In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function.

In our work we have established certain indefinite integrals involving Fibonacci polynomials, Lucas polynomials, Bernoulli polynomials and Hermite polynomials.

The essential ingredients of the recipe are the Bernoulli polynomials and their relatives.

Terao, The Shi arrangements and the Bernoulli polynomials, Bull.

The Bernoulli polynomials are defined by the relation

Further properties of Bernoulli polynomials can be found in (1).

Qi in [11] and [13]-[15], defined the generalization of Bernoulli polynomials as

In this paper we introduce the generalized poly-Eulerian polynomials and from this, we investigate the classical relationship involving generalized poly-Eulerian polynomials and Bernoulli polynomials.

For some details on the

Bernoulli polynomials and the Bernoulli numbers, see for example (8), (9).

Bernoulli polynomials play an important role in various expansions and approximation formulas which are useful both in analytic theory of numbers and in classical and numerical analysis.

In the usual notation, the n-th

Bernoulli polynomials were defined by

Liu, The generalized central factorial numbers and higher order Norlund Euler-

Bernoulli polynomials, (Chinese) Acta Math.