LetAbe a prime ring with maximal right ring of quotientsQand with extended centroidC. Further, letSbe a set, let αS→Abe a map, letmbe a positive integer and letEiFiSm-1, be maps. We study functional identities of the formfor alls1,s2,⋯,smϵS(where, etc). If Sαis an (m+ l)-free subset ofQ(for example, ifSα=AandAis not algebraic of bounded degreem+1 overC, orSαis a ncentral Lie ideal ofAandAis not algebraic of bounded degreem+ 2 overC), definitive results are obtained.