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The purpose of this paper is to introduce and investigate two new classes of generalized Bernoulli and Apostol-Bernoulli polynomials based on the definition given recently by the authors  . In particular, we obtain a new addition formula for the new class of the generalized Bernoulli polynomials. We also give an extension and some analogues of the Srivastava-Pintér addition theorem  for both classes. Finally, by making use of the new addition formula, we exhibit several interestingdoi:10.5831/hmj.2012.34.3.311 fatcat:6kplkbhhnjhjveea2g3r4n6b6m