In this paper, by using q-deformed bosonic p-adic integral, we give

-Bernoulli numbers and polynomials, we prove Witt`s type formula of

-Bernoulli polynomials and Gauss multiplicative formula for

-Bernoulli polynomials. By using derivative operator to the generating functions of

-Bernoulli polynomials and generalized

-Bernoulli numbers, we give Hurwitz type

-zeta functions and Dirichlet`s type

-L-functions; which are interpolated

-Bernoulli polynomials and generalized

-Bernoulli numbers, respectively. We give generating function of

-Bernoulli numbers with order r. By using Mellin transforms to their function, we prove relations between multiply zeta function and

-Bernoulli polynomials and ordinary Bernoulli numbers of order r and

-Bernoulli numbers, respectively. We also study on

-Bernoulli numbers and polynomials in the space of locally constant. Moreover, we define

-partial zeta function and interpolation function.