An extended generalized \(q\)-extensions for the Apostol type polynomials (Q1667594)

Summary: 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.