Value Sharing of Meromorphic Functions and their Differential-Differences

This paper extends the investigation into the uniqueness problems concerning meromorphic functions and their differences or difference polynomials and explores the conditions under which two transcendental meromorphic functions 𝑓 (𝑧) and 𝑔(𝑧), with hyper-order less than one, they share certain values or small functions. For instance, 𝑓 (𝑧) 𝑛 and share common values together with 𝑔(𝑧) 𝑛 and share common values. Moreover, we address scenarios where 𝑓 (𝑧) 𝑛 and 𝑔 (𝑘) (𝑧 + 𝑐) share values 𝐼𝑀 together with 𝑔(𝑧) 𝑛 and 𝑓 (𝑘) (𝑧 + 𝑐) share values 𝐼𝑀, leading to conclusions about the relationship between the functions.