Bell based Apostol type polynomials and its properties

TL;DR

This paper introduces Bell based Apostol type polynomials of order a, explores their properties, and establishes identities and relations with Stirling numbers, expanding their theoretical framework and potential applications.

Contribution

It defines a new class of polynomials, derives their fundamental properties, and connects them with existing combinatorial numbers, which is a novel contribution.

Findings

Derived generating functions for Bell based Apostol type polynomials.

Established identities including correlation, summation, and derivative formulas.

Connected these polynomials with Stirling numbers.

Abstract

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.