Meromorphic solutions of some $q$-difference equations
Bull. Korean Math. Soc. 2011 Vol. 48, No. 6, 1303-1314
https://doi.org/10.4134/BKMS.2011.48.6.1303
Published online November 1, 2011
BaoQin Chen and ZongXuan Chen
South China Normal University, South China Normal University
Abstract : 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
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