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

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.

Uniqueness for meromorphic solutions of Schwarzian differential equation

<abstract><p>Let $ f $ be a meromorphic function, $ R $ be a nonconstant rational function and $ k $ be a positive integer. In this paper, we consider the Schwarzian differential equation of the form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \left[\frac{f'''}{f'}-\frac{3}{2}\left(\frac{f''}{f'}\right)^{2}\right]^{k} = R(z). \end{align*} $\end{document} </tex-math></disp-formula></p> <p>We investigate the uniqueness of meromorphic solutions of the above Schwarzian differential equation if the meromorphic solution $ f $ shares three values with any other meromorphic function.</p></abstract>

Unicity of Meromorphic Solutions of the Pielou Logistic Equation

This paper mainly considers the unicity of meromorphic solutions of the Pielou logistic equation yz+1=Rzyz/Qz+Pzyz, where Pz,Qz, and Rz are nonzero polynomials. It shows that the finite order transcendental meromorphic solution of the Pielou logistic equation is mainly determined by its poles and 1-value points. Examples are given for the sharpness of our result.

Further Results about a Special Fermat-Type Difference Equation

In this paper, we prove the difference equation Fz3+ΔcFz+c3=1 does not have meromorphic solution of finite order over the complex plane C. We also discuss an application to the unique range set problem.

Meromorphic solutions of difference equations originated from Schwarzian differential equation

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.

Existence of Meromorphic Solutions of First-Order Difference Equations

It is shown that if $$\begin{aligned} f(z+1)^n=R(z,f), \end{aligned}$$f(z+1)n=R(z,f),where R(z, f) is rational in f with meromorphic coefficients and $$\deg _f(R(z,f))=n$$degf(R(z,f))=n, has an admissible meromorphic solution, then either f satisfies a difference linear or Riccati equation with meromorphic coefficients, or the equation above can be transformed into one in a list of ten equations with certain meromorphic or algebroid coefficients. In particular, if $$f(z+1)^n=R(z,f)$$f(z+1)n=R(z,f), where the assumption $$\deg _f(R(z,f))=n$$degf(R(z,f))=n has been discarded, has rational coefficients and a transcendental meromorphic solution f of hyper-order $$<1$$<1, then either f satisfies a difference linear or Riccati equation with rational coefficients, or the equation above can be transformed into one in a list of five equations which consists of four difference Fermat equations and one equation which is a special case of the symmetric QRT map. Solutions to all of these equations are presented in terms of Weierstrass or Jacobi elliptic functions, or in terms of meromorphic functions that are solutions to a difference Riccati equation. This provides a natural difference analogue of Steinmetz’ generalization of Malmquist’s theorem.