Uniqueness of shift and derivatives of meromorphic functions

This paper addresses the uniqueness problem concerning the j-th derivative of a meromorphic function $f(z)$ and the k-th derivative of its shift, $f(z+c),$ where $j,k$ are integers with 0≤ j<k. In this regard, our work surpasses the achievements of [2], as we have improved upon the existing results and provided a more refined understanding of this specific aspect. We give some illustrative examples to enhance the realism of the obtained outcomes. Denote by $E(a,f)$ the set of all zeros of $f-a,$ where each zero with multiplicity m is counted m times. In the paper proved, in particular, the following statement:\\ Let $f(z)$ be a non-constant meromorphic function of finite order, c be a non-zero finite complex number and $j,k$ be integers such that 0≤ j<k. If f⁽ʲ⁾(z) and f⁽ᵏ⁾(z+c) have the same $a-$points for a finite value a(≠ 0) and satisfy conditions E(0,f⁽ʲ⁾(z))⊂ E(0,f⁽ᵏ⁾(z+c)) E(∞,f⁽ᵏ⁾(z+c))⊂ E(∞,f⁽ʲ⁾(z)), then f⁽ʲ⁾(z)≡ f⁽ᵏ⁾(z+c) (Theorem 6).