Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, Turkey
Copyright © 2010 Mehmet Açikgöz et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present a new generating function related to the -Bernoulli
numbers and -Bernoulli polynomials. We give a new construction of these
numbers and polynomials related to the second-kind Stirling numbers and
-Bernstein polynomials. We also consider the generalized -Bernoulli polynomials
attached to Dirichlet's character and have their generating function. We obtain distribution relations for the -Bernoulli polynomials and
have some identities involving -Bernoulli numbers and polynomials related
to the second kind Stirling numbers and -Bernstein polynomials. Finally, we
derive the -extensions of zeta functions from the Mellin transformation of this
generating function which interpolates the -Bernoulli polynomials at negative
integers and is associated with -Bernstein polynomials.