On the uniqueness of certain type of shift polynomials sharing a small function

Abstract

In this article, we consider the uniqueness problem of the shift polynomials $f^{n}(z)(f^{m}(z)-1)\displaystyle\prod_{j=1}^{s} f(z+c_{j})^{\mu _{j}}$ and $f^{n}(z)(f(z)-1)^{m}\displaystyle\prod_{j=1}^{s} f(z+c_{j})^{\mu _{j}}$, where $f(z)$ is a transcendental entire function of finite order, $c_{j} (j=1, 2, ..., s)$ are distinct finite complex numbers and $n(\geq 1),$ $m(\geq 1),$ $s$ and $\mu _{j} (j=1, 2, ..., s)$ are integers. With the concept of weakly weighted sharing and relaxed weighted sharing we obtain some results which extend and generalize some results due to P. Sahoo [Commun. Math. Stat. 3 (2015), 227-238].