Bulletin of the
Korean Mathematical Society BKMS

We consider meromorphic solutions of $q$-difference equations of the form $$ \sum_{j=0}^na_j(z)f(q^jz)=a_{n+1}(z), $$ where $a_0(z),\ldots,a_{n+1}(z)$ are meromorphic functions, $a_0(z)a_n(z)\not\equiv 0$ and $q\in \mathbb{C}$ such that $0<|q|\leq 1$. We give a new estimate on the upper bound for the length of the gap in the power series of entire solutions for the case $0<|q|<1$ and $n=2$. Some growth estimates for meromorphic solutions are also given in the cases $0<|q|<1$ and $|q|=1$. Moreover, we investigate zeros and poles of meromorphic solutions for the case $|q|=1$.

Keywords: $q$-difference equation, growth, type

MSC numbers: Primary 30D35, 39B32