Weakly weighted sharing and unicity of certain shift polynomials

Abstract

In this paper, we discuss the unicity problem of certain shift polynomials. Suppose that c_j (j = 1, …, s) be distinct complex numbers, n, m, s and μ_j (j = 1, …, s) are integers satisfying n + m > 4σ + 14, where σ = μ ₁ + μ ₂ + … μ_s . We prove that if $${p^n}(\gamma ){(p(\gamma ) - 1)^m}\mathop \prod \limits_{j = 1}^s p{(\gamma + {c_j})^{{\mu _j}}}$$ and $${q^n}(\gamma ){(q(\gamma ) - 1)^m}\mathop \prod \limits_{j = 1}^s q{(\gamma + {c_j})^{{\mu _j}}}$$ share ″(α(γ),0)″, then either $$p(\gamma ) \equiv q(\gamma )$$ or $${p^n}{(p - 1)^m}\mathop \prod \limits_{j = 1}^s p{(\gamma + {c_j})^{{\mu _j}}} - {q^n}{(q - 1)^m}\mathop \prod \limits_{j = 1}^s q{(\gamma + {c_j})^{{\mu _j}}}$$. The results obtained greatly improve the results of Saha (Korean J. Math. 28(4)(2020)) and C. Meng (Mathematica Bohemica 139(2014)).