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.