On the Analogs of Bernoulli and Euler Numbers, Related Identities and Zeta and L-Functions

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