On Properties of Meromorphic Solutions for Certain $q$-Difference Equation

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 $|q|\neq 1$.