Uniqueness of differential polynomial of meromorphic function with its q-shift

In the paper, we apply the concept of weighted sharing to study the uniqueness problems of differential polynomial of meromorphic function of zero order with its q-shift. The results of the paper improve and extend some recent results due to H. P. Waghamore and M. M. Manakame [Int. J. Open Problems Compt. Math., 18 (2025), 22-34]. A typical theorem obtained in the paper is as follows: Let P be a polynomial, $f(z)$ be a non-constant meromorphic function of zero-order. Suppose that q is a non-zero complex constant, η ∈ C and n is an integer satisfying n≥ m+3τ+3Ω +6, where m= P, τ =∑ⱼ₌₁ˢμ ⱼ and Ω =∑ⱼ₌₁ˢjμ ⱼ. If fⁿ(z)P(f(z))∏ ⱼ₌₁ˢf⁽ʲ⁾(z)^μ ⱼ and fⁿ(qz+η )P(f(qz+η ))∏ ⱼ₌₁ˢf⁽ʲ⁾(qz+η )^μ ⱼ share $(1,2)$ and (∞,∞), then fⁿ(z)P(f(z))∏ ⱼ₌₁ˢf⁽ʲ⁾(z)^μ ⱼ≡ fⁿ(qz+η )P(f(qz+η ))∏ ⱼ₌₁ˢf⁽ʲ⁾(qz+η )^μ ⱼ. Three other similar theorems are also obtained in the paper.