Let $\mathcal{A}$ be a $(p+q)!$-torsion free semiprime ring. We proved that if $\mathcal{H},\mathcal{D}:\mathcal{A}\to \mathcal{A}$ are two additive mappings fulfilling the algebraic identity $2\mathcal{H}(a^{p+q})=\mathcal{H}(a^p)a^q+a^p\mathcal{D}(a^q)+\mathcal{H}(a^q)a^p+a^q\mathcal{D}(a^p)$ for all $a\in \mathcal{A}$, then $\mathcal{H}$ is a generalized derivation with $\mathcal{D}$ as an associated derivation on $\mathcal{A}$. In addition to that, it is also proved in this article that ${\mathcal{H}}_1$ is a generalized left derivation associated with a left derivation $\delta$ on $\mathcal{A}$ if they fulfilled the algebraic identity $2{\mathcal{H}}_1(a^{p+q})=a^p{\mathcal{H}}_1(a^q)+a^q\delta (a^p)+a^q{\mathcal{H}}_1(a^p)+a^p\delta (a^q)$ for all $a\in \mathcal{A}$. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented.

The goal of this study is to bring out the following conclusion. Let $R$ be a noncommutative prime ring with $2(m+n)!$ torsion freeness and let $m$ and $n$ be fixed, non-negative integers and $d,g$ be Jordan derivations on $R$. If $x^{m+n}d(x)+x^{m}g(x)x^n\in Z(R)$ or $d(x)x^{m+n}+x^{m}g(x)x^n\in Z(R)$ or $x^{n}d(x)x^m+x^{m}g(x)x^n\in Z(R)$ then $d=g=0$ follows for every $x\in R$.