Certain difference polynomials and shared values
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Let $f$ and $g$ be nonconstant meromorphic (entire, respectively) functions in the complex plane such that $f$ and $g$ are of finite order, let $a$ and $b$ be nonzero complex numbers and let $n$ be a positive integer satisfying $n\geq 21$ ($n\geq 12,$ respectively). We show that if the difference polynomials $f^n(z) + af(z+\eta)$ and $g^n(z) + ag(z+\eta)$ share $b$ CM, and if $f$ and $g$ share 0 and $\infty$ CM, where $\eta\neq 0$ is a complex number, then $f$ and $g$ are either equal or at least closely related. The results in this paper are difference analogues of the corresponding results from \cite{ref4}.
Abstract
Let $f$ and $g$ be nonconstant meromorphic (entire, respectively) functions in the complex plane such that $f$ and $g$ are of finite order, let $a$ and $b$ be nonzero complex numbers and let $n$ be a positive integer satisfying $n\geq 21$ ($n\geq 12,$ respectively). We show that if the difference polynomials $f^n(z) + af(z+\eta)$ and $g^n(z) + ag(z+\eta)$ share $b$ CM, and if $f$ and $g$ share 0 and $\infty$ CM, where $\eta\neq 0$ is a complex number, then $f$ and $g$ are either equal or at least closely related. The results in this paper are difference analogues of the corresponding results from \cite{ref4}.
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