Identities with b-generalized derivations and generalized skew derivations on prime rings

Let [Formula: see text] be a prime ring of characteristic different from [Formula: see text], [Formula: see text] be the right Martindale quotient ring of [Formula: see text], [Formula: see text] the extended centroid of [Formula: see text], [Formula: see text] be a noncentral Lie ideal of [Formula: see text], [Formula: see text] and [Formula: see text] be two nonzero [Formula: see text]-generalized derivations of [Formula: see text]. Suppose there exist fixed integers [Formula: see text] such that [Formula: see text], for all [Formula: see text], then either [Formula: see text] satisfies the standard identity [Formula: see text] or there is [Formula: see text] such that [Formula: see text], [Formula: see text], for any [Formula: see text], and one of the following holds: (a) there exists [Formula: see text], (b) [Formula: see text] and [Formula: see text] are [Formula: see text]-linearly dependent. Then, in the second part of the paper we prove a similar result in the case [Formula: see text] and [Formula: see text] are generalized skew derivations of [Formula: see text] such that [Formula: see text], for all [Formula: see text].