Branched continued fraction representations of ratios of Horn's confluent function $\mathrm{H}_6$

Antonova, T. M.; Dmytryshyn, R. I.; Sharyn, S. V.

doi:10.33205/cma.1243021

Cited by 8 publications

(8 citation statements)

References 26 publications

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“…Therefore, truncation error analysis and the computational stability of the branched continued fraction expansions are other directions. These are interesting and somewhat new directions, and there are not many results here (see [25][26][27][28][29][30]).…”

Section: Discussionmentioning

confidence: 99%

“…Let α 2 and γ 3 be real constants satisfying Equation (31), where u k , k ≥ 1 are defined by Equaton (23) and u is a positive number. Then, Equation ( 24) converges uniformly on every compact subset of Equation ( 32) to a function f (z) holomorphic in Θ u , and f (z) is an analytic continuation of Equation (25) in Equation (32).…”

Section: An Application Of Theorem 2 Followsmentioning

confidence: 99%

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On the Analytic Extension of Lauricella–Saran’s Hypergeometric Function FK to Symmetric Domains

Dmytryshyn,

Goran

2024

Symmetry

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In this paper, we consider the representation and extension of the analytic functions of three variables by special families of functions, namely branched continued fractions. In particular, we establish new symmetric domains of the analytical continuation of Lauricella–Saran’s hypergeometric function FK with certain conditions on real and complex parameters using their branched continued fraction representations. We use a technique that extends the convergence, which is already known for a small domain, to a larger domain to obtain domains of convergence of branched continued fractions and the PC method to prove that they are also domains of analytical continuation. In addition, we discuss some applicable special cases and vital remarks.

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Section: Discussionmentioning

confidence: 99%

Section: An Application Of Theorem 2 Followsmentioning

confidence: 99%

On the Analytic Extension of Lauricella–Saran’s Hypergeometric Function FK to Symmetric Domains

Dmytryshyn,

Goran

2024

Symmetry

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“…Theorem 4. Let a and c be real numbers such that 0 < h k ≤ h for all k ≥ 1, where h k , k ≥ 1, are defined by (5), h is a positive number. Then the following is true.…”

Section: Corollarymentioning

confidence: 99%

Про Аналітичне Розширення Гіпергеометричної Функції Горна $H_4$

Dmytryshyn,

Lutsiv,

Dmytryshyn

2024

Carpathian Math. Publ.

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“…In signal and image processing, continued fractions are used for data compression, approximation and reconstruction of signals and images, and noise detection and filtering [32]. The use of continued fractions and BCF in the theory of functions is especially effective for constructing rational approximations of special functions (see [1][2][3][4][5] and also [11-13, 16, 17, 25]).…”

Section: Introductionmentioning

confidence: 99%

Convergence sets and relative stability to perturbations of a branched continued fraction with positive elements

Hladun,

Bodnar,

Rusyn

2024

Carpathian Math. Publ.

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In the paper, the problems of convergence and relative stability to perturbations of a branched continued fraction with positive elements and a fixed number of branching branches are investigated. The conditions under which the sets of elements \[\Omega_0 = ( {0,\mu _0^{(2)}} ] \times [ {\nu _0^{(1)}, + \infty } ),\quad \Omega _{i(k)}=[ {\mu _k^{(1)},\mu _k^{(2)}} ] \times [ {\nu _k^{(1)},\nu _k^{(2)}} ],\]\[i(k) \in {I_k}, \quad k = 1,2,\ldots,\] where $\nu _0^{(1)}>0,$ $0 < \mu _k^{(1)} < \mu _k^{(2)},$ $0 < \nu _k^{(1)} < \nu _k^{(2)},$ $k = 1,2,\ldots,$ are a sequence of sets of convergence and relative stability to perturbations of the branched continued fraction \[\frac{a_0}{b_0}{\atop+}\sum_{i_1=1}^N\frac{a_{i(1)}}{b_{i(1)}}{\atop+}\sum_{i_2=1}^N\frac{a_{i(2)}}{b_{i(2)}}{\atop+}\ldots{\atop+} \sum_{i_k=1}^N\frac{a_{i(k)}}{b_{i(k)}}{\atop+}\ldots\] have been established. The obtained conditions require the boundedness or convergence of the sequences whose members depend on the values $\mu _k^{(j)},$ $\nu _k^{(j)},$ $j=1,2.$ If the sets of elements of the branched continued fraction are sets ${\Omega _{i(k)}} = ( {0,{\mu _k}} ] \times [ {{\nu _k}, + \infty } )$, $i(k) \in {I_k}$, $k = 0,1,\ldots,$ where ${\mu _k} > 0$, ${\nu _k} > 0$, $k = 0,1,\ldots,$ then the conditions of convergence and stability to perturbations are formulated through the convergence of series whose terms depend on the values $\mu _k,$ $\nu _k.$ The conditions of relative resistance to perturbations of the branched continued fraction are also established if the partial numerators on the even floors of the fraction are perturbed by a shortage and on the odd ones by an excess, i.e. under the condition that the relative errors of the partial numerators alternate in sign. In all cases, we obtained estimates of the relative errors of the approximants that arise as a result of perturbation of the elements of the branched continued fraction.

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Branched continued fraction representations of ratios of Horn's confluent function $\mathrm{H}_6$

Cited by 8 publications

References 26 publications

On the Analytic Extension of Lauricella–Saran’s Hypergeometric Function FK to Symmetric Domains

On the Analytic Extension of Lauricella–Saran’s Hypergeometric Function FK to Symmetric Domains

Про Аналітичне Розширення Гіпергеометричної Функції Горна $H_4$

Convergence sets and relative stability to perturbations of a branched continued fraction with positive elements

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