Unicity on entire function concerning its differential-difference operators

2020/09/17 by Huang, Xiao
Mathematics · #30D35 #39A32 #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2009.08066

openalex publication_date 2020/09/17 · openalex created_date 2020/09/21 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the uniqueness of the differential-difference polynomials of entire functions on ℂⁿ. We prove the following result: Let f(z) be a transcendental entire function on ℂⁿ of hyper-order less than 1 and g(z)=b_-1+∑_i=0ⁿb_if^(k_i)(z+η_i), where b_-1 and b_i (i=0…,n) are small meromorphic functions of f on ℂⁿ, k_i≥0 (i=0…,n) are integers, and η_i (i=0…,n) are finite values. Let a₁(z)\not≡∞, a₂(z)\not≡∞ be two distinct small meromorphic functions of f(z) on ℂⁿ. If f(z) and g(z) share a₁(z) CM, and a₂(z) IM. Then either f(z)≡ g(z) or a₁=2a₂=2, f(z)≡ e^2p-2e^p+2, and g(z)≡ e^p, where p(z) is a non-constant entire function on ℂⁿ. Especially, in the case of g(z)=(Δ_ηⁿf(z))^k, we obtain f(z)≡ (Δ_ηⁿf(z))^k.

Citations

• Meromorphic Functions that Share Three or Four Values
• Nevanlinna theory for the difference operator

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