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

Title & Authors
ON THE ANALOGS OF BERNOULLI AND EULER NUMBERS, RELATED IDENTITIES AND ZETA AND L-FUNCTIONS
Kim, Tae-Kyun; Rim, Seog-Hoon; Simsek, Yilmaz; Kim, Dae-Yeoul;

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