Про Нескінченні Залишки Гіллястого Ланцюгового Дробу Ньордунда Для Гіпергеометричних Функцій Аппеля

Hoyenko, N. P.; Hladun, Volodymyr; Manzij, O.

doi:10.15330/cmp.6.1.11-25

Cited by 11 publications

(3 citation statements)

References 3 publications

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“…Numerous studies show that branched continued fraction expansions provide a useful means for representing and extending of special functions, including generalized hypergeometric functions [3,33], Appell's hypergeometric functions [11,20,25], Horn's hypergeometric functions [2,4,5,6,15], Lauricella-Saran's hypergeometric functions [1,12,24], and also some other functions [10,17,18,29]. To render branched continued fractions more useful in computational, one needs to know more about their numerical stability, which is the main concern of this paper.…”

Section: Introductionmentioning

confidence: 99%

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Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$

Dmytryshyn,

Cesarano,

Lutsiv

et al. 2024

Mat. Stud.

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In this paper, we consider some numerical aspects of branched continued fractions as special families of functions to represent and expand analytical functions of several complex variables, including generalizations of hypergeometric functions. The backward recurrence algorithm is one of the basic tools of computation approximants of branched continued fractions. Like most recursive processes, it is susceptible to error growth. Each cycle of the recursive process not only generates its own rounding errors but also inherits the rounding errors committed in all the previous cycles. On the other hand, in general, branched continued fractions are a non-linear object of study (the sum of two fractional-linear mappings is not always a fractional-linear mapping). In this work, we are dealing with a confluent branched continued fraction, which is a continued fraction in its form. The essential difference here is that the approximants of the continued fraction are the so-called figure approximants of the branched continued fraction. An estimate of the relative rounding error, produced by the backward recurrence algorithm in calculating an nth approximant of the branched continued fraction expansion of Horn’s hypergeometric function H4, is established. The derivation uses the methods of the theory of branched continued fractions, which are essential in developing convergence criteria. The numerical examples illustrate the numerical stability of the backward recurrence algorithm.

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Section: Introductionmentioning

confidence: 99%

“…Numerical aspects related to the backward recurrence algorithm for computing the approximants of continued fractions were considered in [7,9,27,28,30]. Some analogous results concerning branched continued fractions can be found in [19,21,22,25,31,32].…”

Section: Introductionmentioning

confidence: 99%

Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$

Dmytryshyn,

Cesarano,

Lutsiv

et al. 2024

Mat. Stud.

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“…At last, in [18], it is indicated which three-and four-term recurrent relations give similar expansions for the Horn's function H 4 . Some interesting and different branched continued fraction representations of other hypergeometric series can be found in [2,3,14,28,29,31], and some special analytic functions of one or several variables in [19,20,21,22,23,30,32].…”

Section: Introductionmentioning

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 convergence of function F4(1,2;2,2;z1,z2) expansion into a branched continued fraction

Hladun¹,

Hoyenko²,

Manziy³

et al. 2022

Math. Model. Comput.

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In the paper, the possibility of the Appell hypergeometric function F4(1,2;2,2;z1,z2) approximation by a branched continued fraction of a special form is analysed. The correspondence of the constructed branched continued fraction to the Appell hypergeometric function F4 is proved. The convergence of the obtained branched continued fraction in some polycircular domain of two-dimensional complex space is established, and numerical experiments are carried out. The results of the calculations confirmed the efficiency of approximating the Appell hypergeometric function F4(1,2;2,2;z1,z2) by a branched continued fraction of special form and illustrated the hypothesis of the existence of a wider domain of convergence of the obtained expansion.

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Про Нескінченні Залишки Гіллястого Ланцюгового Дробу Ньордунда Для Гіпергеометричних Функцій Аппеля

Cited by 11 publications

References 3 publications

Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$

Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$

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

On convergence of function F4(1,2;2,2;z1,z2) expansion into a branched continued fraction

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