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 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 On the convergence of multidimensional S-fractions with independent variables

2018 ◽  
Vol 10 (1) ◽  
pp. 58-64
Author(s):  
O.S. Bodnar ◽  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fraction ◽  
Small Region ◽  
Open Disk ◽  
Convergence Criteria ◽  
Efficient Tool ◽  
Independent Variables ◽  
Multiple Power ◽  
Multiple Power Series ◽  
Branched Continued Fraction

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.

scholarly journals Expansions of the Solutions of the General Heun Equation Governed by Two-Term Recurrence Relations for Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
T. A. Ishkhanyan ◽  
T. A. Shahverdyan ◽  
A. M. Ishkhanyan
Keyword(s):  
Power Series ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Recurrence Relations ◽  
Power Series Expansion ◽  
Heun Equation ◽  
Generalized Hypergeometric Function ◽  
Gamma Functions ◽  
Heun Function ◽  
Finite Sum

We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from those applied before. In general, the coefficients of the expansions obey three-term recurrence relations. However, there exist certain choices of the parameters for which the recurrence relations become two-term. The coefficients of the expansions are then explicitly expressed in terms of the gamma functions. Discussing the termination of the presented series, we show that the finite-sum solutions of the general Heun equation in terms of generally irreducible hypergeometric functions have a representation through a single generalized hypergeometric function. Consequently, the power-series expansion of the Heun function for any such case is governed by a two-term recurrence relation.

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 Subordination for multivalent analytic functions associated with Wright generalized hypergeometric function.

2013 ◽  
Vol 44 (1) ◽  
pp. 61-71
Author(s):  
J. Sokol ◽  
N. Sarkar ◽  
P. Goswami ◽  
J. Dziok
Keyword(s):  
Hypergeometric Function ◽  
Analytic Functions ◽  
Function Theory ◽  
Geometric Function Theory ◽  
Generalized Hypergeometric Function ◽  
Geometric Function ◽  
Differential Subordinations

Recently M. K. Aouf and T. M. Seoudy, (2011, {\it Integral Trans. Spec. Func.} {\bf 22}(6) (2011), 423--430) have introduced families of analytic functions associated with the Dziok--Srivastava operator. In this work we use the Dziok--Raina operator to consider classes of multivalent analytic functions. It is connected with Wright generalized hypergeometric function, see J. Dziok and R. K. Raina (2004, {\it Demonstratio Math.}, {\bf 37}(3) 533--542). Moreover, we present a new result which extends some of the earlier results and give other properties of these classes. We have made use of differential subordinations and properties of convolution in geometric function theory.

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.

Subordination and superordination results associated with the generalized hypergeometric function

2014 ◽  
Vol 64 (5) ◽  
Author(s):  
Nanjundan Magesh ◽  
Jagadeesan Jothibasu ◽  
Sami Murthy
Keyword(s):  
Hypergeometric Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Generalized Hypergeometric Function ◽  
Hadamard Products

AbstractIn the present investigation, we obtain some subordination and superordination results involving Hadamard products for certain normalized analytic functions in the open unit disk. Relevant connections of the results, which are presented in this paper, with various other known results are also pointed out.

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