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-Lfunctions; 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 A-partial zeta function and interpolation function.