Some Identities between the Extended q-Bernstein Polynomials with Weight α and q-Bernoulli Polynomials with Weight (α,β)

Abstract

Using bosonic $p$-adic $q$-integral on ${\mathbb{Z}}_p$, we give some interesting relationships between $q$-Bernoulli numbers with weight ($\alpha$,$\beta$) and $q$-Bernstein polynomials with weight $\alpha$. Also, using $q$-Bernstein polynomials with two variables, we derive some interesting properties associated with $q$-Bernoulli numbers with weight ($\alpha$,$\beta$).