Finite-order Meromorphic Solutions Research Articles

Meromorphic Solutions of Certain Nonlinear Difference Equations

Let $$p_1, p_2$$ be nonzero rational functions, $$\alpha _1, \alpha _2$$ be nonzero constants, and $$P_d(z, f)$$ be a difference polynomial in f of degree d. We prove that every finite order meromorphic solution of nonlinear difference equations $$f^n+P_d(z, f)=p_1e^{\alpha _1 z}+p_2e^{\alpha _2 z}$$ has few poles provided $$n\ge 2$$ and $$d\le n-1$$ . This result shows there exists some difference between the existence of finite order meromorphic solutions of the above equations and its corresponding differential equations. We also give the forms of finite order meromorphic solutions of the above equations under some conditions on $$\alpha _1/\alpha _2$$ .

Meromorphic solutions of certain nonlinear difference equations

Abstract This paper focuses on finite-order meromorphic solutions of nonlinear difference equation f n ( z ) + q ( z ) e Q ( z ) Δ c f ( z ) = p ( z ) {f}^{n}(z)+q(z){e}^{Q(z)}{\text{Δ}}_{c}f(z)=p(z) , where p , q , Q p,q,Q are polynomials, n ≥ 2 n\ge 2 is an integer, and Δ c f {\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.

Uniqueness of Meromorphic Functions Concerning Their Differences and Solutions of Difference PainlevÉ Equations

This paper is devoted to studying some shared value properties for finite-order meromorphic solutions of the difference Painleve IV equation. We also consider sharing value problems for the derivative of a meromorphic function f(z) with its shift \(f(z+c)\) and difference \(\Delta f\).

ON UNICITY OF MEROMORPHIC SOLUTIONS TO DIFFERENCE EQUATIONS OF MALMQUIST TYPE

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 $e_{1}\neq e_{2}$.

On Finite-Order Meromorphic Solutions of a Class Containing a Difference Painlevé I Equation

In this article, we investigate the growth and value distribution of finite-order transcendental meromorphic solutions for some non-linear difference equations. For instance, we show that demanding the existence of a transcendental meromorphic solution of a large class of difference equations containing difference Painleve I reduces the class either to a smaller one which still contains the difference Painleve I, or to a class of first-order equations. In addition, for certain classes of equations, we prove the existence of rational solutions and give their forms.

Finite-order meromorphic solutions and the discrete Painlevé equations

Let w(z) be an admissible finite-order meromorphic solution of the second-order difference equation w ( z + 1 ) + w ( z − 1 ) = R ( z , w ( z ) ) where R(z, w(z)) is rational in w(z) with coefficients that are meromorphic in z. Then either w(z) satisfies a difference linear or Riccati equation or else the above equation can be transformed to one of a list of canonical difference equations. This list consists of all known difference Painlevé equations of the above form, together with their autonomous versions. This suggests that the existence of finite-order meromorphic solutions is a good detector of integrable difference equations.

Existence of finite-order meromorphic solutions as a detector of integrability in difference equations

The existence of sufficiently many finite-order (in the sense of Nevanlinna) meromorphic solutions of a difference equation appears to be a good indicator of integrability. It is shown that, out of a large class of second-order difference equations, the only equation that can admit a sufficiently general finite-order meromorphic solution is the difference Painlevé II equation. The proof given relies on estimates obtained by arguments related to singularity confinement. The existence of meromorphic solutions of a general class of first-order difference equations is also proven by a simple method based on Banach’s fixed point theorem.