F DEHGHAN - Department of Mathematics, Yazd University, Iran
F.M MAALEK GHAINI - Department of Mathematics, Yazd University, Iran

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. These polynomials can be defined by varios methods depending on the applications. In particular, six approaches to the theory of Bernoulli polynomials are known; these are associated with the names of J. Bernoulli, L. Euler, P.E. Appell, A. Hurwitz, E. Lucas and D.H. Lehmer. In this paper we deal with a new determinantal definition for Bernoulli polynomials recently proposed by F. Costabile. Then we express a property of Bernoulli numbers and finally, we consider the applications of Bernoulli polinomials and Bernoulli numbers in numerical analysis.

Bernoulli numbers, Bernoulli polynimials, Extended Euler Formulas, Hessenberg Matrix