Centralizing identities involving generalized derivations in prime rings

Let ℛ {\mathcal{R}} be a prime ring of characteristic not equal to 2, let 𝒰 {\mathcal{U}} be Utumi quotient ring of ℛ {\mathcal{R}} and let 𝒞 {\mathcal{C}} be the extended centroid of ℛ {\mathcal{R}} . Let Δ be a generalized derivation on ℛ {\mathcal{R}} , and let δ 1 {\delta_{1}} and δ 2 {\delta_{2}} be derivations on ℛ {\mathcal{R}} . Let p ⁢ ( v ) {p(v)} be a multilinear polynomial on ℛ {\mathcal{R}} , which is non-central valued on ℛ {\mathcal{R}} . If δ 1 ⁢ ( Δ 2 ⁢ ( p ⁢ ( v ) ) ⁢ p ⁢ ( v ) ) = δ 2 ⁢ ( Δ ⁢ ( p ⁢ ( v ) 2 ) ) {\delta_{1}(\Delta^{2}(p(v))p(v))=\delta_{2}(\Delta(p(v)^{2}))} for all v ∈ ℛ n {v\in\mathcal{R}^{n}} , then we find the complete description of Δ, δ 1 {\delta_{1}} and δ 2 {\delta_{2}} .