Unified Bernoulli-Euler polynomials of Apostol type

Abstract

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