Unified Bernoulli-Euler polynomials of Apostol type

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.