Entire and meromorphic solutions of linear difference equations

2016 ◽  
Vol 56 (1) ◽  
pp. 43-59
Author(s):  
Renukadevi S. Dyavanal ◽  
Madhura M. Mathai
Keyword(s):  
Meromorphic Function ◽  
Difference Equation ◽  
Finite Order ◽  
Difference Equations ◽  
Entire Solution ◽  
Linear Difference Equation ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Difference Polynomial

Abstract In this paper, we shall investigate the existence of finite order entire and meromorphic solutions of linear difference equation of the form $$f^n (z) + p(z)f^{n - 2} (z) + L(z,f) = h(z)$$ where L(z, f) is linear difference polynomial in f(z), p(z) is non-zero polynomial and h(z) is a meromorphic function of finite order. We also consider finite order entire solution of linear difference equation of the form $$f^n (z) + p(z)L(z,f) = r(z)e^{q(z)}$$ where r(z) and q(z) are polynomials.

scholarly journals Meromorphic Solutions of Linear Difference Equations with Polynomial Coefficients

2017 ◽  
Vol 59 (1) ◽  
pp. 159-168
Author(s):  
Y. Zhang ◽  
Z. Gao ◽  
H. Zhang
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Necessary Conditions ◽  
Meromorphic Solution ◽  
Linear Difference Equation ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Polynomial Coefficients ◽  
Difference Analogue ◽  
Borel Exceptional Values

AbstractWe study the growth of the transcendental meromorphic solution f(z) of the linear difference equation:where q(z), p0(z), ..., pn-(z) (n ≥ 1) are polynomials such that p0(z)pn(z) ≢ 0, and obtain some necessary conditions guaranteeing that the order of f(z) satisfies σ(f) ≥ 1 using a difference analogue of the Wiman-Valiron theory. Moreover, we give the form of f(z) with two Borel exceptional values when two of p0(z), ..., pn(z) have the maximal degrees.

On the growth analysis of meromorphic solutions of finite ϕ-order of linear difference equations

Analysis ◽  
2020 ◽  
Vol 40 (4) ◽  
pp. 193-202
Author(s):  
Sanjib Kumar Datta ◽  
Nityagopal Biswas
Keyword(s):  
Difference Equation ◽  
Complex Plane ◽  
Difference Equations ◽  
Growth Analysis ◽  
Linear Difference Equation ◽  
Higher Order ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Unbounded Function ◽  
Growth Properties

AbstractIn this paper, we investigate some growth properties of meromorphic solutions of higher-order linear difference equationA_{n}(z)f(z+n)+\dots+A_{1}(z)f(z+1)+A_{0}(z)f(z)=0,where {A_{n}(z),\dots,A_{0}(z)} are meromorphic coefficients of finite φ-order in the complex plane where φ is a non-decreasing unbounded function. We extend some earlier results of Latreuch and Belaidi [Z. Latreuch and B. Belaïdi, Growth and oscillation of meromorphic solutions of linear difference equations, Mat. Vesnik 66 2014, 2, 213–222].

scholarly journals Growth Analysis of Meromorphic Solutions of Linear Difference Equations with Entire or Meromorphic Coefficients of Finite φ-Order

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 267
Author(s):  
Junesang Choi ◽  
Sanjib Kumar Datta ◽  
Nityagopal Biswas
Keyword(s):  
Difference Equation ◽  
Complex Plane ◽  
Difference Equations ◽  
Growth Analysis ◽  
Meromorphic Solution ◽  
Linear Difference Equation ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Unbounded Function ◽  
Growth Properties

Many researchers’ attentions have been attracted to various growth properties of meromorphic solution f (of finite φ-order) of the following higher order linear difference equation Anzfz+n+...+A1zfz+1+A0zfz=0, where Anz,…,A0z are entire or meromorphic coefficients (of finite φ-order) in the complex plane (φ:[0,∞)→(0,∞) is a non-decreasing unbounded function). In this paper, by introducing a constant b (depending on φ) defined by lim̲r→∞logrlogφ(r)=b<∞, and we show how nicely diverse known results for the meromorphic solution f of finite φ-order of the above difference equation can be modified.

scholarly journals Growth of Meromorphic Solutions of Finite Logarithmic Order of Linear Difference Equations

2015 ◽  
Vol 54 (1) ◽  
pp. 5-20
Author(s):  
Benharrat Belaïdi
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Meromorphic Functions ◽  
Linear Difference Equation ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Logarithmic Order ◽  
Oscillation Of Solutions

Abstract In this paper, we deal with the growth and the oscillation of solutions of the linear difference equation an (z) f (z + n) + an-1 (z) f (z + n - 1) + ··· + a1 (z) f (z + 1) + a0 (z) f (z) = 0; where an(z),···, a0(z) are meromorphic functions of finite logarithmic order such that an(z)a0(z) 6≢ 0.

scholarly journals On zeros and growth of solutions of complex difference equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Min-Feng Chen ◽  
Ning Cui
Keyword(s):  
Entire Function ◽  
Difference Equation ◽  
Finite Order ◽  
Difference Equations ◽  
Meromorphic Solutions ◽  
Complex Difference ◽  
Difference Polynomial ◽  
Complex Difference Equations ◽  
The Difference ◽  
Growth Of Solutions

