On the approximation of an appell hypergeometric function by a branched continued fraction

1998 ◽  
Vol 90 (5) ◽  
pp. 2376-2380
Author(s):  
O. S. Manzii
Keyword(s):  
Hypergeometric Function ◽  
Continued Fraction ◽  
Branched Continued Fraction

scholarly journals On convergence of branched continued fraction expansions of Horn's hypergeometric function $H_3$ ratios

2021 ◽  
Vol 13 (3) ◽  
pp. 642-650
Author(s):  
T.M. Antonova
Keyword(s):  
Uniform Convergence ◽  
Hypergeometric Function ◽  
Compact Subset ◽  
Complex Variable ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Continued Fraction Expansions ◽  
Branched Continued Fraction

The paper deals with the problem of convergence of the branched continued fractions with two branches of branching which are used to approximate the ratios of Horn's hypergeometric function $H_3(a,b;c;{\bf z})$. The case of real parameters $c\geq a\geq 0,$ $c\geq b\geq 0,$ $c\neq 0,$ and complex variable ${\bf z}=(z_1,z_2)$ is considered. First, it is proved the convergence of the branched continued fraction for ${\bf z}\in G_{\bf h}$, where $G_{\bf h}$ is two-dimensional disk. Using this result, sufficient conditions for the uniform convergence of the above mentioned branched continued fraction on every compact subset of the domain $\displaystyle H=\bigcup_{\varphi\in(-\pi/2,\pi/2)}G_\varphi,$ where \[\begin{split} G_{\varphi}=\big\{{\bf z}\in\mathbb{C}^{2}:&\;{\rm Re}(z_1e^{-i\varphi})<\lambda_1 \cos\varphi,\; |{\rm Re}(z_2e^{-i\varphi})|<\lambda_2 \cos\varphi, \\ &\;|z_k|+{\rm Re}(z_ke^{-2i\varphi})<\nu_k\cos^2\varphi,\;k=1,2;\; \\ &\; |z_1z_2|-{\rm Re}(z_1z_2e^{-2\varphi})<\nu_3\cos^{2}\varphi\big\}, \end{split}\] are established.

scholarly journals About the calculation of hypergeometric function F4 (1,2;2,2; z1, z2 ) by the branched continued fraction of a special kind

2021 ◽  
pp. 86-90
Author(s):  
Volodymyr Hladun ◽  
Nataliya Hoyenko ◽  
Levko Ventyk ◽  
Oleksandra Manziy
Keyword(s):  
Hypergeometric Function ◽  
Continued Fraction ◽  
Hypergeometric Series ◽  
Complex Space ◽  
Special Kind ◽  
Partial Sums ◽  
Two Dimensional ◽  
Explicit Formulas ◽  
Recurrent Relations ◽  
Branched Continued Fraction

In the paper, using some recurrent relations, the expansion of the hypergeometric Appel function F4 (1,2;2,2; z1, z2 ) into a branched continued fraction of special form is constructed. Explicit formulas for the coefficients of constructed development are obtained. The structure of the obtained branched continued fraction is investigated. The values of the suitable fractions and the corresponding partial sums of the hypergeometric series at different points of the two-dimensional complex space are calculated. A comparative analysis of the obtained values is carried out, the results of which confirm the efficiency of using branched continued fractions to calculate the values of the hypergeometric function F4 (1,2;2,2; z1, z2 ) in space C2.

scholarly journals Generalized Hypergeometric Function 3F2 Ratios and Branched Continued Fraction Expansions

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 310
Author(s):  
Tamara Antonova ◽  
Roman Dmytryshyn ◽  
Serhii Sharyn
Keyword(s):  
Hypergeometric Function ◽  
Analytic Functions ◽  
Continued Fraction ◽  
Recurrence Relations ◽  
Classical Problem ◽  
Small Region ◽  
Generalized Hypergeometric Function ◽  
Convergence Criteria ◽  
Continued Fraction Expansions ◽  
Branched Continued Fraction

The paper is related to the classical problem of the rational approximation of analytic functions of one or several variables, particulary the issues that arise in the construction and studying of continued fraction expansions and their multidimensional generalizations—branched continued fraction expansions. We used combinations of three- and four-term recurrence relations of the generalized hypergeometric function 3F2 to construct the branched continued fraction expansions of the ratios of this function. We also used the concept of correspondence and the research method to extend convergence, already known for a small region, to a larger region. As a result, we have established some convergence criteria for the expansions mentioned above. It is proved that the branched continued fraction expansions converges to the functions that are an analytic continuation of the ratios mentioned above in some region. The constructed expansions can approximate the solutions of certain differential equations and analytic functions, which are represented by generalized hypergeometric function 3F2. To illustrate this, we have given a few numerical experiments at the end.

