Unified Bernoulli-Euler polynomials of Apostol type

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