Unicity of transcendental meromorphic functions concerning differential-difference polynomials

Abstract

Let $ f $ and $ g $ be two transcendental meromorphic functions of finite order with a Borel exceptional value $ \infty $, let $ \alpha $ $ (\not\equiv 0) $ be a small function of both $ f $ and $ g $, let $ d, k, n, m $ and $ v_j (j = 1, 2, \cdots, d) $ be positive integers, and let $ c_j (j = 1, 2, \cdots, d) $ be distinct nonzero finite values. If $ n\ge \max \{2k+m+\sigma+5, \sigma+2d+3\} $, where $ \sigma = v_1+v_2+\cdots +v_d $, and $ (f^n(z)(f^m(z)-1)\prod _{j = 1}^{d}f^{v_j}(z+c_j))^{(k)} $ and $ (g^n(z)(g^m(z)-1)\prod _{j = 1}^{d}g^{v_j}(z+c_j))^{(k)} $ share $ \alpha $ CM then $ f \equiv tg $, where $ t^m = t^{n+\sigma } = 1. $ This result extends and improves some restlts due to [1,10,14,15,19].

Keywords

• meromorphic functions
• small functions
• unicity
• differences
• differential