Analysis of the p-adic q-Volkenborn integrals: An approach to generalized Apostol-type special numbers and polynomials and their applications

The goal of this paper is to construct some families of generalized Apostol-type special numbers and polynomials attached to the Dirichlet character. Using the p-adic q-Volkenborn integrals including the bosonic and the fermionic p-adic integrals on p-adic integers, we give generating functions for these numbers and polynomials. These numbers and polynomials are associated with some well-known special numbers and polynomials such as the Peters polynomials, the Boole polynomials, the generalized Apostol–Bernoulli numbers and polynomials, the generalized Apostol-Euler numbers and polynomials, the generalized Apostol–Daehee numbers and polynomials, the Stirling numbers and the Bernoulli numbers of the first kind. We investigate some properties of these numbers and polynomials with their generating functions. Using these generating functions and their functional equation, we derive some identities and relations including some special numbers and polynomials. Finally, we give p-adic q-Volkenborn integral representations for these numbers and polynomials with combinatorial sums