On generalized Jordan 1-derivation in rings

Let n ? 1 be a fixed integer and let R be an (n + 1)!-torsion free 1-ring with identity element e. If F, d:R → R are two additive mappings satisfying F(xⁿ⁺¹) = F(x)(x¹)ⁿ + xd(x)(x¹)^n?1 + x²d(x)(x¹)^n?2+ ? +xⁿd(x) for all x ∈ R, then d is a Jordan 1-derivation and F is a generalized Jordan 1-derivation on R.