AbstractLet f be an entire function of finite order, let $n\geq 1$ n ≥ 1 , $m\geq 1$ m ≥ 1 , $L(z,f)\not \equiv 0$ L ( z , f ) ≢ 0 be a linear difference polynomial of f with small meromorphic coefficients, and $P_{d}(z,f)\not \equiv 0$ P d ( z , f ) ≢ 0 be a difference polynomial in f of degree $d\leq n-1$ d ≤ n − 1 with small meromorphic coefficients. We consider the growth and zeros of $f^{n}(z)L^{m}(z,f)+P_{d}(z,f)$ f n ( z ) L m ( z , f ) + P d ( z , f ) . And some counterexamples are given to show that Theorem 3.1 proved by I. Laine (J. Math. Anal. Appl. 469:808–826, 2019) is not valid. In addition, we study meromorphic solutions to the difference equation of type $f^{n}(z)+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}$ f n ( z ) + P d ( z , f ) = p 1 e α 1 z + p 2 e α 2 z , where $n\geq 2$ n ≥ 2 , $P_{d}(z,f)\not \equiv 0$ P d ( z , f ) ≢ 0 is a difference polynomial in f of degree $d\leq n-2$ d ≤ n − 2 with small mromorphic coefficients, $p_{i}$ p i , $\alpha _{i}$ α i ($i=1,2$ i = 1 , 2 ) are nonzero constants such that $\alpha _{1}\neq \alpha _{2}$ α 1 ≠ α 2 . Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.

scholarly journals Note on some representations of general solutions to homogeneous linear difference equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin ◽  
Witold Kosmala ◽  
Zdeněk Šmarda
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Difference Equations ◽  
Fibonacci Sequence ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
Homogeneous Linear

Abstract It is known that every solution to the second-order difference equation $x_{n}=x_{n-1}+x_{n-2}=0$ x n = x n − 1 + x n − 2 = 0 , $n\ge 2$ n ≥ 2 , can be written in the following form $x_{n}=x_{0}f_{n-1}+x_{1}f_{n}$ x n = x 0 f n − 1 + x 1 f n , where $f_{n}$ f n is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

scholarly journals Perron-Type Criterion for Linear Difference Equations with Distributed Delay

2007 ◽  
Vol 2007 ◽  
pp. 1-12
Author(s):  
Jehad O. Alzabut ◽  
Thabet Abdeljawad
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Trivial Solution ◽  
Distributed Delay ◽  
Linear Difference Equation ◽  
Asymptotically Stable ◽  
Linear Difference Equations ◽  
Type Criterion

It is shown that if a linear difference equation with distributed delay of the formΔx(n)=∑k=−d0Δkζ(n+1,k−1)x(n+k−1),n≥1, satisfies a Perron condition then its trivial solution is uniformly asymptotically stable.

scholarly journals Some roughness results concerning reducibility for linear difference equations

1988 ◽  
Vol 11 (4) ◽  
pp. 793-804 ◽  
Author(s):  
Garyfalos Papaschinopoulos
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Exponential Dichotomy ◽  
Coefficient Matrix ◽  
Linear Difference Equation ◽  
Almost Periodic ◽  
Linear Difference Equations ◽  
Joint Property

In this paper we prove first that the exponential dichotomy of linear difference equations is “rough”. Moreover we prove that if the coefficient matrix of a linear difference equation is almost periodic, then the Joint property of having an exponential dichotomy with a projectionPand being reducible withPby an almost periodic kinematics similarity is “rough”.

scholarly journals Investigation of linear difference equations with random effects

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mehmet Merdan ◽  
Şeyma Şişman
Keyword(s):  
Difference Equation ◽  
Random Effects ◽  
Difference Equations ◽  
Negative Binomial ◽  
Random Variables ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Poisson Distributions ◽  
Random Behavior ◽  
Expected Values

Abstract In this study, random linear difference equations obtained by transforming the components of deterministic difference equations to random variables are investigated. Uniform, Bernoulli, binomial, negative binomial (or Pascal), geometric, hypergeometric and Poisson distributions have been used for the random effects for obtaining the random behavior of linear difference equations. The random version of the Z-transform, the RZ-transform, has been used to obtain an approximation for the random linear difference equation. Approximate expected values and variances are calculated by using the RZ-transform. The results have been obtained with Maple and are shown in graphs. It is shown that the random Z-transform is an effective tool for the investigation of random linear difference equations.

scholarly journals Meromorphic solutions of higher order non-homogeneous linear difference equations

2021 ◽  
Vol 13(62) (2) ◽  
pp. 433-450
Author(s):  
Benharrat Belaıdi ◽  
Rachid Bellaama
Keyword(s):  
Lower Bound ◽  
Difference Equation ◽  
Lower Order ◽  
Meromorphic Functions ◽  
Linear Difference Equation ◽  
Complex Numbers ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Early Results ◽  
Homogeneous Linear

In this paper, we investigate the growth of meromorphic solutions of nonhomogeneous linear difference equation A_n(z)f(z + c_n) + · · · + A_1(z)f(z + c_1) + A_0(z)f(z) = A_{n+1}(z), where A_{n+1 (z), · · · , A0 (z) are (entire) or meromorphic functions and c_j (1, · · · , n) are non-zero distinct complex numbers. Under some conditions on the (lower) order and the (lower) type of the coefficients, we obtain estimates on the lower bound of the order of meromorphic solutions of the above equation. We extend early results due to Luo and Zheng.

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