Unified Bernoulli-Euler polynomials of Apostol type

TL;DR

This paper introduces and explores properties of unified Apostol-Bernoulli and Apostol-Euler polynomials, including their arithmetic, derivation, integration, and symmetric identities, connecting them to classical polynomials.

Contribution

It presents a unified framework for Apostol-Bernoulli and Apostol-Euler polynomials, deriving new properties and identities using umbral calculus.

Findings

Established connections to classical Apostol polynomials

Derived new derivation and integration formulas

Obtained symmetric identities using umbral calculus

Abstract

The object of this paper is to introduce and study properties of unified Apostol-Bernoulli and Apostol-Euler polynomials noted by $\left\{\mathfrak{V_{n}}(x;\lambda;\mu)\right\}_{n \geq 0}$. We study some arithmetic properties of $\left\{\mathfrak{V_{n}}(x;\lambda;\mu)\right\}_{n \geq 0}$ as their connection to Apostol-Euler polynomials and Apostol-Bernoulli polynomials. Also, we give derivation and integration representations of $\left\{\mathfrak{V_{n}}(x;\lambda;\mu)\right\}_{n \geq 0}$. Finally, we use the umbral calculus approach to deduce symmetric identities.