Formulas involving moment and interpolation functions for Apostol type polynomials via Nörlund sum and Laplace transform

The goal of this paper is to define a new approach when constructing generating functions for the generalization and unification of the Apostol type Bernoulli polynomials. We apply the Nörlund sum, the Euler operator for derivative, ad the (inverse) Laplace transform to reach this aim. We also aim to prove a functional equation involving the Nörlund sum and the (inverse) Laplace transform. Moreover, by combining these operators with functional equations of the generating functions, we derive some new formulas for the Apostol type polynomials and kth moment of the geometric distribution. Finally, applying these operators, we not only find new the Riemann integral formulas, but also construct interpolation functions related to the Lerch zeta and the Hurwitz zeta functions for these polynomials.