In this paper, the author generalizes the notion of diagonal operators on the space \(H_R=H(B(0,R))\) of functions analytic on the disk \(B(0,R)\), \(0< R\leq \infty\). More precisely, let \((\lambda_n)\subset \mathbb C\) and \(R_1,R_2\) be such that \(0<R_1,R_2\leq \infty\); if for every \(f\in H_{R_1}\) such that \(f(z)=\sum_{n=0}^\infty a_nz^n\) there is \(g\in H_{R_2}\) such that \(g(z)=\sum_{n=0}^\infty \lambda_na_nz^n\), then the operator \(D\colon H_{R_1}\to H_{R_2}\) defined by \(Df=g\) is called a diagonal operator and \((\lambda_n)\) is called its associated sequence. After having studied the properties of such diagonal operators, the author considers the problem of spectral synthesis of operators of this kind, and shows some equivalent conditions for it. The author also proves that such a diagonal operator being synthetic implies that the strong operatorial topology closure of the algebra generated by that operator is the set of all diagonal operators as well as implies the existence of sequences of polynomials with a very special behaviour at the eigenvalues of the operator. These results extends the ones done by the same author and S.Seubert in [J. Math.Anal.Appl.334, 1209–1219 (2007; Zbl 1131.47007)].