An Extended Generalized q -Extensions for the Apostol Type Polynomials

Abstract

Through a modification on the parameters associated with generating function of the $q$-extensions for the Apostol type polynomials of order $\alpha$ and level $m$, we obtain some new results related to a unified presentation of the $q$-analog of the generalized Apostol type polynomials of order $\alpha$ and level $m$. In addition, we introduce some algebraic and differential properties for the $q$-analog of the generalized Apostol type polynomials of order $\alpha$ and level $m$ and the relation of these with the $q$-Stirling numbers of the second kind, the generalized $q$-Bernoulli polynomials of level $m$, the generalized $q$-Apostol type Bernoulli polynomials, the generalized $q$-Apostol type Euler polynomials, the generalized $q$-Apostol type Genocchi polynomials of order $\alpha$ and level $m$, and the $q$-Bernstein polynomials.