In this paper, identities related to (σ,τ)\documentclass[12pt]{minimal}
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\begin{document}$$(\sigma , \tau )$$\end{document}-Lie derivations and (σ,τ\documentclass[12pt]{minimal}
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\begin{document}$$(\sigma , \tau $$\end{document})-derivations are considered. Let m,n≥1\documentclass[12pt]{minimal}
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\begin{document}$$m,n \ge 1$$\end{document} be integers and R\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}$$\end{document} be an M!-torsion free unital ring, where M=max{m,n}\documentclass[12pt]{minimal}
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\begin{document}$$M = \max \{m, n\}$$\end{document}. Suppose that σ,τ:R→R\documentclass[12pt]{minimal}
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\begin{document}$$\sigma , \tau : \mathcal {R} \rightarrow \mathcal {R}$$\end{document} are two endomorphisms such that σ(e)\documentclass[12pt]{minimal}
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\begin{document}$$\sigma (\mathbf e )$$\end{document},τ(e)∈Z(R)\documentclass[12pt]{minimal}
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\begin{document}$$\tau (\mathbf e ) \in Z(\mathcal {R})$$\end{document}, where e\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf e $$\end{document} denotes the unit element of R\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}$$\end{document}. If D:R→R\documentclass[12pt]{minimal}
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\begin{document}$$D:\mathcal {R} \rightarrow \mathcal {R}$$\end{document} is an additive map satisfying D[xn,ym]=[D(xn),σ(ym)]+[τ(xn),D(ym)]\documentclass[12pt]{minimal}
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\begin{document}$$D[x^n, y^m] = [D(x^n), \sigma (y^m)] + [\tau (x^n), D(y^m)]$$\end{document} for all x,y∈R\documentclass[12pt]{minimal}
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\begin{document}$$x, y \in \mathcal {R}$$\end{document}, then D is a (σ,τ\documentclass[12pt]{minimal}
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\begin{document}$$(\sigma , \tau $$\end{document})-Lie derivation. Moreover, we offer a characterization of (σ,τ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma , \tau $$\end{document})-derivations from a C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^{*}$$\end{document}-algebra A\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}$$\end{document} into a Banach A\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}$$\end{document}-bimodule M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document} which reads as follows. Let A\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}$$\end{document} be a unital C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^{*}$$\end{document}-algebra, M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document} be a unital Banach A\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}$$\end{document}-bimodule, and let σ,τ:A→A\documentclass[12pt]{minimal}
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\begin{document}$$\sigma , \tau :\mathcal {A} \rightarrow \mathcal {A}$$\end{document} be continuous endomorphisms such that σ(e)=e=τ(e)\documentclass[12pt]{minimal}
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\begin{document}$$\sigma (\mathbf e ) = \mathbf e = \tau (\mathbf e )$$\end{document}, where e\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf e $$\end{document} denotes the unit element of A\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}$$\end{document}. Suppose that n>1\documentclass[12pt]{minimal}
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\begin{document}$$n > 1$$\end{document} is an integer and d:A→M\documentclass[12pt]{minimal}
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\begin{document}$$d:\mathcal {A} \rightarrow \mathcal {M}$$\end{document} is a linear map satisfying d(an)=∑j=1nτ(a)n-jd(a)σ(a)j-1\documentclass[12pt]{minimal}
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\begin{document}$$d(a^{n}) = \sum _{j = 1}^{n}\tau (a)^{n - j}d(a) \sigma (a)^{j - 1}$$\end{document} for all a∈A\documentclass[12pt]{minimal}
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\begin{document}$$a \in \mathcal {A}$$\end{document}. Then, d is a continuous (σ,τ\documentclass[12pt]{minimal}
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\begin{document}$$(\sigma , \tau $$\end{document})-derivation.
