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Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying $A_{n+1}^{\text{'}}\left(x\right)=(n+1)A_n\left(x\right)$ with $A_0\left(x\right)$ a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, ApostolEuler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.