The uniqueness and value distribution of meromorphic functions with different types of differential-difference polynomials sharing a small function IM

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This paper delves into the uniqueness of finite-order meromorphic functions \(f(z)\) and \(g(z)\) over the extended complex plane, particularly when these functions share a small function \(a(z)\) under specific conditions. The study reveals new insights with significant applications, such as classifying different complexes within \(\mathbb{C}\) based on their uniqueness. The primary goal is to explore the uniqueness of meromorphic functions that share a small function \(a(z)\) in the sense of IM (ignoring multiplicities) while constrained by finite order, alongside certain types of differential-difference polynomials. We focus on two non-constant meromorphic functions \(f(z)\) and \(g(z)\) of finite order, under the assumption that a small function \(a(z)\), relative to \(f(z)\), plays a crucial role in the analysis. The investigation centers on the uniqueness properties of a~specific type of differential-difference polynomial of the form \([f^{n}P[f]H(z,f)]\), where \(P[f]\) is a~differential polynomial of \(f(z)\) and \(H(z, f)\) is a difference polynomial of \(f(z)\), both defined in the equations \(\eqref{*}\) and \(\eqref{**}\), respectively. Importantly, these polynomials do not vanish identically and do not share common zeros or poles with either \(f(z)\) or \(g(z)\). The paper establishes conditions on several parameters, including \(k\), \(n\), \(\overline{d}(P)\), \(\Psi\), \(Q\), \(t\), and \(\xi\), under which the shared value properties between \(f(z)\) and \(g(z)\) lead to two possible outcomes: either \(f(z)\) is a constant multiple of \(g(z)\), or \(f(z)\) and \(g(z)\) satisfy a specific algebraic difference equation. This result contributes to a~deeper understanding of the relationship between shared values and the structural properties of meromorphic functions. As an application, the paper extends several previous results on meromorphic functions, including those by Dyavanal and M. M. Mathai, published in the Ukr. Math. J. (2019). Furthermore, by citing a particular example, we demonstrate that the established results hold true only under specific cases, highlighting the precision of the theorem. Finally, we offer a more compact version of the main theorem as an enhancement, providing a~refined perspective on the uniqueness problem in the context of meromorphic functions.