The theory of composition operators, multiplication operators and weighted composition operators on different spaces of functions have been the subject matter of study for the last several decades. In recent years, many authors studied the products of these operators with differentiation operators on various analytic function spaces. Recently, researchers have initiated the study of the following sum operator, which unifies the theory of products of all these operators: TkΨ,ϕf =k∑j=0ψj· f(j)◦ϕ =k∑j=0D jψj,ϕ f, f ∈ H (D), where H (D) denotes the space of analytic functions on the unit disc D, S (D) denotes the set of all analytic self maps of D, Ψ = (ψj)kj=0be such that ψj ∈ H (D) andϕ ∈ S (D). In this thesis, we characterize the boundedness and the compactness of the op erators T kΨ,ϕ from weighted Bergman spaces Av,p to weighted-type spaces H ∞v(H0v)and weighted Zygmund-type spaces Zw (weighted Bloch-type spaces Bw). Besides characterizing boundedness, completely continuous and compactness of the operator TkΨ,ϕbetween H∞v(H0v) and H∞w (H0w), we obtain essential norm estimates of TkΨ,ϕ. Also,we obtain the boundedness and compactness of the difference TkΨ −TkΦof differentialoperators between H∞v(H0v) and H∞w (H0w). As applications of TkΨ,ϕ, besides obtaining the boundedness, compactness and es sential norm estimates of the weighted differentiation composition operators Dnψ,ϕ:Bv(B0v) → Bw(B0w), Bv(B0v) → H∞w (H0w), H∞v(H0v) → Bw(B0 w), we give new char acterizations and new essential norm estimates of these operators Dnψ,ϕ. Besides, giving examples of bounded, unbounded, compact and non-compact op erators T kΨ,ϕ, we give examples of unbounded weighted differentiation composition operators Dnψ,ϕ: Av,p → Zw(Bw) such that their sum is bounded and also unbounded (non-compact) differential operators TkΨ: H∞v → H∞ w such that their difference is bounded (compact).