On the convergence of multidimensional regular C-fractions with independent variables

In this paper, we investigate the convergence of multidimensional S-fractions with independent variables, which are a multidimensional generalization of S-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. For establishing the convergence criteria, we use the convergence continuation theorem to extend the convergence, already known for a small region, to a larger region. As a result, we have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional S-fraction with independent variables. And, also, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional S-fraction with independent variables. In addition, we have obtained two new convergence criteria for S-fractions as a consequences from the above mentioned results.

In the paper the correspondence between a formal multiple power series and a special type of branched continued fractions, the so-called ‘multidimensional regular C-fractions with independent variables’ is analysed providing with an algorithm based upon the classical algorithm and that enables us to compute from the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional regular C-fraction with independent variables. A few numerical experiments show, on the one hand, the efficiency of the proposed algorithm and, on the other, the power and feasibility of the method in order to numerically approximate certain multivariable functions from their formal multiple power series.

The paper deals with the problem of approximation of functions of several variables by branched continued fractions. We study the correspondence between formal multiple power series and the so-called "multidimensional $S$-fraction with independent variables". As a result, the necessary and sufficient conditions for the expansion of the formal multiple power series into the corresponding multidimensional $S$-fraction with independent variables have been established. Several numerical experiments show the efficiency, power and feasibility of using the branched continued fractions in order to numerically approximate certain functions of several variables from their formal multiple power series.

We establish the conditions of existence and uniqueness of a multidimensional associated fraction with independent variables corresponding to a given formal multiple power series and deduce explicit relations for the coefficients of this fraction. The relationship between the multidimensional associated fraction and the multidimensional J-fraction with independent variables is demonstrated. The convergence of the multidimensional associated fraction with independent variables is investigated in some domains of the space ℂN. We construct the expansions of some functions into the corresponding two-dimensional associated fractions with independent variables and demonstrate the efficiency of approximation of the obtained expansions by convergents.

In this paper, we consider the problem of convergence of an important type of multidimensional generalization of continued fractions, the branched continued fractions with independent variables. These fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. We have established the effective criterion of absolute convergence of branched continued fractions of the special form in the case when the partial numerators are complex numbers and partial denominators are equal to one. This result is a multidimensional analog of the Worpitzky's criterion for continued fractions. We have investigated the polycircular domain of uniform convergence for multidimensional C-fractions with independent variables in the case of nonnegative coefficients of this fraction.

The two-dimensional [formula omitted]-fraction with independent variables for double power series

In this paper the problem of convergence of the important type of a multidimensional generalization of continued fractions, the branched continued fractions with independent variables, is considered. This fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. When variables are fixed these fractions are called the branched continued fractions of the special form. Their structure is much simpler then the structure of general branched continued fractions. It has given a possibility to establish the necessary and sufficient conditions of convergence of branched continued fractions of the special form with the positive elements. The received result is the multidimensional analog of Seidel's criterion for the continued fractions. The condition of convergence of investigated fractions is the divergence of series, whose elements are continued fractions. Therefore, the sufficient condition of the convergence of this fraction which has been formulated by the divergence of series composed of partial denominators of this fraction, is established. Using the established criterion and Stieltjes-Vitali Theorem the parabolic theorems of branched continued fractions of the special form with complex elements convergence, is investigated. The sufficient conditions gave a possibility to make the condition of convergence of the branched continued fractions of the special form, whose elements lie in parabolic domains.

The paper investigates the convergence problem of a special class of branched continued fractions, i.e. the multidimensional S-fractions with independent variables, consisting of \[\sum_{i_1=1}^N\frac{c_{i(1)}z_{i_1}}{1}{\atop+}\sum_{i_2=1}^{i_1}\frac{c_{i(2)}z_{i_2}}{1}{\atop+} \sum_{i_3=1}^{i_2}\frac{c_{i(3)}z_{i_3}}{1}{\atop+}\cdots,\] which are multidimensional generalizations of S-fractions (Stieltjes fractions). These branched continued fractions are used, in particular, for approximation of the analytic functions of several variables given by multiple power series. For multidimensional S-fractions with independent variables we have established a convergence criterion in the domain \[H=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|\arg(z_k+1)|<\pi,\; 1\le k\le N\right\}\] as well as the estimates of the rate of convergence in the open polydisc \[Q=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|z_k|<1,\;1\le k\le N\right\}\] and in a closure of the domain $Q.$

