ON UNICITY OF MEROMORPHIC SOLUTIONS TO DIFFERENCE EQUATIONS OF MALMQUIST TYPE

2015 ◽  
Vol 93 (1) ◽  
pp. 92-98 ◽  
Author(s):  
FENG LÜ ◽  
QI HAN ◽  
WEIRAN LÜ
Keyword(s):  
Uniqueness Theorem ◽  
Finite Order ◽  
Difference Equations ◽  
Complex Numbers ◽  
Finite Complex ◽  
Meromorphic Solutions

In this note, we prove a uniqueness theorem for finite-order meromorphic solutions to a class of difference equations of Malmquist type. Such solutions $f$ are uniquely determined by their poles and the zeros of $f-e_{j}$ (counting multiplicities) for two finite complex numbers $e_{1}\neq e_{2}$.

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 certain nonlinear difference equations

2020 ◽  
Vol 18 (1) ◽  
pp. 1292-1301
Author(s):  
Huifang Liu ◽  
Zhiqiang Mao ◽  
Dan Zheng
Keyword(s):  
Difference Equation ◽  
Finite Order ◽  
Difference Equations ◽  
Nonlinear Difference Equation ◽  
Meromorphic Solutions ◽  
Nonlinear Difference Equations ◽  
Zero Distribution ◽  
Forward Difference

Abstract This paper focuses on finite-order meromorphic solutions of nonlinear difference equation {f}^{n}(z)+q(z){e}^{Q(z)}{\text{Δ}}_{c}f(z)=p(z) , where p,q,Q are polynomials, n\ge 2 is an integer, and {\text{Δ}}_{c}f is the forward difference of f. A relationship between the growth and zero distribution of these solutions is obtained. Using this relationship, we obtain the form of these solutions of the aforementioned equation. Some examples are given to illustrate our results.

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 Study of the growth properties of meromorphic solutions of higher-order linear difference equations

2021 ◽  
Author(s):  
Benharrat Belaïdi ◽  
Rachid Bellaama
Keyword(s):  
Difference Equations ◽  
Meromorphic Functions ◽  
Higher Order ◽  
Complex Numbers ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Order Linear ◽  
Early Results ◽  
Growth Properties ◽  
Homogeneous Linear

AbstractIn this paper, we investigate the growth of meromorphic solutions of homogeneous and non-homogeneous linear difference equations $$\begin{aligned} A_{k}(z)f(z+c_{k})+\cdots +A_{1}(z)f(z+c_{1})+A_{0}(z)f(z)= & {} 0, \\ A_{k}(z)f(z+c_{k})+\cdots +A_{1}(z)f(z+c_{1})+A_{0}(z)f(z)= & {} F, \end{aligned}$$ A k ( z ) f ( z + c k ) + ⋯ + A 1 ( z ) f ( z + c 1 ) + A 0 ( z ) f ( z ) = 0 , A k ( z ) f ( z + c k ) + ⋯ + A 1 ( z ) f ( z + c 1 ) + A 0 ( z ) f ( z ) = F , where $$A_{k}\left( z\right) ,\ldots ,A_{0}\left( z\right) ,$$ A k z , … , A 0 z , $$F\left( z\right) $$ F z are meromorphic functions and $$c_{j}$$ c j $$\left( 1,\ldots ,k\right) $$ 1 , … , k are non-zero distinct complex numbers. Under some conditions on the coefficients, we extend early results due to Zhou and Zheng, Belaïdi and Benkarouba.

scholarly journals Meromorphic solutions of difference equations originated from Schwarzian differential equation

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 2003-2015
Author(s):  
Shuang-Ting Lan ◽  
Zhi-Bo Huang ◽  
Chuang-Xin Chen
Keyword(s):  
Differential Equation ◽  
Positive Integer ◽  
Difference Equation ◽  
Rational Function ◽  
Finite Order ◽  
Difference Equations ◽  
Meromorphic Functions ◽  
Meromorphic Solution ◽  
Meromorphic Solutions ◽  
The Difference

Let f (z) be a meromorphic functions with finite order , R(z) be a nonconstant rational function and k be a positive integer. In this paper, we consider the difference equation originated from Schwarzian differential equation, which is of form [?3f(z)?f(z)- 3/2(?2|(z))2]k = R(z)(?f (z))2k. We investigate the uniqueness of meromorphic solution f of difference Schwarzian equation if f shares three values with any meromrphic function. The exact forms of meromorphic solutions f of difference Schwarzian equation are also presented.

scholarly journals The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles

2019 ◽  
Vol 17 (1) ◽  
pp. 1014-1024
Author(s):  
Hong Yan Xu ◽  
Xiu Min Zheng
Keyword(s):  
Fixed Points ◽  
Difference Equations ◽  
Painlevé Equations ◽  
Meromorphic Solutions ◽  
Painleve Equations ◽  
Zeros And Poles

Abstract The purpose of this manuscript is to study some properties on meromorphic solutions for several types of q-difference equations. Some exponents of convergence of zeros, poles and fixed points related to meromorphic solutions for some q-difference equations are obtained. Our theorems are some extension and improvements to those results given by Qi, Peng, Chen, and Zhang.

Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in C2

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 126
Author(s):  
Hong Li ◽  
Hongyan Xu
Keyword(s):  
Positive Integer ◽  
Finite Order ◽  
Difference Equations ◽  
Functional Equations ◽  
Growth Order ◽  
Entire Solutions ◽  
Partial Differential ◽  
Integer Order ◽  
Significant Difference ◽  
Quadratic Trinomial

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), and the partial differential difference equations f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z12=eg(z),f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z2+∂f(z)∂z1+∂f(z)∂z22=eg(z). We establish some theorems about the forms of the finite order transcendental entire solutions of these functional equations. We also list a series of examples to explain the existence of the finite order transcendental entire solutions of such equations. Meantime, some examples show that there exists a very significant difference with the previous literature on the growth order of the finite order transcendental entire solutions. Our results show that some functional equations can admit the transcendental entire solutions with any positive integer order. These results make a few improvements of the previous theorems given by Xu and Cao, Liu and Yang.

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