On Bell based Appell polynomials

Abstract

Recently, several Bell based polynomials such as Bernoulli, Euler, Genocchi and Apostol versions were defined and investigated. The main aim of this paper is to introduce the general family of Bell based Appell polynomials, which includes many new members in addition to the existing ones, and to investigate their properties including determinantal representation, recurrence relation, derivative formula, addition formula, shift operators and differential equation. Furthermore, we introduce 2-iterated Bell-Appell polynomials and investigate their similar properties. With the help of this 2-iterated family, we also obtain the closed form summation formulae between the usual and the generalized versions of the Bell based Appell polynomials. Finally, we introduce the Bell based Bernoulli-Euler, Bernoulli-Genocchi, Euler-Genocchi and Stirling-Appell polynomials of the second kind as special cases of 2-iterated Bell based Appell polynomials and state the corresponding results.