δ-Jordan derivations of prime rings

Let [Formula: see text] be a prime ring of characteristic different from [Formula: see text], [Formula: see text] its Utumi quotient ring, [Formula: see text] its extended centroid, [Formula: see text] a noncentral multilinear polynomial over [Formula: see text], [Formula: see text] and [Formula: see text] two generalized derivations of [Formula: see text] such that [Formula: see text]. We describe all possible forms of [Formula: see text] and [Formula: see text] in the case [Formula: see text] for all [Formula: see text]. In particular we prove that [Formula: see text] must be a derivation of [Formula: see text] unless when one of the following cases occurs: (1) there exist [Formula: see text] such that [Formula: see text], [Formula: see text], for any [Formula: see text], with [Formula: see text]; (2) [Formula: see text] is central valued on [Formula: see text] and there exist [Formula: see text] and [Formula: see text] such that [Formula: see text], [Formula: see text] for all [Formula: see text], with [Formula: see text].