bell polynomials for computing closed formulae of numbers and polynomials

Exponential partial bell polynomials are a main tool for computing explicit formula for a large family of numbers and polynomials. using these polynomials, we review most of the polynomials already studied in the book of djordjevic-milovanovic (2014). this study allows to unify the writing of these polynomials and obtain other extensions. consequently the obtained results find their application in number theory; notably for the arithmetical properties of power products, fermat numbers, mersenne numbers and fermat last theorem.