In the ring of formal power series the authors study the subring generated by and together with the differential operator . Using their calculus of generalized fractions and residues they show that the Bernoulli numbers of order and Bernoulli polynomials can be realized as the residue of some suitable generalized fractions. Furthermore, they derive formulas of complete summations similar to
where are given rational numbers and are positive numbers. Various known identities are derived, see K. Dilcher [J. Number Theory 60, 23-41 (1996; Zbl 0863.11011)] or M. Eie [Trans. Am. Math. Soc. 348, 1117-1136 (1996; Zbl 0864.11043)].