# ON THE ANALOGS OF BERNOULLI AND EULER NUMBERS, RELATED IDENTITIES AND ZETA AND L-FUNCTIONS

• Kim, Tae-Kyun (DIVISION OF GENERAL EDUCATION-MATHEMATICS KWANGWOON UNIVERSITY) ;
• Rim, Seog-Hoon (DEPARTMENT OF MATHEMATICAL EDUCATION KYUNGPOOK NATIONAL UNIVERSITY) ;
• Simsek, Yilmaz (UNIVERSITY OF AKDENIZ FACULTY OF ART AND SCIENCE DEPARTMENT OF MATHEMATICS) ;
• Kim, Dae-Yeoul (NATIONAL INSTITUTE FOR MATHEMATICAL SCIENCE)
• Published : 2008.03.31
• 165 13

#### Abstract

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.

#### Keywords

Bernoulli numbers and polynomials;zeta functions

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