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. Hürwitz [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. 1 – Short review of classical approaches 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 various 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 ([2], 1690), L. Euler ([4], 1738), P.E. Appell ([1], 1882), A. Hürwitz ([7], 1890), E. Lucas ([8], 1891) and D.H. Lehmer ([7], 1988). The term Bernoulli polynomials was used first in 1851 by Raabe [10] in connection with