Weighted differential ($q$-tri)dendriform algebras

In this paper, we first introduce a weighted derivation on algebras over an operad $\mathcal{P}$, and prove that for the free $\mathcal{P}$-algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential ($q$-tri)dendriform algebras and study some basic properties of them. Then Novikov-(tri)dendriform algebras are initiated, which can be induced from differential ($q$-tri) dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts.