Characterization of $(delta‎, ‎varepsilon)$-double derivation on rings ‎and ‎algebras

This paper is an attempt to prove the following result:Let $n>1$ be an integer and let $mathcal{R}$ be a $n!$-torsion-free ring with the identity element. Suppose that $d, delta, varepsilon$ are additive mappings satisfyingbegin{equation}d(x^n) = sum^{n}_{j=1}x^{n-j}d(x)x^{j-1}+sum^{n-1}_{j=1}sum^{j}_{i=1}x^{n-1-j}Big(delta(x)x^{j-i}varepsilon(x)+varepsilon(x)x^{j-i}delta(x)Big)x^{i-1}quadend{equation}for all $x in mathcal{R}$. If $delta(e) = varepsilon(e) = 0$, then $d$ is a Jordan $(delta, varepsilon)$-double derivation. In particular, if $mathcal{R}$ is a semiprime algebra and further, $delta(x) varepsilon(x) + varepsilon(x) delta(x) = frac{1}{2}Big[(delta varepsilon + varepsilon delta)(x^2) - (delta varepsilon(x) + varepsilon delta(x))x - x (delta varepsilon(x) + varepsilon delta(x))Big]$ holds for all $x in mathcal{R}$, then $d - frac{delta varepsilon + varepsilon delta}{2}$ is a derivation on $mathcal{R}$.