Some formulas for the generalized Apostol-type polynomials and numbers

In this paper, we perform a further investigation for the unified family of the generalized Apostol-Bernoulli, Euler and Genocchi polynomials and numbers introduced by El-Desouky and Gomaa (2014). By using the generating function methods and summation transform techniques, we establish some new formulas for this family of polynomials and numbers, and give some illustrative special cases.