Unified Bernoulli-Euler polynomials of Apostol type

Abstract

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