complicated. Therefore, two approaches are used for the construction of the branched continued fractions, which corresponds to the FMPS: to overlay additional conditions on the elements of the branched continued fraction [3,6] or to variate constructions of the fraction [1,[7][8][9][10]. We give here a few facts and definitions that are used.…”