Incomplete $q$-Chebyshev Polynomials

Abstract

In this paper, we get the generating functions of $q$-Chebyshev polynomials using  $\eta _{z}$ operator, which is $\eta _{z}\left( f(z)\right)=f(qz)$ for any given function $f\left( z\right) $. Also considering explicit formulas of $q$-Chebyshev polynomials, we give new generalizations of $q$-Chebyshev polynomials called incomplete $q$-Chebyshev polynomials of the first and second kind. We obtain recurrence relations and several properties of these polynomials. We show that there are connections between incomplete $q$-Chebyshev polynomials and the some well-known polynomials.