A new approach to Bernoulli polynomials

Six approaches to the theory of Bernoulli polynomials are known; these are associated with the names of J. Bernoulli [2], L. Euler [4], E. Lucas [8], P. E. Appell [1], A. H¨urwitz [6] and D. H. Lehmer [7]. In this note we deal with a new determinantal definition for Bernoulli polynomials recently proposed by F. Costabile [3]; in particular, we emphasize some consequent procedures for automatic calculation and recover the better known properties of these polynomials from this new definition. Finally, after we have observed the equivalence of all considered approaches, we conclude with a circular theorem that emphasizes the direct equivalence of three of previous approaches