An analog of the Vorpits'kii convergence criterion for branched continued fractions of special form

We study branched continued fractions of a special form with inequivalent variables. We establish a multidimensional analog of the Vorpits'kii convergence criterion for continued fractions. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 35–38.     

Convergence criteria for branched continued fractions with nonnegative components

For branched continued fractions with nonnegative components and a fixed or variable number of branchings we establish necessary and sufficient conditions for their approximants to be well-defined. We study necessary and sufficient conditions for convergence that are multivariable analogs of the known Seidel-Stern and Stern criteria for continued fractions with positive elements. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 7–13.     

On the convergence of branched continued fractions

We give a survey of research on the theory of convergence of branched continued fractions. Translated fromMatematychni Metody ta Fizyko-Mechanichni Polya, Vol. 41, No. 1, 1998, pp. 117–126.     

Approximation of the lauricella hypergeometric functionsF D (N) by branched continued fractions

Applying recursion relations for the Lauricella hypergeometric functions F _D ^Nl , we construct an expansion of a ratio of these functions in branched continued fractions. We study the convergence of the resulting expansion in the case of real parameters. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 70–74.     

On the expansion of some functions in a two-dimensional <Emphasis Type="Italic">g</Emphasis>-fraction with independent variables

We expand some functions in a two-dimensional g -fraction with independent variables and show the efficiency of approximations of the obtained expansion by branched continued fractions.     

On the convergence of the even part of the branched continued fraction expansion of a ratio of Lauricella hypergeometric functionsF D (N)

We study the uniform and absolute convergence of the even part of the branched continued fraction that represents a ratio of Lauricella hypergeometric functions.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 7–9.     

On the convergence of periodic integral continued fractions with variable limits of integration

We establish estimates for the “tails” of periodic integral continued fractions with variable upper limits of integration. We prove a theorem on the uniform and absolute convergence of such fractions, and we obtain an estimate of their rate of convergence. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 28–34     

Ideas from Continued Fraction Theory Extended to Padé Approximation and Generalized Iteration

This is a survey of some basic ideas in the convergence theory for continued fractions, in particular value sets, general convergence and the use of modified approximants to obtain convergence acceleration and analytic continuation. The purpose is to show how these ideas apply to some other areas of mathematics. In particular, we introduce {w _k }-modifications and general convergence for sequences of Padé approximants.     

基于Samelson逆的向量值连分式的收敛准则

The aim of this work is to give some criteria on the convergence of vector valued continued fractions defined by Samelson inverse. We give a new approach to prove the convergence theory of continued fractions. First, by means of the modified classical backward recurrence relation, we obtain a formula between the m-th and n-th convergence of vector valued continued fractions. Second, using this formula, we give necessary and sufficient conditions for the convergence of vector valued continued fractions.     

The element sets and value sets of two-dimensional continued fractions

For two-dimensional continued fractions we prove the existence and uniqueness of an optimal sequence of value sets corresponding to an arbitrarily given sequence of element sets. We compute the element set for a given sequence of disk value sets and as a corollary, give the element sets and value sets that are used in convergence criteria for two-dimensional continued fractions. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 55–61.     

On Koch's convergence criterion for branching continued fractions

Extension of the necessary condition for convergence of branching continued fractions with real positive elements to the case when the remainders are complex numbers leads to a multidimensional analog of Koch's theorem. However the part of the theorem that is known in the literature as the Stern-Stolz convergence criterion does not hold in this case.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 10–13.     

Matrix-valued continued fractions

We discuss the properties of matrix-valued continued fractions based on Samelson inverse. We begin to establish a recurrence relation for the approximants of matrix-valued continued fractions. Using this recurrence relation, we obtain a formula for the difference between mth and nth approximants of matrix-valued continued fractions. Based on this formula, we give some necessary and sufficient conditions for the convergence of matrix-valued continued fractions, and at the same time, we give the estimate of the rate of convergence. This paper shows that some famous results in the scalar case can be generalized to the matrix case, even some of them are exact generalizations of the scalar results.     

On branched continued fractions rational interpolation over pyramid-typed grids

In this paper, three-term recurrence relations for branched continued fractions are determined. Based on the algorithm of partial inverse differences in tensor-product-like manner, the finite branched continued fractions can be applied to rational interpolation over pyramid-typed grids in R ³. By means of the three-term recurrence relations, a characterization theorem is valid. Then an error estimation is worked out. Based on the relationship between the partial inverse differences and partial reciprocal ones, and the partial reciprocal derivatives as well, the BCFs osculatory interpolation with its algorithm is stated which shows it feasibility of partial derivable functions in BCFs expansion at one point.     

Multivariate generalized inverse vector-valued rational interpolants

Bivariate rational interpolating functions of the type introduced in [9, 1] are shown to have a natural extension to the case of rational interpolation of vector-valued quantities using the formalism of Graves-Morris [2]. In this paper, the convergence of Stieltjes-type branched vector-valued continued fractions for two-variable functions are constructed by using the Samelson inverse. Based on them, a kind of bivariate vector-valued rational interpolating function is defined on plane grids. Sufficient conditions for existence, characterisation and uniqueness for the interpolating functions are proved. The results in the paper are illustrated with some examples.     

On one criterion for the figured convergence of two-dimensional continued fractions with complex elements

We establish a new criterion for the figured convergence of two-dimensional continued fractions with complex elements. This criterion represents a generalization of the theorems of simple and twin convergence sets for continued fractions.     

Convergence of matrix continued fractions

The aim of this work is to give some criteria on the convergence of matrix continued fractions. We begin by presenting some new results which generalize the links between the convergent elements of real continued fractions. Secondly, we give necessary and sufficient conditions for the convergence of continued fractions of matrix arguments. This paper will be completed by illustrating the theoretical results with some examples.     

Algebraic interpretation of continued fractions

An alternative (equivalent) definition of continued fractions in terms of a group representation is introduced. With this definition, continued fractions are considered as sequences in a topological group, converging (in some sense) to its boundary. This point of view yields an alternative (equivalent) proof for Lane's convergence theorem for periodic continued fractions.     

Repeated modifications of limitk-periodic continued fractions

Summary The advantages of using modified approximants for continued fractions, can be enhanced by repeating the modification process. IfK(a _n /b _n) is limitk-periodic, a natural choice for the modifying factors is ak-periodic sequence of right or wrong tails of the correspondingk-periodic continued fraction, if it exists. If the modified approximants thus obtained are ordinary approximants of a new limitk-periodic continued fraction, we repeat the process, if possible. Some examples where this process is applied to obtain a convergence acceleration are also given.     

‘Classical’ convergence theorems for generalized continued fractions

In this paper the classical convergence theorems by Śleszyński-Pringsheim, Worpitzky and Van Vleck for ordinary continued fractions will be generalized to continued fractions generalizations (along the lines of the Jacobi–Perron algorithm) with four-term recurrence relations.      

矩阵连分式序列的收敛性

本文研究了矩阵连分式的性质，获得了关于矩阵连分式序列收敛性的一些结果．