A classification of generalized skew-derivations on multilinear polynomials in prime rings

In this article, we are intended to examine generalized skew-derivations that act as Jordan homoderivations on multilinear polynomials in prime rings. More specifically, we show that if $F$ is generalized skew-derivation of a prime ring $R$ with associated automorphism $\alpha$ such that the relation $$F(X^2)=F(X)^2+F(X)X+XF(X)$$ holds for all $X\in f(R)$, where $f(x_1,\ldots,x_n)$ is a noncentral valued multilinear polynomial over extended centroid $C$, then either $F=0$ or $F=-id_{R}$ or $F=-id_{R}+\alpha$ (where $id_{R}$ denotes the identity map of $R$).