Some further q-shift difference results on Hayman conjecture

TL;DR

This paper extends classical results on the zeros of differential polynomials to the setting of q-shift difference polynomials for meromorphic functions, and explores their uniqueness when sharing polynomial values.

Contribution

It introduces new zero distribution results for q-shift difference-differential polynomials and addresses their uniqueness when sharing polynomial values, generalizing Hayman's classical work.

Findings

Extended zero distribution results to q-shift difference-differential polynomials.

Established conditions for uniqueness when sharing polynomial values.

Generalized classical differential polynomial results to q-shift difference context.

Abstract

In this paper, we investigate the zero distributions of $q$-shift difference-differential polynomials of meromorphic functions with zero-order that extends and generalizes the classical Hayman results of the zeros of differential polynomials to q-shift difference. We also investigate the uniqueness problem of $q$-shift difference-differential polynomials sharing a polynomial value with finite weight.