Approximation of functions of several variables by multidimensional $S$-fractions with independent variables

Dmytryshyn, R. I.; Sharyn, S. V.

doi:10.15330/cmp.13.3.592-607

Cited by 16 publications

(15 citation statements)

References 21 publications

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“…A discussion of some branched continued fraction expansions of the ratios of different generalizations of Gaussian hypergeometric function and the problems associated with them can found [1,2,3,5,6,17,18]. This all is the more remarkable in the view of fact that branched continued fractions are endowed under certain conditions with wide regions of convergence, good convergence rate, and stability of calculations are an effective tool for approximation of certain analytical functions (see [2,4,8,9,10,11,12,13,14]).…”

Section: Introductionmentioning

confidence: 99%

Three- and four-term recurrence relations for Horn's hypergeometric function $H_4$

Dmytryshyn

Lutsiv

2022

Res. Math.

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Three- and four-term recurrence relations for hypergeometric functions of the second order (such as hypergeometric functions of Appell, Horn, etc.) are the starting point for constructing branched continued fraction expansions of the ratios of these functions. These relations are essential for obtaining the simplest structure of branched continued fractions (elements of which are simple polynomials) for approximating the solutions of the systems of partial differential equations, as well as some analytical functions of two variables. In this study, three- and four-term recurrence relations for Horn's hypergeometric function $H_4$ are derived. These relations can be used to construct branched continued fraction expansions for the ratios of this function and they are a generalization of the classical three-term recurrent relations for Gaussian hypergeometric function underlying Gauss' continued fraction.

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Section: Introductionmentioning

confidence: 99%

Three- and four-term recurrence relations for Horn's hypergeometric function $H_4$

Dmytryshyn

Lutsiv

2022

Res. Math.

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“…to a function f (z) holomorphic in Θ u ; (2) The function f (z) is an analytic continuation of the function on the left side of Equation (1) in Equation (32).…”

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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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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“…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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Approximation of functions of several variables by multidimensional $S$-fractions with independent variables

Cited by 16 publications

References 21 publications

Three- and four-term recurrence relations for Horn's hypergeometric function $H_4$

Three- and four-term recurrence relations for Horn's hypergeometric function $H_4$

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

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

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