On Properties of Meromorphic Solutions for Certain $q$-Difference Equation
Received:November 23, 2021  Revised:May 07, 2022
Key Words: complex $q$-difference equation   transcendental meromorphic function   order of growth   existence  
Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.12261023; 11861023) and the Foundation of Science and Technology Project of Guizhou Province of China (Grant No.[2018]5769-05).
Author NameAffiliation
Lei TAO School of Mathematical Sciences, Guizhou Normal University, Guizhou 550025, P. R. China
School of Mathematics and Big Data, Guizhou Education University, Guizhou 550018, P. R. China 
Jianren LONG School of Mathematics and Big Data, Guizhou Education University, Guizhou 550018, P. R. China 
Hits: 51
Download times: 70
Abstract:
      Let $q$ be a finite nonzero complex number, let the $q$-difference equation $$f(qz)f(\frac{z}{q})=R(z,f(z))=\frac{P(z,f(z))}{Q(z,f(z))}=\frac{\sum_{j=0}^{\tilde{p}}a_j(z)f^{j}(z)}{\sum_{k=0}^{\tilde{q}}b_k(z)f^{k}(z)}\eqno(\dag)$$ admit a nonconstant meromorphic solution $f,$ where $\tilde{p}$ and $\tilde{q}$ are nonnegative integers, $a_j$ with $0\leq j\leq \tilde{p}$ and $b_k$ with $0\leq k\leq \tilde{q}$ are polynomials in $z$ with $a_{\tilde{p}}\not\equiv 0$ and $b_{\tilde{q}}\not\equiv 0$ such that $P(z, f(z))$ and $Q(z, f(z))$ are relatively prime polynomials in $f(z)$ and let $m=\tilde{p}-\tilde{q}\geq 3$. Then, $(\dag)$ has no transcendental meromorphic solution when $|q|=1$, and the lower bound of the lower order of $f$ is obtained when $m \geq 3$ and $|q|\neq 1$.
Citation:
DOI:10.3770/j.issn:2095-2651.2023.01.009
View Full Text  View/Add Comment  Download reader