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

Dmytryshyn, Roman; Goran, Vitaliy

doi:10.3390/sym16020220

Cited by 4 publications

(6 citation statements)

References 34 publications

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“…Note that various definitions of the convergence of branched continued fractions can be found in [15,17].…”

Section: Branched Continued Fraction and Analytic Continuationmentioning

confidence: 99%

“…where 0 < l < 1, to function f (z) holomorphic in P h,l , and f (z) is an analytic continuation of (2) in the domain (17).…”

Section: Branched Continued Fraction and Analytic Continuationmentioning

confidence: 99%

“…If h k < 0 for all k ≥ 1, then the condition ( 7) holds for all p > 0 and 0 < q < 1. Let K be an arbitrary compact set contained in (17). Then K ⊆ H h,l p,q ⊆ P h,l for some p sufficiently small and q sufficiently close to 1, whose H h,l p,q is the domain (8).…”

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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“…Note that various definitions of the convergence of branched continued fractions can be found in [15,17].…”

Section: Branched Continued Fraction and Analytic Continuationmentioning

confidence: 99%

“…where 0 < l < 1, to function f (z) holomorphic in P h,l , and f (z) is an analytic continuation of (2) in the domain (17).…”

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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“…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%

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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“…. ., is bounded, it follows from the estimate(11) and Proposition 1 that the sets (7) form a sequence of convergence sets of the BCF (1), if the condition (8) is satisfied. The sets(7) are sequence of convergence sets of BCF (1…”

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

Cited by 4 publications

References 34 publications

Про Аналітичне Розширення Гіпергеометричної Функції Горна $H_4$

Про Аналітичне Розширення Гіпергеометричної Функції Горна $H_4$

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

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

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