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 On convergence $(2,1,\dots,1)$-periodic branched continued fraction of the special form

2015 ◽  
Vol 7 (2) ◽  
pp. 148-154
Author(s):  
D.I. Bodnar ◽  
M.M. Bubniak
Keyword(s):  
Error Bounds ◽  
Special Form ◽  
Continued Fraction ◽  
Truncation Error ◽  
Branched Continued Fraction

$(2,1,\dots,1)$-periodic branched continued fraction of the special form is defined. Conditions of convergence are established for 2-periodic continued fraction and $(2,1,\dots,1)$-periodic branched continued fraction of the special form. Truncation error bounds are estimated for these fractions under additional conditions.

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 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.

Error Bounds of Continued Fractions for Complex Transport Coefficients and Spectral Functions

1978 ◽  
Vol 33 (11) ◽  
pp. 1380-1382 ◽  
Author(s):  
P. Hänggi
Keyword(s):  
Error Bounds ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Truncation Error ◽  
Transport Coefficients ◽  
Power Spectra ◽  
A Posteriori ◽  
Spectral Functions ◽  
Dynamical Equilibrium ◽  
Complex Continued Fractions

We study the calculation of complex transport coefficients x (ω) and power spectra in terms of complex continued fractions. In particular, we establish classes of dynamical equilibrium and non-equilibrium systems for which we can obtain a posteriori bounds for the truncation error | x (ω) - x(n)(ω)| ≦ c (ω) | x(n)(ω) - x(n-1)(ω)| when the transport coefficient is approximated by its n-th continued fraction approximant x(n)(ω).

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 On some of convergence domains of multidimensional S-fractions with independent variables

2019 ◽  
Vol 11 (1) ◽  
pp. 54-58 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Continuation Theorem ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Connected Set ◽  
Small Domain ◽  
Partial Quotients ◽  
Circular Domains ◽  
Branched Continued Fraction

The convergence of multidimensional S-fractions with independent variables is investigated using the multidimensional generalization of the classical Worpitzky's criterion of convergence, the criterion of convergence of the branched continued fractions with independent variables, whose partial quotients are of the form $\frac{q_{i(k)}^{i_k}q_{i(k-1)}^{i_k-1}(1-q_{i(k-1)})z_{i(k)}}{1}$, and the convergence continuation theorem to extend the convergence, already known for a small domain (open connected set), to a larger domain. It is shown that the union of the intersections of the parabolic and circular domains is the domain of convergence of the multidimensional S-fraction with independent variables, and that the union of parabolic domains is the domain of convergence of the branched continued fraction with independent variables, reciprocal to it.

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.

Continued Fraction Expansions in Scattering Theory and Statistical Non-Equilibrium Mechanics

1978 ◽  
Vol 33 (4) ◽  
pp. 402-417 ◽  
Author(s):  
P. Hänggi ◽  
F. Rösel ◽  
D. Trautmann
Keyword(s):  
Asymptotic Solution ◽  
Error Bounds ◽  
Scattering Theory ◽  
Continued Fraction ◽  
Recursive Algorithms ◽  
Response Functions ◽  
Practical Application ◽  
Scattering Problems ◽  
Continued Fraction Expansions ◽  
Non Equilibrium

We consider several main aspects of the practical application of continued fraction expansions in scattering problems and in the field of equilibrium and non-equilibrium statistical mechanics. We present some recursive algorithms needed for an efficient evaluation of continued fraction coefficients. The method is then applied to the summation of badly converging series which occur in scattering theory and to the asymptotic solution of the Schrödinger equation. In addition, the use of the method for the calculation of response functions, correlations and their derivatives in systems whose time-dependence is described by a master equation is discussed. Finally, the construction of error bounds is investigated.

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