Let $F(x,y)=I+\hspace{-.3cm}\sum\limits_{\substack{p=1\\m+n=p}}^{\infty}\hspace{-.3cm}A_{m,n}x^my^n$ be a formal power series, where the coefficients $A_{m,n}$ are either all matrices or all scalars. We expand $F(x,y)$ into the formal products $\prod\limits_{\substack{p=1\\m+n=p}}^{\infty}\hspace{-.3cm}(I+G_{m,n}x^m y^n)$, $\prod\limits_{\substack{p=1\\m+n=p}}^{\infty}\hspace{-.3cm}(I-H_{m,n}x^m y^n)^{-1}$, namely the \textit{ power product expansion in two independent variables} and \textit{inverse power product expansion in two independent variables} respectively. By developing new machinery involving the majorizing infinite product, we provide estimates on the domain of absolute convergence of the infinite product via the Taylor series coefficients of $F(x,y)$. This machinery introduces a myriad of "mixed expansions", uncovers various algebraic connections between the $(A_{m,n})$ and the $(G_{m,n})$, and uncovers various algebraic connections between the $(A_{m,n})$ and the $(H_{m,n})$, and leads to the identification of the domain of absolute convergence of the power product and the inverse power product as a Cartesian product of polydiscs. This makes it possible to use the truncated power product expansions $\prod\limits_{\substack{p=1\\m+n=p}}^{P}(1+G_{m,n}x^my^n)$,$\prod\limits_{\substack{p=1\\m+n=p}}^{P}(1-H_{m,n}x^my^n)^{-1}$ as approximations to the analytic function $F(x,y)$. The results are made possible by certain algebraic properties characteristic of the expansions. Moreover, in the case where the coefficients $A_{m,n}$ are scalars, we derive two asymptotic formulas for the $G_{m,n},H_{m,n}$, with $m$ {\it fixed}, associated with the majorizing power series. We also discuss various combinatorial interpretations provided by these power product expansions.

In this paper the regular multidimensional $C$-fraction with independent variables, which is a generalization of regular $C$-fraction, is considered. An algorithm of calculation of the coefficients of the regular multidimensional $C$-fraction with independent variables correspondence to a given formal multiple power series is constructed. Necessary and sufficient conditions of the existence of this algorithm are established. The above mentioned algorithm is a multidimensional generalization of the Rutishauser $qd$-algorithm.

Formal series over a group are studied as an algebraic system with componentwise composition and a partial operation of convolution “”. For right-ordered groups a module of formal power series is introduced and studied; these are formal sums with well-ordered supports. Special attention is paid to systems of formal power series (whose supports are well-ordered with respect to the ascending order) that form an -basis, that is, such that every formal power series can be expanded uniquely in this system. -bases are related to automorphisms of the module of formal series that have natural properties of monotonicity and -linearity. The relations and are also studied. Note that in the case of a totally ordered group the system of formal power series forms a skew field with valuation (Mal'tsev-Neumann, 1948-1949.).

Mathematical-thermodynamic solubility model developed by the application of discrete Volterra functional series theory

The paper deals with research of convergence for one of the generalizations of continued fractions -- branched continued fractions of the special form with two branches. Such branched continued fractions, similarly as the two-dimensional continued fractions and the branched continued fractions with two independent variables are connected with the problem of the correspondence between a formal double power series and a sequence of the rational approximants of a function of two variables. Unlike continued fractions, approximants of which are constructed unambiguously, there are many ways to construct approximants of branched continued fractions of the general and the special form. The paper examines the ordinary approximants and one of the structures of figured approximants of the studied branched continued fractions, which is connected with the problem of correspondence. We consider some properties of approximants of such fractions, whose partial numerators are positive and alternating-sign and partial denominators are equal to one. Some necessary and sufficient conditions for figured convergence are established. It is proved that under these conditions from the convergence of the sequence of figured approximants it follows the convergence of the sequence of ordinary approximants to the same limit.

An algorithm for the expansion of a given formal double power series in the associated branched continued fraction with two independent variables is constructed and the conditions for the existence of this expansion are established.

Formal Laurent-Puisieux series (LPS) of the form [EQUATION] are important in calculus and complex analysis. In some Computer Algebra Systems (CASs) it is possible to define an LPS by direct or recursive definition of its coefficients. Since some operations cannot be directly supported within the LPS domain, some systems generally convert LPS to finite truncated LPS for operations such as addition, multiplication, division, inversion and formal substitution. This results in a substantial loss of information. Since a goal of Computer Algebra is --- in contrast to numerical programming --- to work with formal objects and preserve such symbolic information, CAS should be able to use LPS when possible.There is a one-to-one correspondence between formal power series with positive radius of convergence and corresponding analytic functions. It should be possible to automate conversion between these forms. Among CASs only MACSYMA [5] provides a procedure powerseries to calculate LPS from analytic expressions in certain special cases, but this is rather limited.In [2]-[4] we gave an algorithmic approach for computing an LPS for a very rich family of functions. It covers e.g. a high percentage of the power series that are listed in the special series dictionary [1]. The algorithm has been implemented by the author and A. Rennoch in the CAS MATHEMATICA [7], and by D. Gruntz in MAPLE [6].In this note we present some example results of our MATHEMATICA implementation which give insight in the underlying algorithmic procedure.