  This paper is devoted to Ser's and Hasse's series representations for the
zeta-functions, as well as to several closely related results. The notes
concerning Ser's and Hasse's representations are given as theorems, while the
related expansions are given either as separate theorems or as formulae inside
the remarks and corollaries. In the first theorem, we show that the famous
Hasse's series for the zeta-function, obtained in 1930 and named after the
German mathematician Helmut Hasse, is equivalent to an earlier expression given
by a little-known French mathematician Joseph Ser in 1926. In the second
theorem, we derive a similar series representation for the zeta-function
involving the Cauchy numbers of the second kind. In the third theorem, with the
aid of some special polynomials, we generalize the previous results to the
Hurwitz zeta-function. In the fourth theorem, we obtain a similar series with
Gregory's coefficients of higher order. In the fifth theorem, we extend the
results of the third theorem to a class of Dirichlet series. As a consequence,
we obtain several globally convergent series for the zeta-functions. We also
show that Hasse's series may be obtained much more easily by using the theory
of finite differences, and we demonstrate that there exist numerous series of
the same nature. In the sixth theorem, we show that Hasse's series is a simple
particular case of a more general class of series involving the Stirling
numbers of the first kind. All the expansions derived in the paper lead, in
turn, to the series expansions for the Stieltjes constants, including new
series with rational terms for Euler's constant, for the logarithm of the
gamma-function, for the digamma and trigamma functions. Finally, in the
Appendix, we prove an interesting integral representation for the Bernoulli
polynomials of the second kind, formerly known as the Fontana-Bessel
polynomials.
