Let
R be a prime ring of characteristic different from 2,
Q r be its right Martindale quotient ring and
C be its extended centroid. Suppose that
F,
G are generalized skew derivations of
R and
\({f(x_1, \ldots, x_n)}\) is a non-central multilinear polynomial over
C with
n non-commuting variables. If
F and
G satisfy the following condition:
$$F(f(r_1,\ldots, r_n))f(r_1, \ldots,r_n)-f(r_1,\ldots,r_n)G(f(r_1,\ldots, r_n))\in C$$
for all
\({r_1, \ldots, r_n \in R}\), then we describe all possible forms of
F and
G.