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

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 148
Author(s):  
Tamara Antonova ◽  
Roman Dmytryshyn ◽  
Victoriia Kravtsiv
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Continued Fraction ◽  
Recurrence Relations ◽  
Special Functions ◽  
Convergence Criteria ◽  
Several Variables ◽  
Connected Set ◽  
Continued Fraction Expansions ◽  
Branched Continued Fraction

The paper deals with the problem of construction and investigation of branched continued fraction expansions of special functions of several variables. We give some recurrence relations of Horn hypergeometric functions H3. By these relations the branched continued fraction expansions of Horn’s hypergeometric function H3 ratios have been constructed. We have established some convergence criteria for the above-mentioned branched continued fractions with elements in R2 and C2. In addition, it is proved that the branched continued fraction expansions converges to the functions which are an analytic continuation of the above-mentioned ratios in some domain (here domain is an open connected set). Application for some system of partial differential equations is considered.

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

2018 ◽  
Vol 26 (1) ◽  
pp. 18 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Open Disk ◽  
Efficient Tool ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Multiple Power ◽  
Multiple Power Series ◽  
Branched Continued Fraction

In this paper, we investigate the convergence of multidimensional regular С-fractions with independent variables, which are a multidimensional generalization of regular С-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. We have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional regular С-fraction with independent variables. And, in addition, 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 regular С-fraction with independent variables.

scholarly journals The evaluation of certain continued fractions

1937 ◽  
Vol 30 ◽  
pp. vi-x
Author(s):  
C. G. Darwin
Keyword(s):  
Difference Equation ◽  
Hypergeometric Function ◽  
Difference Equations ◽  
Continued Fractions ◽  
Continued Fraction ◽  
The Difference ◽  
Image Position

1. If the approximate numerical value of e is expressed as a continued fraction the result isand it was in finding the proof that the sequence extends correctly to infinity that the following work was done. First the continued fraction may be simplified by setting down the difference equations for numerator and denominator as usual, and eliminating two out of every successive three equations. A difference equation is thus formed between the first, fourth, seventh, tenth … convergents , and this equation will generate another continued fraction. After a little rearrangement of the first two members it appears that (1) implies2. We therefore consider the continued fractionwhich includes (2), and also certain continued fractions which were discussed by Prof. Turnbull. He evaluated them without solving the difference equations, and it is the purpose here to show how the difference equations may be solved completely both in his cases and in the different problem of (2). It will appear that the work is connected with certain types of hypergeometric function, but I shall not go into this deeply.

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

2020 ◽  
Vol 54 (1) ◽  
pp. 3-14
Author(s):  
R. I. Dmytryshyn ◽  
T. M. Antonova
Keyword(s):  
Complex Plane ◽  
Error Bounds ◽  
Continued Fraction ◽  
Truncation Error ◽  
Independent Variables ◽  
Continued Fraction Expansions ◽  
Branched Continued Fraction

The paper deals with the problem of obtaining error bounds for branched continued fraction of the form $\sum_{i_1=1}^N\frac{a_{i(1)}}{1}{\atop+}\sum_{i_2=1}^{i_1}\frac{a_{i(2)}}{1}{\atop+}\sum_{i_3=1}^{i_2}\frac{a_{i(3)}}{1}{\atop+}\ldots$. By means of fundamental inequalities method the truncation error bounds are obtained for the above mentioned branched continued fraction providing its elements belong to some rectangular sets ofa complex plane. Applications are considered for several classes of branched continued fraction expansions including the multidimensional \emph{S}-, \emph{A}-, \emph{J}-fractions with independent variables.

scholarly journals A New Approach to General Interpolation Formulae for Bivariate Interpolation

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Le Zou ◽  
Shuo Tang
Keyword(s):  
Continued Fraction ◽  
Rational Interpolation ◽  
Interpolation Theorem ◽  
Interpolation Scheme ◽  
Bivariate Interpolation ◽  
New Approach ◽  
Multiple Parameters ◽  
Special Cases ◽  
Block Based ◽  
Branched Continued Fraction

General interpolation formulae for bivariate interpolation are established by introducing multiple parameters, which are extensions and improvements of those studied by Tan and Fang. The general interpolation formulae include general interpolation formulae of symmetric branched continued fraction, general interpolation formulae of univariate and bivariate interpolation, univariate block based blending rational interpolation, bivariate block based blending rational interpolation and their dual schemes, and some interpolation form studied by many scholars in recent years. We discuss the interpolation theorem, algorithms, dual interpolation, and special cases and give many kinds of interpolation scheme. Numerical examples are given to show the effectiveness of the method.

A review of branched continued fraction theory for the construction of multivariate rational approximants

1988 ◽  
Vol 4 (2-4) ◽  
pp. 263-271 ◽  
Author(s):  
Annie A.M. Cuyt ◽  
Brigitte M. Verdonk
Keyword(s):  
Continued Fraction ◽  
Rational Approximants ◽  
Branched Continued Fraction
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