A note on the second kind q -Apostol Bernoulli numbers, polynomials, and Zeta function

In this paper we consider a new type of the q -Apostol Bernoulli numbers and polynomials. Firstly, we define the q -Apostol Bernoulli numbers and polynomials by making use of their generating function. Also, we observe many properties, i.e., the recurrence formula, the difference equation, the differential relation.