Automatic continuity of new generalized derivations

Abstract: Let $\mathcal{A}$ and $\mathcal{B}$ be two algebras and let $n$ be a positive integer. A linear mapping $D:\mathcal{A} \rightarrow \mathcal{B}$ is called a \emph{strongly generalized derivation of order $n$} if there exist families of linear mappings $${E_k:\mathcal{A} \rightarrow \mathcal{B}}<em>{k = 1}<sup>{n}$$, $${F_k:\mathcal{A} \rightarrow \mathcal{B}}</em>{k = 1}<sup>{n}$$, $${G_k:\mathcal{A} \rightarrow \mathcal{B}}<em>{k = 1}<sup>{n}$$ and $${H_k:\mathcal{A} \rightarrow \mathcal{B}}</em>{k = 1}<sup>{n}$$ which satisfy $$D(ab) = \sum_{k = 1}^{{n}\left[E_k(a)} F_k(b) + G_k(a)H_k(b)\right]$$ for all $a, b \in \mathcal{A}$. The purpose of this article is to study the automatic continuity of such derivations on Banach algebras and $C<sup>{\ast}$-algebras.