  Many authors studied mix type polynomials and their properties. For their
numerous uses in Statistics, Combinatorial analysis and Number theory. In the
present article, we define a generating function of mix type Bell based Apostol
type polynomials and numbers of order a. Furthermore, we derive some elementary
properties and identities of Bell based Apostol type polynomials of order a
such as correlation formulas, Implicit summation formulas, derivative formulas
and their special cases. Also, we derive some relation with Stirling numbers.
