Unified Bernoulli-Euler polynomials of Apostol type

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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.