2022/11/19 by Yueyang Zhang, Zhang, Yueyang
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #Primary 30D35 #Secondary 34M10
paper · pdf · doi:10.48550/arxiv.2211.10587
openalex publication_date 2022/11/19 · openalex created_date 2022/11/29 · openalex updated_date 2026/07/31
We present a complete description of the form of transcendental meromorphic solutions of the second order differential equation † w''w-w'2+a w'w+b w2=αw+βw'+γ, where a, b, α, β and γ are all rational functions. Together with the Wiman--Valiron theory, we then show that any transcendental meromorphic solution w of equation (†) has hyper-order ς(w)≤ n for some integer n≥ 0. Moreover, if w has finite order σ(w), then 2σ(w) is a positive integer; if β≡γ≡0 and w has infinite order or if γ\not≡0 and w has infinite order, then the hyper-order ς(w) is a positive integer.