On Meromorphic Solutions to a Difference Equation of Tumura-Clunie Type

2025/07/03 by Long, Jianren, Xiang, Xuxu
#30D35 #39A45 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.02572

Abstract

The meromorphic solutions f with ρ₂(f)<1 of the non-linear difference equation fⁿ(z)+P_d(z,f)=p₁e^λ₁z+p₂e^λ₂z+p₃e^λ₃z, are characterized in terms of exponential functions using Nevanlinna theory, under certain conditions on λ_j for j=1,2,3. Here, n>2, P_d(z,f) is a difference polynomial in f of degree ≤ n-1, and λ_j,~p_j\not=0 for ~j=1,2,3. These results improve upon those previously obtained by Chen et al.[Bull. Korean Math. Soc. 61, 745-762 (2024)]. Some examples are provided to illustrate these results. Additionally, if P_d(z,f) is a differential-difference polynomial, then under the supplementary condition N(r,f)=S(r,f), by applying the same proof method, these conclusions still hold.

Citations

• Growth of meromorphic solutions of delay differential equations
• Holomorphic curves with shift-invariant hyperplane preimages
• On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane
• Nevanlinna theory for the difference operator
• Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations
• On the transcendental solutions of a certain type of nonlinear differential equations

Related

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• Tumura-Clunie Differential Equations with Applications to Linear ODE's
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• Solutions of Fermat-type partial differential-difference equations in ℂⁿ