Unification of the three families of generalized Apostol type polynomials on the Umbral algebra

TL;DR

This paper unifies and generalizes Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials using Umbral calculus, introducing new identities, operators, and relations to Stirling numbers.

Contribution

It introduces a unified framework for three families of Apostol type polynomials using Umbral algebra, with new identities and operator definitions.

Findings

Derived new identities for Apostol polynomials

Established relations between these polynomials and Stirling numbers

Extended the generating functions for unification

Abstract

The aim of this paper is to investigate and introduce some new identities related to the unification and generalization of the three families of generalized Apostol type polynomials, which are Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, on the modern theory of the Umbral calculus and algebra. We also introduce some operators. Recently, Ozden constructed generating function of the unification of the Apostol type polynomials (see Ozden [H. Ozden, AIP Conf. Proc. 1281, (2010), 1125-1227.]). By using this generating function, we derive many properties of these polynomials. We give relations between these polynomials and Stirling numbers.