Generalized Apostol Type Polynomials Based on Twin-Basic Numbers

Abstract

In this work, we consider a class of new generating function for \((p,q)\)-analog of Apostol type polynomials of order \(\alpha\) including Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order \(\alpha\). By making use of their generating function, we derive some useful identities. We also introduce the generating functions of \((p,q)\)-analogues of the Stirling numbers of second kind of order \(\tau\) and the Bernstein polynomials by which we construct diverse correlations including aforementioned polynomials and the \((p,q)\)-gamma function.