  Unifying various generalizations of the important notions of Reynolds
operators, the relative cocycle weighted Reynolds operators are studied. Here
cocycle weighted means the weight of the operators is given by a 2-cocycle
rather than by a scaler as in the classical case. We show that the operators
and 2-cocycles uniquely determine each other. We further give a
characterization of relative cocycle weighted Reynolds operators in the context
of pre-Lie algebras. Using a method of Liu, we construct an explicit graded Lie
algebra whose Maurer-Cartan elements are given by a relative cocycle weighted
Reynolds operator. This allows us to construct the cohomology for a relative
cocycle weighted Reynolds operator. This cohomology can also be seen as the
cohomology of a certain pre-Lie algebra with coefficients in a suitable
representation. Then we consider formal deformations of relative cocycle
weighted Reynolds operators from cohomological points of view. Finally, we
introduce the notation of NS-pre-Lie algebras and show NS-pre-Lie algebras
naturally induce pre-Lie algebras and $L$-dendriform algebras.

Rings and Algebras;K-Theory and Homology