Let \(R\) be a prime ring. By a derivation of \(R\), we mean a map \(\delta\colon R\to R\) such that \(\delta(x+y)=\delta(x)+\delta(y)\) and \(\delta(xy)=\delta(x)y+x\delta(y)\) for all \(x,y\in R\). A map \(G\colon R\to R\) is called a generalized derivation if there exists a derivation \(\delta\colon R\to R\) such that \(G(x+y)=G(x)+G(y)\) and \(G(xy)=G(x)y+x\delta(y)\) for all \(x,y\in R\). The derivation \(\delta\) is called the associated derivation of \(G\) and is uniquely determined by the generalized derivation \(G\).
In this paper the authors study a polynomial identity with a pair of generalized derivations. The following theorem is obtained. Let \(R\) be prime ring that is not commutative and such that \(R\ncong M_2(\text{GF}(2))\), \(Q\) be the symmetric Martindale quotient ring of \(R\) and \(C\) be its extended centroid. Let \(D,G\) be two generalized derivations of \(R\), and let \(m,n\) be two fixed positive integers. Then \(D(x^m)x^n=x^nG(x^m)\) for all \(x\in R\) if and only if the following two conditions hold: (i) There exists \(w\in Q\) such that \(D(x)=xw\) and \(G(x)=wx\) for all \(x\in R\); (2) either \(w\in C\), or \(x^m\) and \(x^n\) are \(C\)-dependent for all \(x\in R\).