scholarly journals Representation of a quotient of solutions of a four-term linear recurrence relation in the form of a branched continued fraction

2019 ◽  
Vol 11 (1) ◽  
pp. 33-41 ◽  
Author(s):  
I.B. Bilanyk ◽  
D.I. Bodnar ◽  
L. Buyak
Keyword(s):  
Recurrence Relation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Linear Recurrence ◽  
Partial Quotient ◽  
Term Linear ◽  
Recurrent Relations ◽  
Multiple Power ◽  
Linear Recurrence Relation ◽  
Branched Continued Fraction

The quotient of two linearly independent solutions of a four-term linear recurrence relation is represented in the form of a branched continued fraction with two branches of branching by analogous with continued fractions. Formulas of partial numerators and partial denominators of this branched continued fraction are obtained. The solutions of the recurrence relation are canonic numerators and canonic denominators of $\mathcal{B}$-figured approximants. Two types of figured approximants $\mathcal{A}$-figured and $\mathcal{B}$-figured are often used. A $n$th $\mathcal{A}$-figured approximant of the branched continued fraction is obtained by adding a next partial quotient to the $(n-1)$th $\mathcal{A}$-figured approximant. A $n$th $\mathcal{B}$-figured approximant of the branched continued fraction is a branched continued fraction that is a part of it and contains all those elements that have a sum of indexes less than or equal to $n$. $\mathcal{A}$-figured approximants are widely used in proving of formulas of canonical numerators and canonical denominators in a form of a determinant, $\mathcal{B}$-figured approximants are used in solving the problem of corresponding between multiple power series and branched continued fractions. A branched continued fraction of the general form cannot be transformed into a constructed branched continued fraction. For calculating canonical numerators and canonical denominators of a branched continued fraction with $N$ branches of branching, $N>1$, the linear recurrent relations do not hold. $\mathcal{B}$-figured convergence of the constructed fraction in a case when coefficients of the recurrence relation are real positive numbers is investigated.

scholarly journals On the convergence of multidimensional regular C-fractions with independent variables

2018 ◽  
Vol 26 (1) ◽  
pp. 18 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Open Disk ◽  
Efficient Tool ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Multiple Power ◽  
Multiple Power Series ◽  
Branched Continued Fraction

In this paper, we investigate the convergence of multidimensional regular С-fractions with independent variables, which are a multidimensional generalization of regular С-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. We have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional regular С-fraction with independent variables. And, in addition, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional regular С-fraction with independent variables.

scholarly journals Matrices determined by a linear recurrence relation among entries

2003 ◽  
Vol 373 ◽  
pp. 89-99 ◽  
Author(s):  
Gi-Sang Cheon ◽  
Suk-Geun Hwang ◽  
Seog-Hoon Rim ◽  
Seok-Zun Song
Keyword(s):  
Recurrence Relation ◽  
Linear Recurrence ◽  
Linear Recurrence Relation

scholarly journals Linear recurrences in the degree sequences of monomial mappings

2008 ◽  
Vol 28 (5) ◽  
pp. 1369-1375 ◽  
Author(s):  
ERIC BEDFORD ◽  
KYOUNGHEE KIM
Keyword(s):  
Recurrence Relation ◽  
Linear Recurrence ◽  
Degree Sequences ◽  
Integer Matrix ◽  
Linear Recurrences ◽  
Monomial Map ◽  
Linear Recurrence Relation

AbstractLet A be an integer matrix, and let fA be the associated monomial map. We give a connection between the eigenvalues of A and the existence of a linear recurrence relation in the sequence of degrees.

scholarly journals On convergence of branched continued fraction expansions of Horn's hypergeometric function $H_3$ ratios

2021 ◽  
Vol 13 (3) ◽  
pp. 642-650
Author(s):  
T.M. Antonova
Keyword(s):  
Uniform Convergence ◽  
Hypergeometric Function ◽  
Compact Subset ◽  
Complex Variable ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Continued Fraction Expansions ◽  
Branched Continued Fraction

