This is an excellent paper containing several new representations of the Bernoulli polynomials and the Bernoulli numbers. In the sequel, let n be any nonnegative integer unless otherwise specified, and let S(n,k) be the Stirling numbers of the second kind. It is well known that the Bernoulli numbers , generated by the Taylor expansion are represented by the classical formula The author first obtains an explicit formula for the nth derivative of g(t). Namely, in the finite t-plane punctured at the points 2m i, , where the functions are regular for the considered t and have the representation
He then obtains the nth derivative of the generating function of the Bernoulli polynomials Namely, in the finite t-plane punctured at the points 2m i, , where is the finite difference of the kth order of . In particular, for , he finds the new formula which generalizes the classical representation of given above. He also derives a generalization of the Kronecker-Bergmann formula for to
The author then proceeds to introduce the class of rational functions and to establish an analytic expression of by means of any of the functions . As a corollary, he derives a series of new representations of the Bernoulli numbers . Among other results, he also obtains a representation of as a quotient of two relatively prime polynomials.