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 Name Affiliation 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
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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