A New Approach to <inline-formula><graphic file="1687-1847-2010-951764-i1.gif"/></inline-formula>-Bernoulli Numbers and <inline-formula><graphic file="1687-1847-2010-951764-i2.gif"/></inline-formula>-Bernoulli Polynomials Related to <inline-formula><graphic file="1687-1847-2010-951764-i3.gif"/></inline-formula>-Bernstein Polynomials

Abstract

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.