Unicity of Entire Functions Concerning their $q-$ Derivatives-Difference-Polynomials

TL;DR

This paper investigates the uniqueness of transcendental entire functions of zero-order based on their $q$-shifts and derivatives, establishing conditions under which two functions sharing certain small functions must be identical.

Contribution

It proves a new unicity theorem for entire functions involving $q$-shifts and derivatives, extending previous results in complex analysis.

Findings

If two entire functions share two small functions IM, then they are identical.

The theorem applies to functions of zero-order with respect to their $q$-shifts and derivatives.

The result generalizes classical unicity theorems to the setting of $q$-difference and differential polynomials.

Abstract

In this paper, we study the unicity of entire functions concerning their $q-$shifts and $k-$th derivatives and prove: Let $f(z)$ be a transcendental entire function of zero-order, and $g(z)$ define as in (1.1). Let $a(z), b(z)$ be two distinct small functions of $f(z)$. If $f(z)$ and $g(z)$ share $a(z), b(z)$ IM, then $f(z)\equiv g(z)$.