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