Parametric kinds of generalized Apostol-Bernoulli polynomials and their properties
The purpose of this paper is to define generalized Apostol--Bernoulli polynomials with including a new cosine and sine parametric type of generating function using the quasi-monomiality properties and trigonometric functions. In this study, the Apostol-Bernoulli polynomials with three variable are defined with two new generating functions cosine and sine parameters. Then, we investigate multiplicative and derivative operators, diffrential equations, some summation formulas and partial differential equations for these polynomials. Moreover, we introduce Gould--Hopper--Apostol--Bernoulli type polynomials, Hermite--Appell--Apostol--Bernoulli type polynomials and truncated exponential Apostol--Bernoulli type polynomials. Finally, the special cases of these new polynomials are investigated, and the corresponding results are expressed.