The value distribution and uniqueness of meromorphic functions concerning certain types of differential-difference polynomials

In this paper, we study the distribution of zeros and the uniqueness of certain type of differential-difference polynomial of the form (fn(z)P[f](z)H(z,f))(k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(f<^>n(z)P[f](z)H(z,f))<^>{(k)}$$\end{document} sharing a small function with the notion of weakly weight and relaxed weight, where f be transcendental meromorphic function of finite order, P[f] is a differential polynomial of f defined as (1) and H(z, f) is a difference polynomial of f defined as (2). We introduce a constructor U(z, w) to show the number of omitted values of P[f]. As some applications, we extend many previous results of entire functions due to Banerjee and Majumder (Anal Math, 43:415-444, 2017) and Majumder and Saha (Ukrainian Math J, 73:791-810, 2021) to the case of meromorphic functions.