0] [member of] (0, 1/2) is a unique zero point (in the interval (0,1/2)) of the Bernoulli polynomial [B.

Finally, we specially note that if k is even then the corresponding Bernoulli polynomial [B.

2k](t) the Bernoulli polynomial and {t} = t - [t] the fractional part of t.

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 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.

For some details on the Bernoulli polynomials and the Bernoulli numbers, see for example (8), (9).

k](t), k [greater than or equal to] 0, be periodic functions of period 1, related to the Bernoulli polynomials as

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.

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.