Uniqueness of meromorphic solutions sharing values with a meromorphic function to w(z+1)w(z−1)=h(z)wm(z) $w(z + 1)w(z - 1) = h(z)w^{m}(z)$

Abstract

Abstract For the nonlinear difference equations of the form w(z+1)w(z−1)=h(z)wm(z), $$ w(z + 1)w(z - 1) = h(z)w^{m}(z), $$ where h(z) $h(z)$ is a nonzero rational function and m=±2,±1,0 $m = \pm 2, \pm 1,0$, we show that its transcendental meromorphic solution is mainly determined by its zeros, 1-value points and poles except for some special cases. Examples for the sharpness of these results are given.

Keywords

• Meromorphic solutions
• Difference Painlevé equation
• Uniqueness