q -Bernstein Polynomials Associated with q -Stirling Numbers and Carlitz's q -Bernoulli Numbers

Abstract

Recently, Kim (2011) introduced $q$-Bernstein polynomials which are different $q$-Bernstein polynomials of Phillips (1997). In this paper, we give a $p$-adic $q$-integral representation for $q$-Bernstein type polynomials and investigate some interesting identities of $q$-Bernstein type polynomials associated with $q$-extensions of the binomial distribution, $q$-Stirling numbers, and Carlitz's $q$-Bernoulli numbers.