Estimation of the Rates of Pointwise and Uniform Convergence of Branched Continued Fractions with Inequivalent Variables

Bodnar, D. І.; Bilanyk, I. B.

doi:10.1007/s10958-022-06062-w

Cited by 7 publications

(8 citation statements)

References 10 publications

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“…Further investigations can be continued in various directions. First, we can extend the domains of convergence of the branched continued fraction expansions with real and complex coefficients in their elements using parabolic domains of convergence [11,12] and angular domains of convergence [9,10]. Other research directions are the truncation errors analysis and the computational stability of the branched continued fraction expansions.…”

Section: Discussionmentioning

confidence: 99%

“…It shows that branched continued fraction (4) converges for all z ∈ Y L , Y L ⊂ H p,q . Hence, by [2, Theorem 3], the convergence of ( 4) is uniform on compact subsets of (9). The proof of (B) is analogous to the proof of Theorem 2 (B) in [2], hence, it is omitted.…”

Section: Branched Continued Fraction and Analytic Continuationmentioning

confidence: 97%

“…Let arg(z 1 ) = ϕ, n be an arbitrary natural number, and z be an arbitrary fixed point in the domain (9). Let us prove that Re(G…”

Section: Branched Continued Fraction and Analytic Continuationmentioning

confidence: 99%

“…1 (z) = 0 for all n ≥ 1 and z ∈ H p,q , i.e. that each approximant ( 12) is a holomorphic function in (9). Now, let K be an arbitrary compact subset of (9), then there exists an open bi-disk…”

Section: Branched Continued Fraction and Analytic Continuationmentioning

confidence: 99%

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Про Аналітичне Розширення Гіпергеометричної Функції Горна $H_4$

Dmytryshyn,

Lutsiv,

Dmytryshyn

2024

Carpathian Math. Publ.

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Section: Discussionmentioning

confidence: 99%

Section: Branched Continued Fraction and Analytic Continuationmentioning

confidence: 97%

“…Let arg(z 1 ) = ϕ, n be an arbitrary natural number, and z be an arbitrary fixed point in the domain (9). Let us prove that Re(G…”

Section: Branched Continued Fraction and Analytic Continuationmentioning

confidence: 99%

Section: Branched Continued Fraction and Analytic Continuationmentioning

confidence: 99%

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Про Аналітичне Розширення Гіпергеометричної Функції Горна $H_4$

Dmytryshyn,

Lutsiv,

Dmytryshyn

2024

Carpathian Math. Publ.

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“…Nevertheless, the problems of improving and developing new methods of researching the convergence of such and similar branched continued fractions are open. Along the way, let us note the recent interesting and promising ideas regarding the study of the convergence of branched continued fractions proposed in papers [9,10,11].…”

Section: Discussionmentioning

confidence: 99%

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

Antonova

Dmytryshyn

Sharyn

2023

Constructive Mathematical Analysis

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In this paper, we derive some branched continued fraction representations for the ratios of the Horn's confluent function $\mathrm{H}_6.$ The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. We establish the estimates of the rate of convergence for the branched continued fraction expansions in some region $\Omega$ (here, region is a domain (open connected set) together with all, part or none of its boundary). It is also proved that the corresponding branched continued fractions uniformly converge to holomorphic functions on every compact subset of some domain $\Theta,$ and that these functions are analytic continuations of the ratios of double confluent hypergeometric series in $\Theta.$ At the end, several numerical experiments are represented to indicate the power and efficiency of branched continued fractions as an approximation tool compared to double confluent hypergeometric series.

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On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function H4(a,b;c,d;z1,z2) Ratios

Antonova

Dmytryshyn

Lutsiv

et al. 2023

Axioms

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The paper deals with the problem of representation of Horn’s hypergeometric functions by branched continued fractions. The formal branched continued fraction expansions for three different Horn’s hypergeometric function H4 ratios are constructed. The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. It is proven that the branched continued fraction, which is an expansion of one of the ratios, uniformly converges to a holomorphic function of two variables on every compact subset of some domain H,H⊂C2, and that this function is an analytic continuation of this ratio in the domain H. The application to the approximation of functions of two variables associated with Horn’s double hypergeometric series H4 is considered, and the expression of solutions of some systems of partial differential equations is indicated.

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Estimation of the Rates of Pointwise and Uniform Convergence of Branched Continued Fractions with Inequivalent Variables

Cited by 7 publications

References 10 publications

Про Аналітичне Розширення Гіпергеометричної Функції Горна $H_4$

Про Аналітичне Розширення Гіпергеометричної Функції Горна $H_4$

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

On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function H4(a,b;c,d;z1,z2) Ratios

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