In this paper, by using q-deformed bosonic p-adic integral, we give $\lambda$ -Bernoulli numbers and polynomials, we prove Witt's type formula of $\lambda$ -Bernoulli polynomials and Gauss multiplicative formula for $\lambda$ -Bernoulli polynomials. By using derivative operator to the generating functions of $\lambda$ -Bernoulli polynomials and generalized $\lambda$ -Bernoulli numbers, we give Hurwitz type $\lambda$ -zeta functions and Dirichlet's type $\lambda$ -L-functions; which are interpolated $\lambda$ -Bernoulli polynomials and generalized $\lambda$ -Bernoulli numbers, respectively. We give generating function of $\lambda$ -Bernoulli numbers with order r. By using Mellin transforms to their function, we prove relations between multiply zeta function and $\lambda$ -Bernoulli polynomials and ordinary Bernoulli numbers of order r and $\lambda$ -Bernoulli numbers, respectively. We also study on $\lambda$ -Bernoulli numbers and polynomials in the space of locally constant. Moreover, we define $\lambda$ -partial zeta function and interpolation function.