Modification and unification of the Apostol-type numbers and polynomials and their applications

In this paper, we construct generating functions for modification and unification of the Apostol-type polynomials Yn,β(v)(x;k,a,b) of order v. By using these generating functions, we derive many new identities related to the generalized Stirling type numbers of the second kind, array-type polynomials, Eulerian polynomials and the modification and unification of the Apostol-type polynomials and numbers. We give many applications related to these numbers and polynomials and PDEs.