The paper deals with the problem of convergence of the branched continued fractions with two branches of branching which are used to approximate the ratios of Horn's hypergeometric function $H_3(a,b;c;{\bf z})$. The case of real parameters $c\geq a\geq 0,$ $c\geq b\geq 0,$ $c\neq 0,$ and complex variable ${\bf z}=(z_1,z_2)$ is considered. First, it is proved the convergence of the branched continued fraction for ${\bf z}\in G_{\bf h}$, where $G_{\bf h}$ is two-dimensional disk. Using this result, sufficient conditions for the uniform convergence of the above mentioned branched continued fraction on every compact subset of the domain $\displaystyle H=\bigcup_{\varphi\in(-\pi/2,\pi/2)}G_\varphi,$ where \[\begin{split} G_{\varphi}=\big\{{\bf z}\in\mathbb{C}^{2}:&\;{\rm Re}(z_1e^{-i\varphi})<\lambda_1 \cos\varphi,\; |{\rm Re}(z_2e^{-i\varphi})|<\lambda_2 \cos\varphi, \\ &\;|z_k|+{\rm Re}(z_ke^{-2i\varphi})<\nu_k\cos^2\varphi,\;k=1,2;\; \\ &\; |z_1z_2|-{\rm Re}(z_1z_2e^{-2\varphi})<\nu_3\cos^{2}\varphi\big\}, \end{split}\] are established.

scholarly journals About the calculation of hypergeometric function F4 (1,2;2,2; z1, z2 ) by the branched continued fraction of a special kind

2021 ◽  
pp. 86-90
Author(s):  
Volodymyr Hladun ◽  
Nataliya Hoyenko ◽  
Levko Ventyk ◽  
Oleksandra Manziy
Keyword(s):  
Hypergeometric Function ◽  
Continued Fraction ◽  
Hypergeometric Series ◽  
Complex Space ◽  
Special Kind ◽  
Partial Sums ◽  
Two Dimensional ◽  
Explicit Formulas ◽  
Recurrent Relations ◽  
Branched Continued Fraction

In the paper, using some recurrent relations, the expansion of the hypergeometric Appel function F4 (1,2;2,2; z1, z2 ) into a branched continued fraction of special form is constructed. Explicit formulas for the coefficients of constructed development are obtained. The structure of the obtained branched continued fraction is investigated. The values of the suitable fractions and the corresponding partial sums of the hypergeometric series at different points of the two-dimensional complex space are calculated. A comparative analysis of the obtained values is carried out, the results of which confirm the efficiency of using branched continued fractions to calculate the values of the hypergeometric function F4 (1,2;2,2; z1, z2 ) in space C2.

scholarly journals Closed Form Continued Fraction Expansions of Special Quadratic Irrationals

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Daniel Fishman ◽  
Steven J. Miller
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Recurrence Relation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Fibonacci Numbers ◽  
Continued Fraction Expansions

We derive closed form expressions for the continued fractions of powers of certain quadratic surds. Specifically, consider the recurrence relation with , , a positive integer, and (note that gives the Fibonacci numbers). Let . We find simple closed form continued fraction expansions for for any integer by exploiting elementary properties of the recurrence relation and continued fractions.

General anharmonic oscillators

1978 ◽  
Vol 364 (1717) ◽  
pp. 265-275 ◽  
Keyword(s):  
Eigenvalue Problem ◽  
Asymptotic Behaviour ◽  
Recurrence Relation ◽  
Quantum Number ◽  
Anharmonic Oscillator ◽  
Linear Recurrence ◽  
Anharmonic Oscillators ◽  
Anharmonicity Constant ◽  
Linear Recurrence Relation

The eigenvalue problem of the general anharmonic oscillator (Hamiltonian H 2 μ ( k, λ ) = -d 2 / d x 2 + kx 2 + λx 2 μ , ( k, λ ) is investi­gated in this work. Very accurate eigenvalues are obtained in all régimes of the quantum number n and the anharmonicity constant λ . The eigenvalues, as functions of λ , exhibit crossings. The qualitative features of the actual crossing pattern are substantially reproduced in the W. K. B. approximation. Successive moments of any transition between two general anharmonic oscillator eigenstates satisfy exactly a linear recurrence relation. The asymptotic behaviour of this recursion and its consequences are examined.

scholarly journals On constructing a shortest linear recurrence relation

1997 ◽  
Vol 42 (11) ◽  
pp. 1554-1558 ◽  
Author(s):  
M. Kuijper ◽  
J.C. Willems
Keyword(s):  
Recurrence Relation ◽  
Linear Recurrence ◽  
Linear Recurrence Relation

Convolution Identities for Stirling Numbers of the First Kind

Integers ◽  
2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Takashi Agoh ◽  
Karl Dilcher
Keyword(s):  
Recurrence Relation ◽  
Stirling Numbers ◽  
Linear Recurrence ◽  
Linear Recurrence Relation

AbstractWe derive several new convolution identities for the Stirling numbers of the first kind. As a consequence we obtain a new linear recurrence relation which generalizes known relations.

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