Abstract: We investigate the uniqueness of finite-order transcendental meromorphic solutions to difference Painlevé Ⅲ equations. We suppose w is a finite-order transcendental meromorphic solution to difference Painlevé Ⅲ equation. If w shares two distinct finite values with another meromorphic function w and they have the same poles (counting multiplicities), we conclude that w ≡ w under certain conditions.

Key words: difference Painlevé Ⅲ equation, transcendental meromorphic solutions, share values