The zero distribution and uniqueness of difference-differential polynomials (Q2845483)

The study of value distribution of difference-differential polynomials is the main objective of the paper. The results of the paper extend a number of known theorems. The following is a typical result of the paper: Let \(f\) be a transcendental entire function with hyper-order less than \(1\). Suppose that \(P(z)\) is a polynomial of degree \(n\) in \(z\) with constant coefficients and \(t\) is the number of distinct zeros of \(P(z)\). If \(n \geq t(k + 1) + 1\), then \([P(f)f(z + c)]^{(k)} - a(z)\) has infinitely many zeros, where \(c\) (\(\neq 0\)) is a complex number and \(a(z)\) is a small function of \(f\). The authors also prove some related uniqueness theorems.