  By applying the p-adic q-Volkenborn Integrals including the bosonic and the
fermionic p-adic integrals on p-adic integers, we define generating functions,
attached to the Dirichlet character, for the generalized Apostol-Bernoulli
numbers and polynomials, the generalized Apostol-Euler numbers and polynomials,
generalized Apostol-Daehee numbers and polynomials, and also generalized
Apostol-Changhee numbers and polynomials. We investigate some properties of
these numbers and polynomials with their generating functions. By using these
generating functions and their functional equation, we give some identities and
relations including the generalized Apostol-Daehee and Apostol-Changhee numbers
and polynomials, the Stirling numbers, the Bernoulli numbers of the second
kind, Frobenious-Euler polynomials, the generalized Bernoulli numbers and the
generalized Euler numbers and the Frobenious-Euler polynomials. By using the
bosonic and the fermionic p-adic integrals, we derive integral represantations
for the generalized Apostol-type Daehee numbers and the generalized
Apostol-type Changhee numbers.
